# Generalized Higher Order (ph, , o, p, p, th, m)-Invexities in Parametric Optimality Conditions for Discrete Minmax Fractional Programming.

Byline: Ram U. VermaAbstract: First several new classes of higher order (ph, , o, p, p, th, m)-invexities are introduced, and then a set of higher-order parametric necessary optimality conditions and several sets of higher order sufficient optimality conditions for a discrete minmax fractional programming problem applying various higher order (ph, , o, p, p, th, m)-invexity constraints are established. The obtained results are new and generalize a wide range of results in the literature.

Keywords: Discrete minmax fractional programming, (ph, , o, p, p, th, m)-invex functions, necessary optimality conditions, sufficient optimality conditions.

1. INTRODUCTION

In this communication, first several new classes of generalized second-order (ph, , o, p, p, th, m)-invex functions are introduced, and then these are applied to establish a set of second-order necessary optimality conditions leading to several sets of second-order sufficient optimality conditions and theorems for the following discrete minmax fractional programming problem:

(Equation)

Subject to (Eq.) X, where X is an open convex subset of Rn (n-dimensional Euclidean space), (Eq.) and (Eq.) are real-valued functions defined on X, are real-valued functions defined on X, and for each (Eq.) 0 for all x satisfying the constraints of (P).

The first part of this presentation deals with several new notions of the generalized second order (ph, , o, p, p, th, m)-invexities, which generalize/unify most of the existing generalized invexities and variants in the literature. Then some second-order optimality conditions for our principal problem (P) are established.

The obtained results can be generalized to its semiinfinite counterparts as well. Furthermore, our results can be applied to the new notion (developed in Chinchuluun and Pardalos [1], Pitea and Postalache [2- 4]) of multitime multiobjective variational problems.

Zalmai [13-15] introduced and investigated some significant results in a series of publications, while the results of Verma and Zalmai [11] and Verma [9] are significant to our problem on hand. For more details to this context, we refer the reader [5-16].

The results thus obtained here in this communication are new and application-oriented to context of results available in the literature.

2. PRELIMINARIES

Verma and Zalmai [11] introduced the notion of the generalized (ph, , p, th, m)-invexities, and further applied to establish a class of second order parametric necessary optimality conditions as well as sufficient optimality conditions for a discrete minmax fractional programming problem using the general frameworks for the (ph, , p, th, m)-invexities. In this section, we first generalize the notion of the generalized (ph, , p, th, m)-invexities, and then recall some important auxiliary results for the problem (P) on hand.

Definition 2.1. Let f be a differentiable real-valued function defined on Rn. Then f is said to be n-invex (invex with respect to) at y if there exists a function

(Equations)

Let f be a twice differentiable real-valued function defined on (Eq.)n. Now we introduce the new classes of generalized second-order hybrid invex functions which seem to be application-oriented to developing a new optimality-duality theory for nonlinear programming based on second-order necessary and sufficient optimality conditions. We shall abbreviate "second- order invex" as sonvex. Let (Eq.) be a twice differentiable function.

Definition 2.2. The function f is said to be (strictly)

(Equations)

Definition 2.3. The function f is said to be (strictly) (Equations)-pseudosonvex at (Eq.) if there

(Equations)

Definition 2.4. The function (Eq.) is said to be (prestrictly) (Equations)-quasisonvex at (Eq.)

(Equations)

Here we present some examples for our new notions of generalized invex functions.

Example 1. The function f is said to be (prestrictly) (Equations)-quasisonvex at (Eq.) if there exist functions

(Equations)

Example 2. The function f is said to be (prestrictly) (Equations)-quasisonvex at (Eq.) if there exist functions

(Equations)

We recall the following results on the second order optimality conditions to the context of the main results to be established in the next section. Theorem 2.1. [11] Let (Eq.) be an optimal solution of

(Equations)

3. SUFFICIENT OPTIMALITY CONDITIONS

In this section, we present several second-order sufficiency results in which various generalized (Equations)-sonvexity assumptions are imposed on the individual as well as certain combinations of the problem functions.

For the sake of the compactness, we shall use the following notations during the statements as well as the proofs of sufficiency theorems:

(Equations)

where F (assumed to be nonempty) is the feasible set of (P), defined by (Equations) In addition, assume that any one of the following six sets of conditions holds:

(Equations)

Now proceeding as in the proof of part (a) and using this inequality instead of (3.6), we arrive at (3.8), which leads to the desired conclusion that x is an optimal solution of (P). (c) - (e): The proofs are similar to those of parts (a) and (b).

(Equations)

As seen in the proof of part (a), this inequality leads to the desired conclusion that (Eq.) is an optimal solution of (P).

(Equations)

As shown in the proof of Theorem 3.1, this inequality leads to the conclusion that (Eq.) is an optimal solution to (P).

(Equations)

4. CONCLUDING REMARKS

We established several results applying the new notion of higher order (ph, , o, p, p, th, m)-invexities, which generalizes/unifies most of the existing generalized invexities and its variants in the literature, and then we proved some results on second-order optimality conditions for our principal problem (P).

The obtained results to the context of discrete minmax fractional programming offer further applications to other fields of research endeavors relating to discrete fractional programming problems, including the publications [1-4] relating to the multitime multiobjective variational problems.

REFERENCES

[1] Chinchuluun A, Pardalos PM. A survey of recent developments in multiobjective optimization. Annals of Operations Research 2007; 154: 29-50. http://dx.doi.org/10.1007/s10479-007-0186-0

[2] Pitea A, Postolache M. Duality theorems for a new class of multitime multiobjective variational problems. Journal of Global Optimization 2012; 54(1): 47-58. http://dx.doi.org/10.1007/s10898-011-9740-z

[3] Pitea A, Postolache M. Minimization of vectors of curvilinear functionals on the second order jet bundle: Necessary conditions. Optimization Letters 2012; 6(3): 459-70. http://dx.doi.org/10.1007/s11590-010-0272-0

[4] Pitea A, Postolache M. Minimization of vectors of curvilinear functionals on the second order jet bundle: Sufficient efficiency conditions. Optimization Letters 2012; 6(8): 1657- 69. http://dx.doi.org/10.1007/s11590-011-0357-4

[5] Srivastava MK, Bhatia M. Symmetric duality for multiobjective programming using second order (F, ) -convexity. Opsearch 2006; 43: 274-95.

[6] Srivastava KK, Govil MG. Second order duality for multiobjective programming involving (Eq.) -type I functions. Opsearch 2000; 37: 316-26.

[7] Verma RU. Weak (Eq.) efficiency conditions for multiobjective fractional programming. Applied Mathematics and Computation 2013; 219: 6819-927. http://dx.doi.org/10.1016/j.amc.2012.12.087

[8] Verma RU. A generalization to Zalmai type second order univexities and applications to parametric duality models to discrete minimax fractional programming. Advances in Nonlinear Variational Inequalities 2012; 15(2): 113-23.

[9] Verma RU. Second order (Eq.) invexity frameworks and efficiency conditions for multiobjective fractional programming. Theory and Applications of Mathematics and Computer Science 2012; 2(1): 31-47.

[10] Verma RU. Role of second order B (Eq.)-invexities and parametric sufficient conditions in semiinfinite minimax fractional programming. Transactions on Mathematical Programming and Applications 2013; 1(2): 13-45.

[11] Verma RU, Zalmai GJ. Generalized second-order parametric optimality conditions in discrete minmax fractional programming. Transactions on Mathematical Programming and Applications 2014; 2(12): 1-20.

[12] Yang XM, Yang XQ, Teo KL, Hou SH. Second order duality for nonlinear programming. Indian J. Pure Appl. Math. 2004; 35: 699-708.

[13] Zalmai GJ. General parametric sufficient optimality conditions for discrete minmax fractional programming problems containing generalized (Eq.) -V-invex functions and arbitrary norms. Journal of Applied Mathematics and Computing 2007; 23(1-2): 1-23. http://dx.doi.org/10.1007/BF02831955

[14] Zalmai GJ. Hanson-Antczak-type generalized invex functions in semiinfinte minmax fractional programming, Part I: Sufficient optimality conditions. Communications on Applied Nonlinear Analysis 2012; 19(4): 1-36.

[15] Zalmai GJ. Hanson - Antczak - type generalized (Eq.)-V- invex functions in semiinfinite multiobjective fractional programming Part I, Sufficient efficiency conditions. Advances in Nonlinear Variational Inequalities 2013; 16(1): 91-114.

[16] Zeidler E. Nonlinear Functional Analysis and its Applications III, Springer-Verlag, New York, 1985. http://dx.doi.org/10.1007/978-1-4612-5020-3

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Author: | Verma, Ram U. |
---|---|

Publication: | Journal of Basic & Applied Sciences |

Article Type: | Report |

Date: | Dec 31, 2016 |

Words: | 1521 |

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