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Optimality conditions and duality in minmax fractional programming, part I: necessary and sufficient optimality conditions.

1. INTRODUCTION

In this paper, we present a set of necessary optimality conditions and a multitude of global parametric sufficient optimality results under a variety of generalized (F, b, [phi], [rho], [theta])univexity conditions for the following continuous minmax fractional programming problem involving nondifferentiable functions:

(P) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to

[G.sub.j](x) + [parallel] [C.sub.j]x [[parallel].sub.c(j)] [less than or equal to] 0, j [member of] [q.bar], [H.sub.k](x) = 0, k [member of] [r.bar], x [member of] X,

where X is a nonempty open convex subset of [R.sup.n] (n-dimensional Euclidean space), Y is a compact metrizable topological space, f (*, y), g(*, y), y [member of] Y, [G.sub.j], j [member of] [q.bar] [equivalent to] {1,2, ..., q}, and [H.sub.k], k [member of] [r.bar], are real-valued functions defined on X, for each y [member of] Y and j [member of] [q.bar], A(y), B(y), and [C.sub.j] are respectively l x n, m x n, and [n.sub.j] x [rho] matrices, [parallel] * [[parallel].sub.a], [parallel] * [[parallel].sub.a], and [parallel] * [[parallel].sub.c(j)] are arbitrary norms on [R.sup.e], [R.sup.m], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], respectively, and for each y [member of] Y, g(x, y) - [parallel] B(y)x [[parallel].sub.b] > 0 for all x satisfying the constraints of (P).

Evidently, (P) is a general prototype optimization model that contains as special cases a very large number of standard, well-known mathematical programming problems some of which have been extensively investigated in the related literature. More specifically, the following collection of seventeen problems can be viewed as particular cases of (P):

(P1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where F (assumed to be nonempty) is the feasible set of (P), that is,

F = {x [member of] X: [G.sub.j](x) + [parallel] [C.sub.j]x [[parallel].sub.c(j)] [less than or equal to] 0, j [member of] [q.bar], [H.sub.k](x) = 0, k [member of] [r.bar]};

(P2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(P3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(P4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to

[G.sub.j](x) + <x, [M.sub.j]x>1/2 [less than or equal to] 0, j [member of] [q.bar], [H.sub.k](x) = 0, k [member of] [r.bar], x [member of] X,

where f, g, [G.sub.j], j [member of] [q.bar], [H.sub.k], k [member of] [r.bar], and X are as defined in the description of (P), K(y), L(y), y [member of] Y, and [M.sub.j], j [member of] [q.bar], are [rho] x [rho] symmetric positive semidefinite matrices, and <u, v> denotes the inner (scalar) product of the v-dimensional vectors u and v, that is,

<u, v> = [v summation over (i = 1)] [u.sub.i] [v.sub.i], where [u.sub.i] and [v.sub.i] are the ith components of u and v, respectively;

(P5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where G is the feasible set of (P4), that is,

G = {x [member of] X: [G.sub.j](x) + <x, [M.sub.j]x>1/2 [less than or equal to] 0, j [member of] [q.bar], [H.sub.k](x) = 0, k [member of] [r.bar]};

(P6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(P7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(P8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(P9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(P10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where for each i [member of] [p.bar], [f.sub.i] and [g.sub.i] are real-valued functions defined on X, and [K.sub.i] and [L.sub.i] are n x [rho] symmetric positive semidefinite matrices;

(Pll) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(P12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(P13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(P14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(P15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(P16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(P17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

H = {x [member of] X: [G.sub.j](x) [less than or equal to] 0, j [member of] [q.bar], [H.sub.k](x) = 0, k [member of] [r.bar]}.

The problem (P4) is obtained from (P) by choosing [parallel] * [parallel].sub.a], [[parallel] * [[parallel].sub.b], y [member of] Y, and [parallel] * [[parallel].sub.c(j)], j [member of] [q.bar], to be the [l.sub.2]-norm [parallel] * [[parallel].sub.2], and defining K(y) = A[(y).sup.T] A(y), L(y) = B[(y).sup.T] B(y),y [member of] Y, and [M.sub.j] = [C.sup.T.sub.j][C.sub.j], j [member of] [q.bar]. Similarly, (P6) is obtained from (P) by setting Y = {1,2,...,p}, f(x, i) = [f.sub.i](x), g(x, i) = [g.sub.i](x), A(i) = [A.sub.i] and B(i) = [B.sub.i], i [member of] [p.bar].

It appears that the basic continuous minmax programming problem (P3) was investigated in a serious manner for the first time by Danskin [16] who established necessary optimality conditions for this problem and clearly demonstrated its utility in providing realistic models for some significant real-world problems. Following the pioneering work of Danskin, many authors have been studying various aspects of different types of continuous minmax programming problems in the past four decades. For a wealth of information about several facets of continuous minmax problems, including optimality conditions, duality relations, solution algorithms, and areas of applications, the reader is referred to [5]-[7], [10, 11, 17, 27], [34]-[37], [39, 44], [47]-[50], [54, 55], [57]-[61].

The discrete minmax optimization problems (P6), (P10), and (P14) are known as generalized fractional programming problems in the literature of mathematical programming, and have been the focus of considerable attention in the past two decades. The salient features of these problems that appear to be responsible for this surge of interest are their capability to provide realistic models for some significant real-world problems (they have been encountered in multiobjective programming [1], approximation theory [2, 3, 33, 58], goal programming [12, 32], facility location planning [4], and economics [56], among others), their mathematical tractability (they can be transformed to equivalent parametric nonlinear programming problems with nonfractional objective functions), and their generality (they contain, as special cases, three important classes of nonlinear programming problems, namely, problems with fractional, discrete max, and conventional objective functions).

The notion of duality for generalized linear fractional programming was initially considered by von Neumann [56] in the context of an economic equilibrium problem. More recently, a number of optimality criteria, duality results, and computational algorithms for several classes of generalized linear and nonlinear fractional programming problems have appeared in the related literature. [lambda] fairly extensive list of references pertaining to various aspects of these problems is given in [63].

Perhaps, one of the most extensively studied and widely used special cases of (P) is the basic fractional programming model (P16). Various aspects of this popular optimization model have been investigated vigorously in the past four decades culminating in the accumulation of a relatively large literature. One of the principal reasons for such a continual immense interest in this particular optimization paradigm appears to be its capability in providing realistically representative models for a number of significant classes of problems in the fields of operations research, management science, and economics. This feature is primarily due to the fact that in many areas, including resource allocation, transportation, production planning, sequencing, inventory control, financial planning, maintenance and replacement scheduling, and reliability assessment, ratios such as profit/capital,

profit/revenue, return/cost, return/risk, cost/time, profit/time, etc., can serve as useful measures of system performance. Proper characterization and utilization of these measures often requires optimization of certain ratios which, in turn, gives rise to the formulation of fractional programming problems. In noneconomic situations, fractional programming problems have arisen in information theory, stochastic programming, numerical analysis, approximation theory, cluster analysis, graph theory, multifacility location theory, decomposition of large-scale mathematical programming problems, and goal programming, among others. For comprehensive surveys and extensive lists of references dealing with several aspects of fractional programming, including modeling properties, actual and potential areas of applications, optimality conditions, duality formulations, sensitivity and stability analysis, and computational algorithms, the reader is referred to [14, 15, 46, 51, 52, 53].

The rest of this paper is organized as follows. In Section 2, we recall a number of definitions and auxiliary results which will be needed in the sequel. In Section 3, we combine a Dinkelbach-type indirect parametric approach with some nonsmooth analytic techniques to establish a set of necessary optimality conditions for (P). In Section 4, we begin our discussion of sufficient optimality conditions where we formulate and prove numerous sets of sufficiency criteria under a variety of generalized (F, b, [phi], [rho], [theta])-univexity assumptions that are placed on the individual as well as certain combinations of the problem functions. Utilizing two partitioning schemes, in Section 5 we establish several sets of generalized sufficiency results each of which is in fact a family of such results whose members can easily be identified by appropriate choices of certain sets and functions. Finally, in Section 6 we summarize our main results and also point out some further research opportunities arising from certain modifications of the principal problem model considered in this paper.

It is evident that the necessary optimality conditions as well as all the sufficient optimality results obtained for (P) are also applicable, when appropriately specialized, to each one of the seventeen classes of optimization problems designated above as (P1)-(P17), which are particular cases of (P). Since in most cases these results can easily be modified and restated for each one of the seventeen problems, we shall not state them explicitly.

Optimization problems containing norms arise naturally in many areas of the decision sciences, applied mathematics, and engineering. They are encountered most frequently in facility location problems, approximation theory, and engineering design. [lambda] number of these problems have already been investigated in the related literature. Likewise, optimization problems involving square roots of positive semidefinite quadratic forms have arisen in stochastic programming, multifacility location problems, and portfolio selection problems, among others. [lambda] fairly extensive list of references pertaining to several aspects of these two classes of problems is given in [62].

2. PRELIMINARIES

For the purpose of formulating and proving various collections of generalized sufficiency criteria for (P), in this study we shall use a new class of generalized convex functions, called (F, b, [phi], [rho], [theta])-univex functions, which will be defined later in this section. This class of functions may be viewed as a combination of several previously defined types of generalized convex functions. Its main ingredients are invex functions, ([eta], [rho])-invex functions, F-convex functions, and univex functions which were introduced in [22], [28], [23], and [8], respectively. These functions were proposed as generalizations of the class of differentiable convex functions.

Prior to giving the definitions of the new classes of generalized convex functions, it will be useful for purposes of reference and comparison to recall the definitions of the principal components of these functions mentioned above. We shall keep this review to a bare minimum because our primary objective is only to put a number of interrelated generalized convexity concepts into proper perspective. For this reason, we shall only reproduce the essential forms of the definitions without elaborating on their refinements, variants, special cases, and other manifestations. For full discussions of the consequences and applications of the underlying ideas, the reader may consult the original sources. We begin by defining an invex function which occupies a pivotal position in a vast array of generalized convex functions some of which are specified in the following definitions.

Definition 2.1. Let f be a differentiable real-valued function defined on the open convex set X. Then f is said to be [eta]-invex (invex with respect to [eta]) at y if there exists a function [eta]: X x X [right arrow] [R.sup.n] such that for each x [member of] X,

f(x) - f(y) [greater than or equal to] <[nabla]f(y), [eta](x, y)),

where [nabla]f(y) = ([partial derivative]f(y)/[partial derivative][y.sub.1], [partial derivative]f(y)/ [partial derivative][y.sub.2], ..., [partial derivative]f(y)/ [partial derivative][y.sub.n]) is the gradient of f at y; f is said to be [eta]-invex on X if the above inequality holds for all x, y [member of] X.

From this definition it is clear that every differentiable real-valued convex function is invex with [eta](x, y) = x - y. This generalization of the concept of convexity was originally proposed by Hanson [22] who showed that for a nonlinear programming problem of the form

Minimize f(x) subject to ([g.sub.i] (x) < 0, i [member of] [m.bar], x [member of] [R.sup.n],

where the differentiable functions f, [g.sub.i]: [R.sup.n] [right arrow] R, i [member of] [m.bar], are invex with respect to the same function [eta]: [R.sup.n] x [R.sup.n] [right arrow] [R.sup.n], the Karush-Kuhn-Tucker necessary optimality conditions are also sufficient. The term invex (for invariant convex) was coined by Craven [13] to signify the fact that the invexity property, unlike convexity, remains invariant under bijective coordinate transformations.

In a similar manner, one can readily define [eta]-pseudoinvex and [eta]-quasiinvex functions as generalizations of differentiable pseudoconvex and quasiconvex functions.

The notion of invexity has been generalized in several directions. For our present purposes, we shall need a simple extension of invexity, namely, [rho]-invexity which was originally defined in [28].

Let [rho] be a function from X x X to [R.sup.n], and let h be a differentiable real-valued function defined on X.

Definition 2.2. The function h is said to be ([eta], [rho])-invex at [x.sup.*] if there exists [rho] [member of] R such that for each x [member of] X,

h(x) - h([x.sup.*]) [greater than or equal to] ([nabla]h([x.sup.*]), [rho](x, [x.sup.*])) + [rho] [parallel] x - [x.sup.*] [[parallel].sup.2] .

Definition 2.3. The function h is said to be ([eta], [rho])-pseudoinvex at [x.sup.*] [member of] X if there exists [rho] [member of] R such that for each x [member of] X,

([nabla]h([x.sup.*]), [eta](x, [x.sup.*])> [greater than or equal to] [rho] [parallel] x - [x.sup.*] [[parallel].sup.2] [right arrow] h(x) [greater than or equal to] h([x.sup.*]).

Definition 2.4. The function h is said to be ([eta], [rho])-quasiinvex at [x.sup.*] [member of] X if there exists [rho] [member of] R such that for each x [member of] X,

h(x) [less than or equal to] h([x.sup.*]) [right arrow] ([nabla]h([x.sup.*]), [eta](x, [x.sup.*])> [less than or equal to] [rho] [parallel] x -[x.sup.*] [[parallel].sup.2].

The notion of invexity has been extended in many ways. For more information pertaining to various generalizations of invex functions and their applications along with extensive lists of relevant references, the reader is referred to [9, 19, 20, 29, 38, 40, 42, 45].

Another generalization of an invex function is an F-convex function which is defined in terms of a sublinear function, that is, a function that is subadditive and positively homogeneous.

Definition 2.5. [lambda] function F: X [right arrow] R is said to be sublinear (superlinear) if F(x + y) [less than or equal to] ([greater than or equal to])F(x) + F(y) for all x, y [member of] X, and F(ax) = aF(x) for all x [member of] X and a [member of] [R.sub.+] [equivalent] [0, [infinity]).

Now combining the definitions of F-convex and ([eta], [rho])-invex functions given in [23] and [28], respectively, we can define (F, [rho])-convex, (F, [rho])-pseudoconvex, and (F, [rho])-quasiconvex functions.

Let g be a real-valued differentiable function defined on X, and assume that for each x, y [member of] X, the function F(x, y; *): X [right arrow] R is sublinear.

Definition 2.6. The function g is said to be (F, [rho])-convex at y if there exists a real number [rho] such that for each x [member of] X,

g(x) - g(y) [greater than or equal to] F(x, y; [nabla]g(y)) + [rho] [parallel] x - y [[parallel].sup.2].

Definition 2.7. The function g is said to be (F, [rho])-pseudoconvex at y if there exists a real number [rho] such that for each x [member of] X,

F(x, y; [nabla]g(y)) [greater than or equal to] - [rho] [parallel] x - y [[parallel].sup.2] [right arrow] g(x) [greater than or equal to] g(y).

Definition 2.8. The function g is said to be (F, [rho])-quasiconvex at y if there exists a real number [rho] such that for each x [member of] X,

g(x) [less than or equal to] g(y) [right arrow] F(x,y; [nabla]g(y)) [less than or equal] - [rho] [parallel] x - y [[parallel].sup.2]

Evidently, if in Definitions 2.6-2.8 we choose F(x, y; [nabla]g(y)) = <[nabla]g(y),[eta](x, y)>,where [eta]: X x X [right arrow] [R.sup.n] is a given function, and set [rho] = 0, then we see that they reduce to the definitions of [eta]-invexity, [eta]-pseudoinvexity, and [eta]-quasiinvexity for the function g.

The foregoing classes of generalized convex functions have been utilized for establishing numerous sets of sufficient optimality conditions and a variety of duality results for several categories of static and dynamic optimization problems. For recent surveys and syntheses of these results, the reader is referred to [31, 43].

Another significant generalization of the notion of invexity, called univexity, which subsumes a number of previously proposed types of generalized convex functions, was recently given in [8]. We recall the definitions of univex, pseudounivex, and quasiunivex functions.

Let h be a differentiable real-valued function defined on X, let [rho] be a function from X x X to [R.sup.n], let [PHI] be a real-valued function defined on X, and let b be a function from X x X to [R.sub.+] \{0} [equivalent to] (0, [infinity]).

Definition 2.9. The function h is said to be univex at y with respect to [eta], [PHI] and b if for each x [member of] X,

b(x,y) [PHI] (h(x) - h(y)) [greater than or equal to] <[nabla]h(y),n(x, y)>.

Definition 2.10. The function h is said to be pseudounivex at y with respect to [eta], [PHI]and b if for each x [member of] X,

<[nabla]h(y),n(x,y)> [greater than or equal to] 0 [right arrow] b(x, y) [PHI] (h(x) - h(y)) [greater than or equal to] 0.

Definition 2.11. The function h is said to be quasiunivex at y with respect to [eta], [PHI] and b if for each x [member of] X,

b(x, y) [PHI] (h(x) - h(y)) [less than or equal to] 0 [right arrow] <[nabla]h(y),[eta](x, y)> [less than or equal to] 0.

Finally, we are in a position to give our definitions of generalized (F, b, [phi], [rho], [theta])-univex functions. They are formulated by combining Definitions 2.5-2.11.

Let [x.sup.*] [member of] X and assume that the function F: X [right arrow] R is differentiable at [x.sup.*].

Definition 2.12. The function F is called (strictly) (F, b, [phi], [rho], [theta])-univex at [x.sup.*] if there exist functions b: X x X [right arrow] [R.sub.+] \{0} [equivalent to] (0, [infinity]),[phi]: R [right arrow] R, [rho]: X x X [right arrow] R, and [theta]: X x X [right arrow] [R.sup.n], and a sublinear function F(x, [x.sup.*]; *): [R.sup.n] [right arrow] R such that for each x [member of] X(x = [x.sup.*]),

[phi](F(x) - F([x.sup.*]))(>) [greater than or equal to] F(x, [x.sup.*]; b(x, [x.sup.*])[nabla]F([x.sup.*])) + [rho](x, [x.sup.*]) [parallel] [theta](x, [x.sup.*]) [[parallel].sup.2].

Definition 2.13. The function F is said to be (strictly) (F, b, [phi], [rho], [theta])-pseudounivex at [x.sup.*] if there exist functions b: X x X [right arrow] [R.sub.+]\{0} = (0, oc),[phi]: R [right arrow] R,[rho]: X x X [right arrow] R, and [theta]: X x X [right arrow] [R.sup.n], and a sublinear function F(x,[x.sup.*]; *): [R.sup.n] [right arrow] R such that for each x [member of] X(x [not equal to] [x.sup.*]),

F(x, [x.sup.*]; b(x, [x.sup.*])[nabla]F([x.sup.*])) [greater than or equal to] [rho](x, [x.sup.*])[parallel][theta](x, [x.sup.*])[[parallel].sup.2] [right arrow] [phi] (F (x) - F ([x.sup.*])) (>) [greater than or equal to] 0.

Definition 2.14. The function F is said to be (prestrictly)(F, b, [phi], [rho], [theta])-quasiunivex at [x.sup.*] if there exist functions b: X x X [right arrow] [R.sub.+]\{0}[equivalent to] (0, [infinity]),[phi]: R [right arrow] R, [rho]: X x X [right arrow] R, and [theta]: X x X [right arrow] [R.sup.n], and a sublinear function F(x, [x.sup.*]; *): [R.sup.n] [right arrow] R such that for each x [member of] X,

[phi]{F(x) - F([x.sup.*]))(<) [less than or equal to] 0 [right arrow] F(x,[x.sup.*]; b(x, [x.sup.*])[nabla]F([x.sup.*]) [less than or equal to] [rho](x, [x.sup.*]) [parallel][theta](x, [x.sup.*])[[parallel].sup.2].

From the above definitions it is clear that if F is (F, b, [phi], [rho], [theta])-univex at [x.sup.*], then it is both (F, b, [phi], [rho], [theta])-pseudounivex and (F, b, [phi], [rho], [theta])-quasiunivex at [x.sup.*], if F is (F, b, [phi], [rho], [theta])quasiunivex at [x.sup.*], then it is prestrictly (F, b, [phi], [rho], [theta])-quasiunivex at [x.sup.*], and if F is strictly (F, b, [phi], [rho], [theta])-pseudounivex at [x.sup.*], then it is(F, b, [phi], [rho], [theta])-quasiunivex at [x.sup.*].

In the proofs of the sufficiency theorems, sometimes it may be more convenient to use certain alternative but equivalent forms of the above definitions. These are obtained by considering the contrapositive statements. For example, (F, b, [phi], [rho], [theta])-quasiunivexity can be defined in the following equivalent way:

F is said to be (F, b, [phi], [rho], [theta])-quasiunivex at [x.sup.*] if there exist functions b: X x X [right arrow] [R.sub.+]\{0} [equivalent to] (0, [infinity]), [phi]: R [right arrow] R, [rho]: X x X [right arrow] R, and 0: X x X - [R.sup.n], and a sublinear function F(x, [x.sup.*]; *): [R.sup.n] [right arrow] R such that for each x [member of] X,

F x, [x.sup.*]; b(x, [x.sup.*])[nabla]F([x.sup.*])) > - [rho](x, [x.sup.*]) [parallel] [theta] (x, [x.sup.*]) [[parallel].sup.2] [right arrow] [phi] (F(x) - F([x.sup.*])) > 0.

Needless to say that the new classes of generalized convex functions specified in Definitions 2.12- 2.14 contain a great variety of special cases that can easily be identified by appropriate choices of the functions F, b, [phi], [rho], and [theta]. In particular, they subsume all the generalized convex functions specified in Definitions 2.1-2.11.

3. NECESSARY OPTIMALITY CONDITIONS

In this section, we establish a set of necessary optimality conditions for (P). We derive these conditions by combining the optimality results of [24] for a nonsmooth nonlinear programming problem with a Dinkelbach-type [18] indirect parametric approach. The auxiliary problem which makes this approach possible has the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [lambda] is a parameter. The relationships between this problem and (P) that are useful for our present purposes are stated in the following lemma whose proof is straightforward and hence omitted.

Lemma 3.1. Let [[lambda].sup.*] be the optimal value of (P),and let v([lambda]) be the optimal value of (P[lambda]) for any fixed [lambda] [member of] R such that (P[lambda]) has an optimal solution. Then the following statements are valid:

(a) If [x.sup.*] is an optimal solution of (P), then it is an optimal solution of (P[[lambda].sup.*]) and v([[lambda].sup.*]) = 0.

(b) If (P[bar.[lambda]]) has an optimal solution [bar.x}] for some [bar.[lambda]] [member of] R with v([bar.[lambda]]) = 0, then [bar.x] is an optimal solution of (P) and [bar.[lambda]] = [[lambda].sup.*].

Throughout this paper, the gradient and the subdifferential are always taken with respect to the first variable x. So [nabla]f (x, y) and [partial derivative]([parallel] A(y)x [parallel]) are the abbreviations for [[nabla].sub.x]f (x, y) and [[partial derivative].sub.x]([parallel] A(y)x [parallel]), respectively. We also posit the following assumptions which will remain in effect throughout the sequel:

(A1) For each x [member of] X, the functions f(x, *) and g(x, *) are continuous on Y.

(A2) Each entry of the matrices A(y) and B(y) is a continuous real-valued function defined on Y.

The following auxiliary result will be needed in the proof of the necessary optimality conditions for (P).

Lemma 3.2. [21] Let Q be a metrizable compact topological space, and for each [alpha] [member of] Q, let [f.sub.[alpha]] be a real-valued Lipschitz function defined on [R.sup.n] with the same Lipschitz constant. Let [x.sup.*] [member of] [R.sup.n] and assume that the mapping [alp[ha] [right arrow] [f.sub.[alpha]]([x.sup.*]) is continuous, and the set-valued mapping ([alpha], x) [right arrow] [partial derivative][f.sub.[alpha]](x) is upper semicontinuous at ([alpha], [x.sup.*]) for all [alpha] [member of] Q. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and co(S) indicates the convex hull of the set S.

We also need a formula for the subdifferential of a norm function.

Lemma 3.3. [25] Let the function f: [R.sup.n] [right arrow] R be defined by f (x) = [parallel] Ax [[parallel].sub.a], where A is an m x n matrix and [parallel] * [[parallel].sub.a] is an arbitrary norm. Then the subdifferential [partial derivative]f([x.sup.*]) of the convex function f at [x.sup.*] is given by

[partial derivative]f([x.sup.*]) = {[[zeta].sup.T]A: [parallel] [zeta] [[parallel].sup.*.sub.a] [less than or equal to] 1, <[zeta], A[x.sup.*]> = [parallel] A[x.sup.*] [[parallel].sub.a]},

where [parallel] * [[parallel].sup.*.sub.a] is the dual to the norm [parallel] * [[parallel].sub.a], that is, [parallel] [delta] [[parallel].sup.*.sub.a] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Using the formula above, we can prove the following lemma.

Lemma 3.4. Let F(x, y) = [parallel] A(y)x [[parallel].sub.a]. Then the set-valued map (x, y) [right arrow] [partial derivative]F(x,y) is upper semicontinuous.

Proof. Suppose to the contrary that the statement is not true. Then there exist an [member of] > O and a sequence ([x.sub.n], [y.sub.n]) [right arrow] ([bar.x], [bar.y]) such that [partial derivative]([parallel]A([y.sub.n])[x.sub.n][[parallel].sub.a]) is not a subset of [partial derivative]([parallel] A([bar.y])[bar.x] [[parallel].sub.a)] + [phi]B, where B is the unit ball in [R.sup.n]. By Lemma 3.3, there exists [[xi].sub.n] [member of] [R.sup.l] with [parallel] [[xi].sub.n] [[parallel].sup.*.sub.a] [less than or equal to] 1 and [parallel] A([y.sub.n])[x.sub.n] [[parallel].sub.a] = <[[xi].sub.n], A([y.sub.n])[x.sub.n]> such that [[xi].sup.T.sub.n]A ([y.sub.n]) [not member of] [partial derivative]([parallel] A([bar.y])[bar.x] [[parallel].sub.a] + [member of]B. Without loss of generality, we may assume that [[xi].sub.n] [right arrow] [[xi].sub.0]. Since each entry of A(y) is continuous, we conclude that [[xi].sup.T.sub.n]A([y.sub.n]) [right arrow] [[xi].sup.T.sub.0] A([bar.y]), [parallel] [[xi].sub.0] [[parallel].sup.*.sub.a] [less than or equal to] 1, and [parallel] A([bar.y])[bar.x] [[parallel].sub.a] = <[[xi].sub.0], A([bar.y])[bar.x]>. Therefore, [[xi].sup.T.sub.0] A([bar.y]) [member of] [partial derivative]([parallel] A([bar.y])[bar.x] [[parallel].sub.a]), which contradicts the fact that [[xi].sup.T.sub.n] A([y.sub.n]) [not member of] [partial derivative]([parallel] A([bar.y])[bar.x] [[parallel].sub.a]) + [member of]B. Therefore, the set-valued map (x, y) [right arrow] [partial derivative]F(x, y) is upper semicontinuous.

For x [member of] X, let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Next, with the help of Lemmas 3.1, 3.3, and 3.4, and the optimality results of [24] we prove the following theorem which is the main result of this section.

Theorem 3.1. Let [x.sup.*] be an optimal solution of (P). Assume that the functions f (*, y), g(*, y), y [member of] Y, [G.sub.j], j [member of] [q.bar], and [H.sub.k], k [member of] [r.bar], are continuously differentiable at [x.sup.*], and that any one of the constraint qualifications specified in [24, 30] holds at [x.sup.*]. Then there exist [[lambda].sup.*] [member of] R, (p*, [[bar.y].sup.*], [u.sup.*], [[alpha].sup.*], [[beta].sup.*]) [member of] K([x.sup.*]), [v.sup.*] [member of] [R.sup.q.sub.+] [equivalent to] {v [member of] [R.sup.q]: v [greater than or equal to] 0}, [w.sup.*] [member of] [R.sup.r], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.1)

[[v.sup.*].sub.j] [[G.sub.j]([x.sup.*]) + [parallel] [C.sub.j][x.sup.*] [[parallel].sub.c(j)] =0, j [member of] [q.bar], (3.2)

[parallel] [alpha].sup.*i] [[parallel].sup.*.sub.a] [less than or equal to] 1, [parallel] [[beta].sup.*i] [[parallel].sup.*.sub.b] [less than or equal to] i [member of] [[p.bar].sup.*] (3.3)

[parallel] [[gamma].sup.*j] [[parallel].sup.*.sub.c(j)] [less than or equal to] 1, j [member of] [q.bar] (3.4)

<[[alpha].sup.*i], A([y.sup.*i])[x.sup.*]) = [parallel] A([y.sup.*i])[x.sup.*] [[parallel].sub.a], <[[beta].sup.*i], B([[y.sup.*i]) [x.sup.*]) = [parallel] B([[y.sup.*i])[x.sup.*] [[parallel].sub.b], i [member of] [[p.bar].sup.*], (3.5)

<[[gamma].sup.*j], [C.sub.j][[lambda].sup.*]) = [parallel] [C.sub.j][[lambda].sup.*] [[parallel].sub.c(j)], j [member of] [q.bar]. (3.6)

Proof. Since x is an optimal solution of (P), by Lemma 3.1, it is an optimal solution of (P[[lambda].sup.*]), where [[lambda].sup.*] is the optimal value of (P) at [x.sup.*]. By Theorem 3.3 in [24], the Lagrange multiplier set of the problem (P[[lambda].sup.*]) is not empty. Therefore, there exist [v.sup.*] [member of] [R.sup.q.sub.+] and [w.sup.*] [member of] [R.sup.r], such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and (3.2) holds. By Lemmas 3.2 and 3.4, we know that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.7)

From Lemma 3.1 it is easy to see that [Y.sub.0]([x.sup.*])= [bar.Y]([x.sup.*]). By Caratheodory's theorem [25] and Lemma 3.3, there exist (p*, [[bar.y].sup.*], [u.sup.*], [[alpha].sup.*], [[beta].sup.*]) [member of] K([x.sup.*]) and [[gamma].sup.*j] [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], j [member of] [q.bar], as specified in the statement of the theorem, such that (3.1) -(3.6) hold.

To give the reader a glimpse of what is involved in the process of modifying and restating the results of this paper for (P1) - (P17), we next specialize Theorem 3.1 for (P6).

Theorem 3.2. Let [x.sup.*] be an optimal solution of (P6) and assume that the functions [f.sub.i], [g.sub.i], i [member of] [p.bar], [G.sub.j], j [member of] [q.bar], and [H.sub.k], k [member of] [r.bar], are continuously differentiable at [x.sup.*], and that any one of the constraint qualifications specified in [24, 30] holds at [x.sup.*]. Then there exist [[lambda].sup.*] [member of] R, [u.sup.*] [member of] U [equivalent to] {u [member of] [R.sup.p]: u [greater than or equal to] 0, [p summation over (i=1) [u.sub.i] = 1}, [v.sup.*] [member of] [R.sup.q.sub.+], [w.sup.*] [member of] [R.sup.r], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The form and contents of the necessary optimality conditions given in Theorem 3.1 provide clear guidelines for formulating numerous sets of sufficient optimality conditions and many duality models for (P). The rest of this paper is devoted to investigating various sets of sufficiency criteria for (P). The sufficiency results established here may be viewed as substantial extensions and improvements of those obtained for (P4) in [64]. [lambda] vast number of duality results for (P) which are based on Theorem 3.1 and the ensuing sufficiency results are discussed in [65, 66].

4. SUFFICIENT OPTIMALITY CONDITIONS

In this section, we present a number of sufficiency results in which various generalized (F, b, [phi], [rho], [theta])-univexity assumptions are imposed on the individual as well certain combinations of the problem functions. In formulating these results, we shall use a streamlined version of the necessary conditions given in Theorem 3.1. Specifically, we use an altered version of these conditions obtained by dropping (3.5) and (3.6), and modifying (3.2) accordingly. The resulting reduced set of equations and inequalities will lead to relatively shorter statements and proofs for many of the sufficiency principles that will be developed in this study.

In the proofs of our sufficiency theorems, we shall make frequent use of two auxiliary results, namely, the generalized Cauchy inequality and an alternative expression for the objective function of (P). These results are formally stated in the following lemmas.

Lemma 4.1. [26] For each a, b [member of] [R.sup.m, <a, b> [less than or equal to] [parallel] a [parallel] * [parallel] b [parallel].

Lemma 4.2. For each x [member of] X,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

it is clear that for any [rho] [member of] [n+1.bar], u [member of] U, [y.sup.i] [member of] Y, i [member of] [p.bar], we have

[phi](x) [greater than or equal to] [f(x, [y.sup.i] + [parallel] A([y.sup.i])x [[parallel].sub.a]]/[g(x,y) - [parallel] B([y.sup.i]) x [[parallel].sub.b]]],

and hence

[phi](x)[u.sub.i][g (x,[y.sup.i]) - [parallel] B ([y.sup.i])x [[parallel].sub.b]] [greater than or equal to] [u.sub.i][f (x,[y.sup.i]) + [parallel] A ([y.sup.i])x [[parallel].sub.a]],

which implies

[phi](x) [greater than or equal to] [[p summation over (i=1) [u.sub.i] [f (x),[y.sup.i]) + [parallel] A ([y.sup.i]) x [[parallel].sub.a]]/[[p summation over (i=1) [u.sub.i] [g (x,[y.sup.i]) + [parallel] B ([y.sup.i])x [[parallel].sub.b]]]]

From this inequality we conclude that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On the other hand, since the set Y is compact, there is [y.sup.*] [member of] Y, such that

[phi](x) = [f(x, [y.sup.*]) + [parallel] A([y.sup.*])x [[parallel].sub.a]]/[g(x, [y.sup.*]) - [parallel] B([y.sup.*])x [[parallel].sub.b]],

and, therefore, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now the assertion of the lemma follows from the last two inequalities.

To simplify the ensuing presentation, we use the following list of symbols:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We begin our discussion of sufficiency criteria for (P) with a collection of results in which separate (F, b, [phi], [rho], [theta])-univexity conditions are imposed on the functions [A.sub.i] (*, [bar.y], [alpha]) and [B.sub.i](*, [bar.y], [beta]), i [member of] [p.bar], whereas different types of generalized (F, b, [phi], [rho], [theta])-univexity assumptions are placed on certain combinations of the constraint functions.

Let

K = {(p, [bar.y], u, [alpha], [beta]):1 [less than or equal to] p [less than or equal to] n + 1; [bar.y] = ([y.sup.1], [y.sup.2], ..., [y.sup.p]), [y.sup.i] [member of] Y; u [member of] [R.sup.p.sub.+],

[p summation over (i = 1) = 1; [alpha] = ([[alpha].sup.1], [[alpha].sup.2], ..., [[alpha].sup.p]), [[alpha].sup.i] [member of] [R.sup.l]; [beta] = ([[beta].sup.1], [[beta].sup.2], ..., [[beta].sup.p]), [[beta].sup.i] [member of] [R.sup.m]}.

Theorem 4.1. Let [x.sup.*] [member of] F, let [[lambda].sup.*] = [phi]([x.sup.*]) [greater than or equal to] 0, and assume that the functions f (*, y), g(*, y), y [member of] Y, [G.sub.j], j [member of] [q.bar], and [H.sub.k], k [member of] [r.bar], are differentiable at [x.sup.*], and that there exist (p, [[bar.y].sup.*], [u.sup.*], [[alpha].sup.*], [[beta].sup.*]) [member of] K, [v.sup.*] [member of] [R.sup.q.sub.+], [w.sup.*] [member of] [R.sup.r], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], j [member of] [q.bar], such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4.1)

[[u.sup.*].sup.i] {f ([x.sup.*],[[y.sup.*i]) + <[[alpha].sup.*i], A([[y.sup.*i])[x.sup.*]> - [[lambda].sup.*][g([x.sup.*],[[y.sup.*i]) - <[[beta].sup.*i], B([[y.sup.*i])[x.sup.*]]} = 0, I [member of] [p.bar] (4.2)

[V*.sub.j][[G.sub.j]([x.sup.*]) + <[[gamma].sup.*j], [C.sub.j][x.sup.*]]> = 0, j [member of] [q.bar], (4.3)

[parallel] [[alpha].sup.*i] [[parallel].sup.*.sub.a] [less than or equal to] 1, [parallel] [[beta].sup.*i] [[parallel].sup.*.sub.b] [less than or equal to] 1, i [member of] [p.bar] (4.4)

[parallel] [[gamma].sup.*j] [[parallel]sup.*.sub.c(j)] [less than or equal to] 1, j [member of] [q.bar]. (4.5)

Assume, furthermore, that any one of the following six sets of conditions holds:

(a) (i) for each i [member of] [p.bar], [A.sub.i] (*, [[bar.y].sup.*], [[alpha].sup.*]) is (F, b, [phi], [[bar.[rho]].sub.i], [theta])-univex and [B.sub.i] [A.sub.i] (*, [[bar.y].sup.*], [[beta].sup.*]) is (F, b, [phi], [[rho].sub.i], [theta])-univex at [x.sup.*], [phi] is superlinear, and [phi](a) [greater than or equal to] 0 [right arrow] a [greater than or equal to] 0;

(ii) for each j [member of] [J.sub.+] = [J.sub.+] ([v.sup.*]), [C.sub.j](*, [[gamma].sup.*]) is (F, b, [[phi].sub.j], [[rho].sub.j], [theta])-quasiunivex at [x.sup.*], [[phi].sub.j] is increasing, and [[phi].sub.j] (0) = 0;

(iii) for each k [member of] [K.sub.*] = [K.sub.*]([w.sup.*]), [D.sub.k](*, [w.sup.*]) is (F, b, [[phi].sub.k], [[rho].sub.k], [theta])-quasiunivex at [x.sup.*] and [[phi].sub.k] (0) = 0;

(iv) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(b) (i) for each i [member of] [bar.p], [A.sub.i](*, [y.sup.*], [[alpha].sup.*]) is (F, b, [phi], [[rho].sub.i], [theta]) -univex and [B.sub.i](*, [y.sup.*], [[beta].sup.*]) is (F, b, [phi], [[rho].sub.i], [theta])-univex at [x.sup.*],f> is superlinear, and [phi] [greater than or equal to] (a) > 0 [right arrow] a [greater than or equal to] 0;

(ii) C(*, [v.sup.*], [[gamma].sup.*]) is (F, b, [phi], [rho], [theta])-quasiunivex at [x.sup.*], [phi] is increasing, and [phi](0) = 0;

(iii) for each k [member of] [K.sub.*], [D.sub.k](*,[w.sup.*]) is (F, b, [[phi].sub.k], [[rho].sub.k], [theta])-quasiunivex at [x.sup.*] and [[phi].sub.k](0) = 0;

(iv) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(c) (i) for each i [member of] [p.bar], [A.sub.i](*, [y.sup.*], [[alpha].sup.*]) is (F, b, [phi], [[rho].sub.i], [theta])-univex and [B.sub.i](*, [y.sup.*], [[beta].sup.*]) is (F, b, [phi], [[rho].sub.i], [theta])-univex at [x.sup.*],f> is superlinear, and [phi](a) [greater than or equal to] 0 [right arrow] a [greater than or equal to] 0;

(ii) for each j [member of] [J.sub.+], [C.sub.j](*, [[gamma].sup.*]) is (F, b, [[phi].sub.j], [[rho].sub.j], [theta])-quasiunivex at [x.sup.*], [[phi].sub.j] is increasing, and [[phi].sub.j] (0) = 0;

(iii) D(*,[w.sup.*]) is (F, b, [phi], [rho], [theta])-quasiunivex at [x.sup.*] and [phi](0) = 0;

(iv) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(d) (i) for each i [member of] [p.bar], [A.sub.i](*, [[bar.y].sup.*], [[alpha].sup.*]) is (F, b, [phi], [[rho].sub.i], [theta]) -univex and [B.sub.i](*, [[bar.y].sup.*], [[beta].sup.*]) is (F, b, [phi], [[rho].sub.i], [theta])-univex at [x.sup.*], [phi] is superlinear, and [phi](a) [greater than or equal to] 0 [right arrow] a [greater than or equal to] 0;

(ii) C(*, [v.sup.*],[gamma] *) is (F, b, [phi], [rho], [theta])-quasiunivex at [x.sup.*], [phi] is increasing, and [phi] (0) = 0;

(iii) D(*, [w.sup.*]) is (F, b, [phi], [rho], [theta])-quasiunivex at [x.sup.*] and [phi] (0) = 0;

(iv) p* (x, [x.sup.*]) + p(x, [x.sup.*]) + p(x, [x.sup.*]) [greater than or equal to] 0 for all x [member of] F;

(e) (i) for each i [member of] [p.bar], [A.sub.i] (*, [[bar.y].sup.*], [[alpha].sup.*]) is (F, b, [phi],[[rho].sub.i], [theta])-univex and [B.sub.i] (*, [[bar.y].sup.*], [[beta].sup.*]) is (F, b, [phi],[[rho].sub.i], [theta])-univex at [x.sup.*], [phi] is superlinear, and [phi](a) [greater than or equal to] 0 [right arrow] a [greater than or equal to] 0;

(ii) G (*, [v.sup.*], [w.sup.*], [[gamma].sup.*]) is (F, b, [phi], [rho], [theta])-quasiunivex at [x.sup.*], [phi] is increasing, and [phi](0) = 0;

(iii) p* (x, [x.sup.*]) + p(x, [x.sup.*]) [greater than or equal to] 0 for all x [member of] F;

(f) the Lagrangian-type function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is (F, b, [phi], [rho], [theta])-pseudounivex at [x.sup.*] and [phi](a) [greater than or equal to] 0 [right arrow] a [greater than or equal to] 0.

Then [x.sup.*] is an optimal solution of (P).

Proof. Let x be an arbitrary feasible solution of (P).

(a): Using the hypotheses specified in (i), we have for each i [member of] [p.bar],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In as much as [[lambda].sup.*] [greater than or equal to] 0, [u.sup.*] [greater than or equal to] 0, [p summation over (i = 1)] [[u.sup.*].sub.i] = 1, [phi] is superlinear, and F(x, [x.sup.*]; *) is sublinear, we [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.6)

Since for each j [member of] [J.sub.+],

[G.sub.j](x) + < [[gamma].sup.*j], [C.sub.j] x > [less than or equal to] [G.sub.j] (x) + [parallel] [[gamma].sup.*j] [[parallel].sub.*c(j)] [parallel] [C.sub.j] x [[parallel].sub.c(j)]) (by Lemma 4.1)

[less than or equal to] [G.sub.j](x) + [parallel] [C.sub.j] x [[parallel].sub.c(j)] (by (4.5))

[less than or equal to] 0(since x [member of] F)

= [G.sub.j]([x.sup.*]) + ([[gamma].sup.*j], [C.sub.j][x.sup.*]> (by (4.3)),

in view of the properties of the functions [[phi].sub.j], j [member of] [q.bar], we get

[[phi].sub.j] ([G.sub.j] (x) + ([[gamma].sup.*j], [C.sub.j] x > - [[G.sub.j] ([x.sup.*]) + ([[gamma].sup.*j], [C.sub.j] [x.sup.*] >]) [less than or equal to] 0,

which by (ii) implies that

F(x, [x.sup.*]; b(x, [x.sup.*])[[nabla][G.sub.j]([x.sup.*]) + [C.sup.T.sub.j] [[gamma].sup.*j]]) [less than or equal to] - [[rho].sub.j] (x, [x.sup.*]) [parallel] [theta](x, [x.sup.*]) [[parallel].sup.2].

As [[v.sup.*].sub.j] [greater than or equal to] 0 for each j [member of] [q.bar], [[v.sup.*].sub.j] = 0 for each j [member of] [q.bar]\[J.sub.+] (complement of [J.sub.+] relative to [q.bar]), and F(x, [x.sup.*]; *) is sublinear, the above inequalities yield

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.7)

In a similar manner, we can show that (iii) leads to the following inequality:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.8)

Combining (4.1), (4.6), (4.7), and (4.8), and using (4.2), (iv), and the sublinearity of F(x, [x.sup.*]; *), we obtain

[phi]([p summation over (i = 1)] [[u.sup.*].sub.i] {f (x, [[y.sup.*i]) + ([[alpha].sup.*i], A([[y.sup.*i])x > - [[lambda].sup.*][g(x, [[y.sup.*i]), < [[beta].sup.*i], B([[y.sup.*i])x>]}) [greater than or equal to] 0.

But [phi] [greater than or equal to] (a) > 0 [right arrow] a [greater than or equal to] 0 and hence we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.9)

Now using (4.9) and Lemma 4.2, we see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since x [member of] F was arbitrary, we conclude from this inequality that [x.sup.*] is an optimal solution of (P).

(b): As shown in part (a), for each j [member of] [J.sub.+], we have

[G.sub.j](x) + ([[gamma].sup.*j],[C.sub.j]x) [less than or equal to] [G.sub.j]([x.sup.*]) + ([[gamma].sup.*j], [C.sub.j][x.sup.*]),

and hence using the properties of the function [phi], we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which in view of (ii) implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now proceeding as in the proof of part (a) and using this inequality instead of (4.7), we arrive at (4.9), which leads to the desired conclusion that [x.sup.*] is an optimal solution of (P).

(c) - (e): The proofs are similar to those of parts (a) and (b).

(f): By our (F, b, [phi], [rho], [theta])-pseudounivexity assumption and the sublinearity of f(x, [x.sup.*]; *), (4.1) implies that

[phi](L(x, [[bar.y].sup.*], [[lambda].sup.*], [u.sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) - L(x, [[bar.y].sup.*], [[lambda].sup.*], [u.sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*])) [greater than or equal to] 0.

But [phi] (a) [greater than or equal to] 0 [right arrow] a [greater than or equal to] 0 and hence we have

L(x, [[bar.y].sup.*], [[lambda].sup.*], [u.sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) [greater than or equal to] L(x, [[bar.y].sup.*], [[lambda].sup.*], [u.sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*], [[lambda].sup.*]).

Because x, [x.sup.*] [member of] F, [v.sup.*] [greater than or equal to] 0, and (4.2) and (4.3) hold, the right-hand side of the above inequality is equal to zero, and so we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (by Lemma 4.1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (by the feasibility of x),

which is precisely (4.9). As seen in the proof of part (a), this inequality leads to the desired conclusion that [x.sup.*] is an optimal solution of (P).

The following variant of Theorem 4.1 can be proved in an identical manner.

Theorem 4.2. Let [x.sup.*] [member of] F, let [[lambda].sup.*] = [phi]([x.sup.*]) [greater than or equal to] 0, and assume that the functions f (*,y), g(*,y), y [member of] Y, [G.sub.j], j [member of] [q.bar], and [H.sub.k], k [member of] [r.bar], are differentiable at [x.sup.*], and that there exist (p, [[bar.y].sup.*], [u.sup.*], [[alpha].sup.*], [[beta].sup.*]) [member of] K, [v.sup.*] [member of] [R.sup.q.sub.+], [w.sup.*] [member of] [R.sup.r], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], j [member of] [q.bar], such that (4.2) - (4.5) and the following inequality hold:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4.10)

where F(x, [x.sup.*]; *): [R.sup.n] [right arrow] R is a given sublinear function. Furthermore, assume that any one of the six sets of conditions specified in Theorem 4.1 is satisfied. Then [x.sup.*] is an optimal solution of(P).

Although the proofs of Theorems 4.1 and 4.2 are essentially the same, their contents are somewhat different. This can easily be seen by comparing (4.1) with (4.10). We observe that any octuple ([x.sup.*], [[lambda].sup.*], [u.sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) that satisfies (4.1)-(4.5) also satisfies (4.2)(4.5), and (4.10), but the converse is not necessarily true. Moreover, (4.1) is a system of n equations, whereas (4.10) is a single inequality. Evidently, from a computational point of view, (4.1) is preferable to (4.10) because of the dependence of the latter on the feasible set of (P).

In Theorem 4.1, separate (F, b, [phi], [rho], [theta])-univexity assumptions were imposed on the functions [A.sub.i](*, [[bar.y].sup.*], [[alpha].sup.*]) and [B.sub.i](*, [[bar.y].sup.*], [[beta].sup.*]), i [member of] [p.bar]. It is possible to formulate a great variety of additional sets of sufficient optimality conditions in which various generalized (F, b, [phi], [rho], [theta])-univexity requirements are placed on certain combinations of these and the constraint functions. However, it turns out that a great majority of these results are subsumed in the generalized sufficiency criteria that will be established in the next section and, therefore, they will not be discussed separately.

5. GENERALIZED SUFFICIENCY CRITERIA

In this section, we discuss several families of sufficient optimality results under various generalized (F, b, [phi], [rho], [theta])-univexity hypotheses imposed on certain combinations of the problem functions. This is accomplished by employing a certain partitioning scheme which was originally proposed in [41] for the purpose of constructing generalized dual problems for nonlinear programming problems. For this we need some additional notation.

Let {[J.sub.0], [J.sub.1], ..., [J.sub.m]} and {[K.sub.0], [K.sub.1], ..., [K.sub.m]} be partitions of the index sets [q.bar] and [r.bar], respectively; thus, [J.sub.[mu]] [subset] [q.bar] for each [mu] [member of] [m.bar] [union] {0}, [J.sub.[mu]] [intersection] [J.sub.v] = 0 for each [mu], v [member of] [m.bar] [union] {0} with [mu] [not equal to] v, and [m union over ([mu] = 0)] [J.sub.[mu]] = [q.bar]. Obviously, similar properties hold for {[K.sub.0], [K.sub.1], ..., [K.sub.m]}.

Moreover, if [m.sub.1] and [m.sub.2] are the numbers of the partitioning sets of [q.bar] and [r.bar], respectively, then m = max{[m.sub.1], [m.sub.2]} and [J.sub.[mu]] = 0 or [K.sub.[mu]] = 0 for [mu] min{[m.sub.1], [m.sub.2]}.

In addition, we use the real-valued functions [[PHI].sub.i] (*, [bar.y], [lambda], v, w, [alpha], [beta], [gamma]), i [member of] [p.bar], [PHI] (*, [bar.y], [lambda], u, v, w, [alpha], [beta], [gamma]), and [[lambda].sub.t] (*, v, w, [gamma]) defined, for fixed [bar.y], [lambda], u, v, w, [alpha], [beta], and [gamma],on X as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Making use of the sets and functions defined above, we can now formulate our first collection of generalized sufficiency results for (P) as follows.

Theorem 5.1. Let [x.sup.*] [member of] F, let [[lambda].sup.*] = [phi]([x.sup.*]) [greater than or equal to] 0, and assume that the functions f (*,y),g(*,y), y [member of] Y, [G.sub.j], j [member of] [q.bar], and [H.sub.k], k [member of] [r.bar], are differentiable at [x.sup.*], and that there exist (p, [[bar.y].sup.*], [u.sup.*], [[alpha].sup.*], [[beta].sup.*]) [member of] K, [v.sup.*] [member of] [R.sup.q.sub.+], [w.sup.*] [member of] [R.sup.r], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], j [member of] [q.bar], such that (4.1) - (4.5) hold. Assume, furthermore, that any one of the following four sets of hypotheses is satisfied:

(a) (i) [PHI](*, [y.sup.*],[[lambda].sup.*],[u.sup.*],[v.sup.*],[w.sup.*], [[alpha].sup.*], [[beta].sup.*],[[gamma].sup.*]) is prestrictly (F, b, [bar.[phi]], [bar.[rho]], [theta])-quasiunivex at [x.sup.*] and

[phi] (a) [greater than or equal to] 0 [right arrow] a [greater than or equal to] 0;

(ii) for each t [member of] [m.bar], [[lambda].sub.t](*, [v.sup.*], [w.sup.*], [[gamma].sup.*]) is strictly (F, b, [[phi].sub.t], [[rho].sub.t], [theta])-pseudounivex at [x.sup.*], [[phi].sub.t] is increasing, and [[phi].sub.t] (0) = 0;

(iii) [bar.[rho]](x, [x.sup.*]) + [m summation over (t = 1)] [[rho].sub.t](x, [x.sup.*]) [greater than or equal to] 0 for all x [member of] F;

(b) (i) [PHI] (*, [y.sup.*], [[lambda].sup.*], [u.sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) is (F, b, [bar.[phi]], [bar.[rho]], [theta])-pseudounivex at [x.sup.*] and [phi] (a) [greater than or equal to] 0 [right arrow] a [greater than or equal to] 0;

(ii) for each t [member of] [m.bar], [[lambda].sub.t](*, [v.sup.*], [w.sup.*], [[gamma].sup.*]) is (F, b, [phi], [rho], [theta])-quasiunivex at [x.sup.*], [phi]t is increasing, and [[phi].sub.t] (0) = 0;

(iii) p(x, [x.sup.*]) + 2 P>t(x, [x.sup.*]) > 0 for all x [member of] F;

(c) (i) [PHI] (*, [y.sup.*], [[lambda].sup.*], [u.sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) is prestrictly (F, b, [bar.[phi]], [bar.[rho]], [theta])-quasiunivex at [x.sup.*] and [phi] (a) [greater than or equal to] 0 [right arrow] a [greater than or equal to] 0;

(ii) for each t [member of] [m.bar], [[lambda].sub.t](*, [v.sup.*], [w.sup.*], [[gamma].sup.*]) is (F, b, [phi], [rho], [theta])-quasiunivex at [x.sup.*], [[phi].sub.t] is increasing, and [[phi].sub.t] (0) = 0;

(iii) [bar.[rho]] (x, [x.sup.*]) + [m summation over (t = 1)] [[rho].sub.t](x, [x.sup.*]) > 0 for all x [member of] F;

(d) (i) [PHI](*, [y.sup.*], [[lambda].sup.*], [u.sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) is prestrictly (F, b, [bar.[phi]], [bar.[rho]], [theta])-quasiunivex at [x.sup.*] and [phi] (a) [greater than or equal to] 0 [right arrow] a [greater than or equal to] 0;

(ii) for each t [member of] [bar.[m.sub.1]], [[lambda].sub.t](*, [v.sup.*], [w.sup.*], [[gamma].sup.*]) is (F, b, [phi], [rho], [theta])-quasiunivex at [x.sup.*], for each t [member of] [bar.[m.sub.2]] [not equal to] 0, [[lambda].sub.t](*, [v.sup.*], [w.sup.*], [[gamma].sup.*]) is strictly (F, b, [phi], [rho], [theta])-pseudounivex at [x.sup.*], and for each t [member of] [m.bar], [[phi].sub.t] is increasing and [[phi].sub.t] (0) = 0, where {[bar.[m.sub.1]], [bar.[m.sub.2]]} is a partition of [m.bar];

(iii) [bar.[rho]](x, [x.sup.*]) + [m summation over (t = 1)] [[rho].sub.t](x, [x.sup.*]) [greater than or equal to] 0 for all x [member of] F.

Then [x.sup.*] is an optimal solution of(P).

Proof. Let x be an arbitrary feasible solution of (P).

(a): Since F(x, [x.sup.*]; *) is sublinear and b(x, [x.sup.*]) > 0, from (4.1) it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since for each t [member of] [m.bar],

[[lambda].sub.t](x, [v.sup.*], [w.sup.*], [[gamma].sup.*])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(by Lemma 4.1 and the nonnegativity of [v.sup.*])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[less than or equal to] 0 (by the feasibility of x)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(by (4.3) and the feasibility of [x.sup.*])

[[lambda].sub.t]([x.sup.*], [v.sup.*], [w.sup.*], [[gamma].sup.*])

and hence [[phi].sub.t] ([[lambda].sub.t](x, [v.sup.*], [w.sup.*], [[gamma].sup.*]) - [[lambda].sub.t](x, [v.sup.*], [w.sup.*], [[gamma].sup.*])) [less than or equal to] 0, it follows from (ii) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Summing over t and using the sublinearity of F(x, [x.sup.*]; *), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5.2)

Combining (5.1) and (5.2), and using (iii) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.3)

which by virtue of (i) implies that

[bar.[phi]] ([PHI] (x, [[bar.y].sup.*], [[lambda].sup.*], [u.sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) - [PHI] (x, [[bar.y].sup.*], [[lambda].sup.*], [u.sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*])) [greater than or equal to] 0.

But [bar.[phi]] (a) [greater than or equal to] 0 [right arrow] a [greater than or equal to] 0, and hence we get

[PHI] (x, [[bar.y].sup.*], [[lambda].sup.*], [u.sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) > [PHI] (x, [[bar.y].sup.*], [[lambda].sup.*], [u.sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) = 0,

where the equality follows from (4.2), (4.3), and the feasibility of x . Therefore, we have

0 [less than or equal to] [PHI] (x, [[bar.y].sup.*], [[lambda].sup.*], [u.sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(by Lemma 4.1 and the feasibility of x)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(by the feasibility of x) .

Now using this inequality and Lemma 4.2, as in the proof of Theorem 4.1, we obtain [phi]([x.sup.*]) [less than or equal to] [phi](x). Since x was arbitrary, we conclude that [x.sup.*] is an optimal solution of (P).

(b): Proceeding as in the proof of part (a), we see that (ii) leads to the following inequality:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now combining this inequality with (5.1) and using (iii), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The rest of the proof is identical to that of part (a).

(c) - (d): The proofs are similar to those of parts (a) and (b).

Theorem 5.2. Let [x.sup.*] [member of] F, let [[lambda].sup.*] = [phi]([x.sup.*]) [greater than or equal to] 0, and assume that the functions f (*, y), g (*, y)); y [member of] Y, [G.sub.j], j [member of] [q.bar], and [H.sub.k], k [member of] [r.bar], are differentiable at [x.sup.*], and that there exist (p, [[bar.y].sup.*], [u.sup.*], [[alpha].sup.*], [[[beta].sup.*]) [member of] K, [v.sup.*] [member of] [R.sup.q.sub.+], [w.sup.*] [member of] [R.sup.r], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that (4.1)-(4.5) hold. Assume, furthermore, that any one of the following seven sets of hypotheses is satisfied:

(a) (i) for each i [member of] [I.sub.+] [equivalent to] [I.sub.+](u *), [[PHI].sub.i] (*, [y.sup.*], [[lambda].sup.*],[v.sup.*], [w.sup.*],[[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) is (F, b, [[bar.[phi]].sub.i], [[bar.[rho]].sub.i], [theta])-pseudounivex at [x.sup.*], [[bar.[phi]].sub.i] is strictly increasing, and [[bar.[phi]].sub.i] (0) = 0;

(ii) for each t [member of] [m.bar], [[lambda].sub.t](*, [v.sup.*], [w.sup.*], [[gamma].sup.*]) is (F, b, [[phi].sub.t], [[rho].sub.t], [theta])-quasiunivex at [x.sup.*], [[phi].sub.t] is increasing, and [[phi].sub.t] (0) = 0;

(iii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(b) (i) for each i [member of] [I.sub.+], [[PHI].sub.i] (*, [y.sup.*], [[lambda].sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) is prestrictly (F, b, [[bar.[phi]].sub.i], [[bar.[rho]].sub.i], [theta])-quasiunivex at [x.sup.*], [[bar.[phi]].sub.i] is strictly increasing, and [[bar.[phi]].sub.i] (0) = 0;

(ii) for each t [member of] [m.bar], [[lambda].sub.t](*, [v.sup.*], [w.sup.*], [[gamma].sup.*]) is strictly (F, b, [[phi].sub.t], [[rho].sub.t], [theta])-pseudounivex at [x.sup.*], [[phi].sub.t] is increasing, and [[phi].sub.t] (0) = 0;

(iii)[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(c) (i) for each i [member of] [I.sub.+], [[PHI].sub.i] (*, [y.sup.*], [[lambda].sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) is prestrictly (F, b, [[bar.[phi]].sub.i], [[bar.[rho]].sub.i], [theta])-quasiunivex at [x.sup.*], [[bar.[phi]].sub.i] is strictly increasing, and [[bar.[phi]].sub.i] (0) = 0;

(ii) for each t [member of] [m.bar], [[lambda].sub.t](*, [v.sup.*], [w.sup.*], [[gamma].sup.*]) is (F, b, [[phi].sub.t], [[rho].sub.t], [theta])-quasiunivex at [x.sup.*], [[phi].sub.t] is increasing, and [[phi].sub.t] (0) = 0;

(iii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(d) (i) for each i [member of] [I.sub.1+], [[PHI].sub.i] (*, [y.sup.*], [[lambda].sup.*],[v.sup.*],[w.sup.*],[[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) is (F, b, [[bar.[phi]].sub.i], [[bar.[rho]].sub.i], [theta])-pseudounivex at [x.sup.*], for each i [member of] [I.sub.2+], [[PHI].sub.i] (*, [y.sup.*], [[lambda].sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) is prestrictly (F, b, [[bar.[phi]].sub.i], [[bar.[rho]].sub.i], [theta])-quasiunivex at [x.sup.*], and for each i [member of] [I.sub.+], [[bar.[phi]].sub.i] is strictly increasing and [[bar.[phi]].sub.i] (0) = 0, where, {[I.sub.1+], I.sub.2+]} is a partition of [I.sub.+];

(ii) for each t [member of] [M.bar], [[lambda].sub.t](*, [v.sup.*], [w.sup.*], [[gamma].sup.*]) is strictly (F, b, [[phi].sub.t], [[rho].sub.t], [theta])-pseudounivex at [x.sup.*], [[phi].sub.t] is increasing, and [[phi].sub.t] (0) = 0;

(iii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(e) (i) for each i [member of] [I.sub.1+] [not equal] 0, [[PHI].sub.i] (*, [y.sup.*], [[lambda].sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) is (F, b, [[bar.[phi]].sub.i], [[bar.[rho]].sub.i], [theta])-pseudounivex at [x.sup.*], for each i [member of] [I.sub.2+], [[PHI].sub.i] (*, [y.sup.*], [[lambda].sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) is prestrictly (F, b, [[bar.[phi]].sub.i], [[bar.[rho]].sub.i], [theta])-quasiunivex at [x.sup.*],and for each i [member of] [I.sub.+], [[bar.[phi]].sub.i] is strictly increasing and f>i (0) = 0, where,{[I.sub.1+], [I.sub.2+]} is a partition of [I.sub.+];

(ii) for each t [member of] [m.bar], [[lambda].sub.t](*, [v.sup.*], [w.sup.*], [[gamma].sup.*]) is (F, b, [[phi].sub.t], [[rho].sub.t], [theta]-quasiunivex at [x.sup.*], [[phi].sub.t] is increasing, and [[phi].sub.t](0) = 0;

(iii)[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(f) (i) for each i [member of] [I.sub.+], [[PHI].sub.i] (*, [y.sup.*], [[lambda].sup.*],[v.sup.*],[w.sup.*],[[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) is prestrictly (F, b, [[bar.[phi]].sub.i], [[bar.[rho]].sub.i], [theta])-quasiunivex at [x.sup.*], [[bar.[phi]].sub.i] is strictly increasing, and [[bar.[phi]].sub.i](0) = 0;

(ii) for each t [member of] [[m.bar].sub.1] [not equal to] 0, [[lambda].sub.t](*, [v.sup.*], [w.sup.*], [[gamma].sup.*]) is strictly (F, b, [[bar.[phi]].sub.t], [[rho].sub.t], [theta])-pseudounivex at [x.sup.*], for each t [member of] [[m.bar].sub.2], [[lambda].sub.t](*, [v.sup.*], [w.sup.*], [[gamma].sup.*]) is (F, b, [[phi].sub.t], [[rho].sub.t], [theta])-quasiunivex at [x.sup.*], and for each t [member of] [m.bar], [[phi].sub.t] is increasing and [[phi].sub.t](0) = 0; where {[[m.bar].sub.1], {[[m.bar].sub.2]} is a partition of [m.bar];

(iii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(g) (i) for each i [member of] [I.sub.1+], [[PHI].sub.i] (*, [y.sup.*], [[lambda].sup.*],[v.sup.*],[w.sup.*],[[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) is (F, b, [[bar.[phi]].sub.i], [[bar.[rho]].sub.i], [theta])-pseudounivex at [x.sup.*],for each i [member of] [I.sub.2+], [[PHI].sub.i] (*, [y.sup.*], [[lambda].sup.*], [v.sup.*],[w.sup.*],[[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) is prestrictly (F, b, [[bar.[phi]].sub.i], [[bar.[rho]].sub.i], [theta])-quasiunivex at [x.sup.*], and for each i [member of] [I.sub.+], [[bar.[phi]].sub.i] is strictly increasing and [[bar.[phi]].sub.i] (0) = 0, where,{[I.sub.+], [I.sub.2+]} is a partition of [I.sub.+];

(ii) for each t [member of] [[m.bar].sub.1], [[lambda].sub.t](*, [v.sup.*], [w.sup.*], [[gamma].sup.*]) is strictly (F, b, [[phi].sub.t], [[rho].sub.t], [theta])-pseudounivex at [x.sup.*], for each t [[m.bar].sub.1], [[lambda].sub.t](*, [v.sup.*], [w.sup.*], [[gamma].sup.*]) is (F, b, [[phi].sub.t], [[rho].sub.t], [theta])-quasiunivex at [x.sup.*], and for each t [member of] [m.bar], [[phi].sub.t] is increasing and [phi]t(0) = 0; where {[[m.bar].sub.2]} is a partition of [m.bar];

(iii)[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(iv) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then x is an optimal solution of(P).

Proof. (a): Suppose to the contrary that x is not an optimal solution of (P). Then there is a feasible solution [bar.x] of (P) such that [phi]([bar.x]) < [phi]([x.sup.*]) = [[lambda].sup.*], which implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence, for each i [member of] [p.bar], we have

f([bar.x], [[y.sup.*i]) + [parallel] [[y.sup.*i] [bar.x]) [[parallel].sub.a] - [[lambda].sup.*][g ([bar.x], [[y.sup.*i]) - [parallel] B([[y.sup.*i]) [bar.x] [[parallel].sub.b]] < 0. (5.4)

Keeping in mind that [[lambda].sup.*] and [v.sup.*] are nonnegative, we see that for each i [member of] [I.sub.+],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (by Lemma 4.1 and the feasibility of x)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (by the feasibility of x)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and the feasibility of [x.sup.*])

= [[PHI].sub.i] ([x.sup.*], [[bar.y].sup.*], [[lambda].sup.*],[v.sup.*], [w.sup.*],[[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]),

and so using the properties of the function [[bar.[phi]].sub.i], we get

[[bar.[phi]].sub.i] ([[PHI].sub.i] ([bar.x], [[bar.y].sup.*], [[lambda].sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*])) - [[PHI].sub.i] ([x.sup.*], [[bar.y].sup.*], [[lambda].sup.*], [v.sup.*], [w.sup.*],[[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) < 0,

which in view of (i) implies that for each i [member of] [I.sub.+],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [u.sup.*] [greater than or equal to] 0, [u.sup.*] = 0 for each i [member of] [p.bar]\[I.sub.+], [p summation over (i=1)] [[u.sup.*].sub.i] = 1, and F([bar.x], [x.sup.*]; *) is sublinear, the above

inequalities yield

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5.5)

As seen in the proof of Theorem 4.1, our assumptions in (ii) lead to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which when combined with (5.1), results into

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In view of (iii), this inequality contradicts (5.5). Hence, [x.sup.*] is an optimal solution of (P). (b) - (g): The proofs are similar to that of part (a).

Each one of the twenty-two sets of conditions given in Theorems 5.1 and 5.2, and in their modified versions obtained by replacing (4.1) with (4.10), can be viewed as a family of sufficient optimality conditions whose members can easily be identified by appropriate choices of the partitioning sets [J.sub.[mu]] and [K.sub.[mu]], [mu] [member of] [q.bar] [union] {0}.

In the remainder of this section, we present another collection of sufficiency results which are somewhat different from those stated in Theorems 5.1 and 5.2. These results are formulated by utilizing a partition of [p.bar] in addition to those of [q.bar] and [r.bar], and by placing appropriate generalized (F, b, [phi], [rho], [theta])-univexity requirements on certain combinations of the problem functions.

Let {[I.sub.0], [I.sub.1], ..., [I.sub.l]} be a partition of [p.bar] such that L = {0,1, 2, ..., l} [subset] M = {0,1, ..., m}, and let the function [[PI].sub.t](*, [bar.y], [lambda], u, v, w, [alpha], [beta], [gamma]): X [right arrow] R be defined, for fixed [bar.y], [lambda], u, v, w, [alpha], [beta], [gamma], and [gamma], by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 5.3. Let [x.sup.*] [member of] F, let [[lambda].sup.*] = [phi]([x.sup.*]) [greater than or equal to] 0, and assume that the functions f (*, y),g(*, y), y [member of] Y, [G.sub.j], j [member of] [q.bar], and [H.sub.k], k [member of] [r.bar], are differentiable at [x.sup.*], and that there exist (p, [[bar.y].sup.*], [u.sup.*], [[alpha].sup.*], [[beta].sup.*]) [member of] K, [v.sup.*] [member of] [R.sup.q.sub.+], [w.sup.*] [member of] [R.sup.r], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that (4.1) - (4.5) hold. Assume, furthermore, that any one of the following seven sets of hypotheses is satisfied:

(a) (i) for each t [member of] L, [[PI].sub.t](*, [[bar.y].sup.*], [[lambda].sup.*], [u.sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) is strictly (F, b, [[phi].sub.t], [[rho].sub.t], [theta])-pseudounivex at [x.sup.*], [[phi].sub.t] is increasing, and [[phi].sub.t] (0) = 0;

(ii) for each t [member of] M \ L, [[lambda].sub.t](*, [v.sup.*], [w.sup.*], [[gamma].sup.*]) is (F, b, [[phi].sub.t], [[rho].sub.t], [theta])-quasiunivex at [x.sup.*], [[phi].sub.t] is increasing, and [[phi].sub.t] (0) = 0;

(iii) [summation over (t [member of] M)] [[rho].sub.t](x, [x.sup.*]) [greater than or equal to] 0 for all x [member of] F;

(b) (i) for each t [member of] L, [[PI].sub.t](*, [[bar.y].sup.*], [[lambda].sup.*], [u.sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) is (F, b, [[phi].sub.t], [[rho].sub.t], [theta])-quasiunivex at x, [[phi].sub.t] is increasing, and [[phi].sub.t](0) = 0;

(ii) for each t [member of] M \ L, [[lambda].sub.t] (*, [v.sup.*], [w.sup.*], [[gamma].sup.*]) is strictly (F, b, [[phi].sub.t], [[rho].sub.t], [theta])-pseudounivex at x, [[phi].sub.t] is increasing, and [[phi].sub.t](0) = 0;

(iii) [summation over (t [member of] M)] [[rho].sub.t](x, [x.sup.*]) [greater than or equal to] 0 for all x [member of] F;

(c) (i) for each t [member of] L, [[PI].sub.t](*, [[bar.y].sup.*], [[lambda].sup.*], [u.sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) is (F, b, [[phi].sub.t], [[rho].sub.t], [theta])-quasiunivex at x, [[phi].sub.t] is increasing, and [[phi].sub.t](0) = 0;

(ii) for each t [member of] M \ L, [[lambda].sub.t] (*, [v.sup.*], [w.sup.*], [[gamma].sup.*]) is (F, b, [[phi].sub.t], [[rho].sub.t], [theta])-quasiunivex at [x.sup.*], [[phi].sub.t] is increasing, and [[phi].sub.t](0) = 0;

(iii) [summation over (t [member of] M)] [[rho].sub.t](x, [x.sup.*]) [greater than or equal to] 0 for all x [member of] F;

(d) (i) for each t [member of] [L.sub.1], [[PI].sub.t](*, [[bar.y].sup.*], [[lambda].sup.*], [u.sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) is strictly (F, b, [[phi].sub.t], [[rho].sub.t], [theta])-pseudounivex at [x.sup.*], for each t [member of] [L.sub.2], [[PI].sub.t](*, [[bar.y].sup.*], [[lambda].sup.*], [u.sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) is (F, b, [[phi].sub.t], [[rho].sub.t], [theta])-quasiunivex at [x.sup.*], and for each t [member of] L, [[phi].sub.t] is increasing and [[phi].sub.t](0) = 0, where{[L.sub.1],[L.sub.2]} is a partition of L;

(ii) for each t [member of] M \ L, [[lambda].sub.t] (*, [v.sup.*], [w.sup.*], [[gamma].sup.*]) is strictly (F, b, [[phi].sub.t], [[rho].sub.t], [theta])-pseudounivex at x, [[phi].sub.t] is increasing, and [[phi].sub.t](0) = 0;

(iii) [summation over (t [member of] M)] [[rho].sub.t](x, [x.sup.*]) [greater than or equal to] 0 for all x [member of] F;

(e) (i) for each t [member of] [L.sub.1] [not equal to] [[PI].sub.t](*, [[bar.y].sup.*], [[lambda].sup.*], [u.sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) is strictly (F, b, [[phi].sub.t], [[rho].sub.t], [theta])-pseudounivex at [x.sup.*], for each t [member of] [L.sub.2], [[PI].sub.t](*, [[bar.y].sup.*], [[lambda].sup.*], [u.sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) is (F, b, [[phi].sub.t], [[rho].sub.t], [theta])-quasiunivex at [x.sup.*], and for each t [member of] L, [[phi].sub.t] is increasing and [[phi].sub.t](0) = 0, where {[L.sub.1],[L.sub.2]} is a partition of L;

(ii) for each t [member of] M \ L, [[lambda].sub.t] (*, [v.sup.*], [w.sup.*], [[gamma].sup.*]) is (F, b, [[phi].sub.t], [[rho].sub.t], [theta])-quasiunivex at [x.sup.*], [[phi].sub.t] is increasing, and [[phi].sub.t](0) = 0;

(iii) [summation over (t [member of] M)] [[rho].sub.t](x, [x.sup.*]) [greater than or equal to] 0 for all x [member of] F;

(f) (i) for each t [member of] L, [[PI].sub.t](*, [[bar.y].sup.*], [[lambda].sup.*], [u.sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) is (F, b, [[phi].sub.t], [[rho].sub.t], [theta])-quasiunivex at x, [[phi].sub.t] is increasing, and [[phi].sub.t](0) = 0;

(ii) for each t [member of] [(M\L).sub.1] [not equal to] [[lambda].sub.t] (*, [v.sup.*], [w.sup.*], [[gamma].sup.*]) is strictly (F, b, [[phi].sub.t], [[rho].sub.t], [theta])-pseudounivex at [x.sup.*], for each t [member of] [(M\L).sub.2], [[lambda].sub.t] (*, [v.sup.*], [w.sup.*], [[gamma].sup.*]) is (F, b, [[phi].sub.t], [[rho].sub.t], [theta])-quasiunivex at [x.sup.*], and for each t [member of] L, [[phi].sub.t] is increasing and [[phi].sub.t](0) = 0, where {[(M\L).sub.1], [(M\L).sub.2]} is a partition of M \ L;

(iii) [summation over (t [member of] M)] [[rho].sub.t](x, [x.sup.*]) [greater than or equal to] 0 for all x [member of] F;

(g) (i) for each t [member of] [L.sub.1], [[PI].sub.t](*, [[bar.y].sup.*], [[lambda].sup.*], [u.sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) is strictly (F, b, [phi], [rho], [theta]) -pseudounivex at [x.sup.*], for each t [member of] [L.sub.2], [[PI].sub.t](*, [[bar.y].sup.*], [[lambda].sup.*], [u.sup.*], [v.sup.*], [w.sup.*], [[alpha].sup.*], [[beta].sup.*], [[gamma].sup.*]) is (F, b, [phi], [rho], [theta])-quasiunivex at [x.sup.*],and for each t [member of] L, [[phi].sub.t] is increasing and [[phi].sub.t](0) = 0, where {[L.sub.1],[L.sub.2]} is a partition of L;

(ii) for each t [member of] [(M \ L).sub.1], [[lambda].sub.t](x, [v.sup.*], [w.sup.*], [[gamma].sup.*]) is strictly (F, b, [[phi].sub.t], [[rho].sub.t], [theta])-pseudounivex at [x.sup.*],for each t [member of] [(M \ L).sub.2], [[lambda].sub.t](x, [v.sup.*], [w.sup.*], [[gamma].sup.*]) is (F, b, [[phi].sub.t], [[rho].sub.t], [theta])-quasiunivex at [x.sup.*], and for each t [member of] M \ L, [[phi].sub.t] is increasing and [phi]t(0) = 0, where {[(M \ L).sub.1], [(M \ L).sub.2]} is a partition of M \ L;

(iii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(iv) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then [x.sup.*] is an optimal solution of (P).

Proof. (a): Suppose to the contrary that [x.sup.*] is not an optimal solution of (P). As seen in the proof of Theorem 5.2, this supposition leads to the inequalities

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (by (5.6))

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.7)

As shown in the proof of Theorem 4.1, for each t [member of] M\L, [lambda]t(x, [v.sup.*], [w.sup.*], [[gamma].sup.*]) [less than or equal to] [lambda]t(x, [v.sup.*], [w.sup.*], [[gamma].sup.*]), and so [[phi].sub.t] [lambda]t(x, [v.sup.*], [w.sup.*], [[gamma].sup.*]) - [lambda]t (x, [v.sup.*], [w.sup.*], [[gamma].sup.*]) [less than or equal to] 0, which in view of (ii) implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.8)

Now combining (5.7) and (5.8) and using the sublinearity of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which contradicts (4.1). Therefore, [x.sup.*] is an optimal solution of (P).

(b) - (g): The proofs are similar to that of part (a).

The fourteen families of sufficiency results stated in Theorem 5.3, and in its modified version obtained by replacing (4.1) with (4.10), contain a very large number of special cases and variants which can readily be identified by appropriate choices of the partitioning sets [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and the sets L and M.

6. CONCLUDING REMARKS

Using a Dinkelbach-type [8] parametric approach, in this paper we have established a set of necessary optimality conditions and numerous sets of sufficient optimality criteria for a continuous minmax fractional programming problem containing arbitrary norms (and square roots of positive semidefinite quadratic forms). In formulating these criteria, we have exploited the structure of the subdifferentials of the arbitrary norms in conjunction with the basic properties of various types of generalized (F, b, [phi], [rho], [theta])-univex functions. Since all the optimality conditions and especially the sufficiency results proved for the prototype problem (P) can easily be modified and restated for each one of the seventeen special cases of (P) designated as (P1)-(P17) in Section 1, they collectively subsume a truly vast number of sufficiency results previously obtained for several classes of nonlinear programming problems by an assortment of ad hoc methods. Furthermore, the style of presentation adopted in this paper as well as the main results derived here will prove useful in studying other related categories of nonlinear programming problems and utilizing other types of generalized convexity concepts. In particular, the results of the present study can be extended to the following semiinfinite version of (P):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where X, Y, f, g, A(y), B(y), [parallel] * [[parallel].sub.a], and [parallel] * [[parallel].sub.b] are as defined in the description of (P), [T.sub.j] and [S.sub.k] are compact subsets of complete metric spaces, for each j [member of] [q.bar], [[??].sub.j](*, t) is a real-valued function defined on X for all t [member of] [T.sub.j], for each k [member of] [r.bar], [[??].sub.k] (x, *) is a real-valued function defined on X for all s [member of] [S.sub.k], for each j [member of] [q.bar] and k [member of] [r.bar], [G.sub.j](x, *) and [[??].sub.k] (x, *) are continuous real-valued functions defined, respectively, on [T.sub.j] and [S.sub.k] for all x [member of] X, and for each j [member of] [q.bar] and t [member of] [T.sub.j], [C.sub.j] (t) is an [n.sub.j] x n matrix and [parallel] * [[parallel].sub.c(j)] is an arbitrary norm on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We shall investigate this semiinfinite programming problem in subsequent papers.

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Northern Michigan University

G. J. Zalmai

Department of Mathematics and Computer Science

Marquette, MI 49855, USA

E-mail address: gzalmai@nmu.edu

Northern Michigan University

Qinghong Zhang

Department of Mathematics and Computer Science

Marquette, MI 49855, USA

E-mail address: qzhang@nmu.edu

Received: February 05, 2012. Revised: August 01, 2012.

2010 Mathematics Subject Classification: 90C26, 90C30, 90C32, 90C46, 90C47.
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