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Optimizing constrained problems through a T-Cell artificial immune system.

1 INTRODUCTION

We are interested in solving the general non linear programming problem which is defined as follows:

Find [??] = ([x.sub.1], ..., [x.sub.n]) which optimizes ([x.sub.1], ..., [x.sub.n]) subject to:

h.sub.i]([x.sub.1], ..., [x.sub.n]) = 0 i = 1, ...,l [g.sub.j]([x.sub.1], ..., [x.sub.n]) [less than or equal to] 0 j = 1, ..., p

where ([x.sub.1], ..., [x.sub.n]) is the vector of solutions (or decision variables), l is the number of equality constraints and p is the number of inequality constraints (in both cases, constraints could be linear or nonlinear).

In many real-world problems, the decision variables are subject to a set of constraints, and the search has to be bound accordingly. Constrained optimization problems are very common, for example, in engineering applications, and therefore it is important to be able to deal with them efficiently.

Many bio-inspired algorithms (particularly evolutionary algorithms) have been very successful in the solution of a wide variety of optimization problems [27]. But, when they are used to solve constrained optimization problems, they need a special method to incorporate the problem's constraints into their fitness function. Evolutionary algorithms (EAs) often use exterior penalty functions in order to do this [25]. However, penalty functions require the definition of accurate penalty factors and performance is highly dependent on them.

Recently, several researchers have proposed novel constraint-handling techniques for EAs [4, 17, 22]. These approaches have been able to outperform penalty functions and can handle all types of constraints (linear, nonlinear, equality, inequality).

The main motivation of the work presented in this paper is to explore the capabilities of a new AIS model in the context of constrained global optimization. The proposed model is based on the process that suffers the T-Cell. We also propose a dynamic tolerance factor and several mutation operators that allow us to deal with different types of constraints.

2 MAIN MODELS OF AIS

According to [11] the main models of Artificial Immune System are: Negative Selection, Clonal Selection and Immune Network Models. They are briefly described next.

Forrest et al. [23] proposed the Negative Selection model for detection of changes. This model is based on the discrimination principle that the immune system adopts to distinguish between self and nonself. This model generates random detectors and discards the detectors that are unable to recognize themselves. Thus, it maintains the detectors that identify any nonself. It performs a probabilistic detection and it is robust because it searches any foreign action instead of a particular action.

The Immune Network Model was proposed by Jerne [15], and it is a mathematical model of the immune system. In this case, the dynamics of the lymphocytes are simulated by differential equations. This model assumes that lymphocytes are an interconnected network. Several models have been derived from it [14, 1].

Clonal Selection is based on the way in which both B-cells and T-cells adapt in order to match and kill the foreign cells [11]. Clonal Selection involves: 1) the AIS' ability to adapt its B-cells to new types of antigens and 2) the affinity maturation by hypermutation. CLONALG proposed by Nunes de Castro and Von Zuben [20, 21] was originally used to solve pattern recognition and multimodal optimization problems, and there are a few extensions of this algorithm for constrained optimization. CLONALG works in the following way: first, it creates a random population of antibodies, it sorts it according to some fitness function, it clones them, it mutates each clone, it selects the fittest antibodies and clones it and replaces the worst antibodies for antibodies that are randomly generated.

3 AIS FOR CONSTRAINED PROBLEMS

Models presented in 2 have been used in several types of problems, but particularly, the use of artificial immune systems to solve constrained (numerical) optimization problems is scarce. The only previous related work that we found in the specialized literature is the following:

Hajela and Yoo [26, 27] have proposed a hybrid between a Genetic Algorithm (GA) and an AIS for solving constrained optimization problems. This approach works on two populations. The first is composed by the antigens (which are the best solutions), and the other by the antibodies (which are the worst solutions). The idea is to have a GA embedded into another GA. The outer GA performs the optimization of the original (constrained) problem. The second GA uses as its fitness function a Hamming distance so that the antibodies are evolved to become very similar to the antigens, without becoming identical. An interesting aspect of this work was that the infeasible individuals would normally become feasible as a consequence of the evolutionary process performed. This approach was tested with some structural optimization problems.

Kelsey and Timmis [16] proposed an immune inspired algorithm based on the clonal selection theory to solve multimodal optimization problems. Its highlight is the mutation operator called Somatic Contiguous Hypermutation, where mutation is applied on a subset of contiguous bits. The length and beginning of this subset is determined randomly.

Coello Coello and Cruz-Cortes [6] have proposed an extension of Hajela and Yoo's algorithm. In this proposal, no penalty function is needed, and some extra mechanisms are defined to allow the approach to work in cases in which there are no feasible solutions in the initial population. Additionally, the authors proposed a parallel version of the algorithm and validated it using some standard test functions reported in the specialized literature.

Balicki [2] made a proposal very similar to the approach of Coello Coello and Cruz-Cortes. Its main difference is the way in which the antibodies' fitness is computed. In this case, Balicki introduces a ranking procedure. This approach was validated using a constrained three-objective optimization problem.

Luh and Chueh [10, 19] have proposed an algorithm (called CMOIA, or Constrained Multi Objective Immune Algorithm) for solving constrained multiobjective optimization problems. In this case, the antibodies are the potential solutions to the problem, whereas antigens are the objective functions. CMOIA transforms the constrained problem into an unconstrained one by associating an interleukine (IL) value with all the constraints violated. IL is a function of both the number of constraints violated and the total magnitude of this constraint violation. Then, feasible individuals are rewarded and infeasible individuals are penalized. Other features of the approach were based on the clonal selection theory and other immunological mechanisms. CMOIA was evaluated using six test functions and two structural optimization problems.

Coello Coello and Cruz-Cortes [7] have proposed an algorithm based on the clonal selection theory for solving constrained optimization problems. The authors experimented with both binary and real-value representation, considering Gaussian-distributed and Cauchy-distributed mutations. Furthermore, they proposed a controlled and uniform mutation operator. This approach was tested with a set of 13 test functions taken from the specialized literature on evolutionary constrained optimization.

4 T-CELL MODEL

This paper presents a novel bio-inspired model based on the T-Cell. In a very simple way, the processes that suffer the T-Cell are the following: first, they are divided in three groups (Virgin Cell, Effector Cells and Memory Cells). Then, the natural immune system generates a huge number of virgin cells. During the immunological response, the T-cells pass through different phases: initiation, reaction and elimination. After the initiation phase, virgin cells becomes effector cells. These react (it means that the cells change in order to improve) and undergo a process called apoptosis. This process eliminates any undesirable cells. The surviving cells become memory cells.

Thus, this model operates on three populations, corresponding to the three groups in which the T-cells are divided: (1) Virgin Cells (VC), (2) Effector Cells (EC) and (3) Memory Cells (MC). Each of them has a specified function. VC has as its main goal to provide diversity. EC tries to explore the conflicting zones of the search space. MC has to explore the neighborhood of the best solutions found so far. VC and EC represent their cells with binary string using Gray coding, MC does the same, but adopting vectors of real values.

4.1 Characteristics of T-Cell

In our proposed model, the constraint-handling method needs to calculate, for each cell (solution) regardless of the population to which it belongs, the following: 1) value of each constraint function, 2) sum of violation constraints (sum-res), it is a positive value determined by the addition of [[g.sub.i].sup.+] for i = 1, ..., p and [absolute value of [h.sub.k] (x)] for k = 1, ..., l and 3) value of objective function (only if the cell is feasible).

When the search process is driven by the value of each constraint and the sum of constraint violations, then the selection mechanisms favors the feasible solutions over the infeasible ones. In this case, it is probable that, in some functions, the search falls into a local optimum. For this reason, we develop a dynamic tolerance factor (DTF). It changes with each new population, since it depends on the value of sum_res. The DTF is calculated by adding the value of each constraint violated in each cell from a particular population (VC or EC). Then, this value is divided by the number of Virgin Cells (for DTF's VC) or three times the number of Effector Cells (for DTF's EC). Thus, the DTF for VC is more flexible than DTF for EC allowing that more infeasible cells being feasible cells, in a virtual way.

When we evalue the population using the DTF, it will be easier to generate solutions that are considered "feasible" (although they may be really infeasible if evaluated with the actual precision required). This allows the exploration of each solution's neighborhood, which otherwise, would not be possible. This DTF is used by both VC and EC. If the value of DTF is lower than 0.0001, we set it to 0.1 and 0.001 for VC and EC, respectively. In contrast, MC adopts a traditional tolerance factor, which is set to 0.0001. The cells within MC need to be evaluated with the traditional tolerance factor because these are the real solutions for the problem.

In order to explore the frontier between feasible and infeasible zones, EC is divided in EC_f and EC_inf. The first is composed by feasible solutions and the other by infeasible solutions. Also, we introduce domain knowledge through the mutation operator, which modify the decision variables involve in the constraint with the highest violation.

Each population that reacts (EC_f, EC_inf and MC) has its own mutation operator. These operators are described next.

The mutation operator for EC_inf works in the following way: first, it identifies the most violated constraint, say c. If this constraint value (c) is larger than sum-res divided the total number of constraints, then we change each bit from each decision variable involve in c with a random probability between 0.01 and 0.2. Otherwise, we change each bit from one decision variable involve in c, randomly selected, with a random probability between 0.01 and 0.2. We use a random probability because after some experiments, we observed that some test functions required different step sizes. If after applying mutation, a cell becomes feasible, it is inserted in EC_f according to an elitist selection.

The mutation operator for EC-f works in the following way: it changes each bit from all decision variables, with a random probability between 0.001 and 0.2. This random probability has the same motivation that the previously.

The mutation operator for MC applies the following equation:

x' = x [+ or -] (N(0,1)lu - ll/[10.sup.m] gen[absolute value of const] [absolute value of dv])[N.sup.(0,1)] (1)

where x and x' are the original and mutated decision variables, respectively. N(0,1) refers to a random number with a uniform distribution between (0,1). lu and ll are the upper and lower limits of x. [absolute value of const] refers to the number of constraints. [absolute value of dv] refers to the number of decision variables of the problem, gen is the current generation number and m is an integer (its value is setted in Section 6).

The replace mechanisms are always applied in an elitist way, both within a population and between different populations. They take into account the value of objective function or the sum of constraint violation, depending on whether the cell is feasible or infeasible, respectively. Additionally, we always consider a feasible cell as better than an infeasible one. Note that before a cell is inserted into another population, it is first evaluated with the tolerance factor of the receptor population.

Therefore, the general structure of our proposed model for constrained problems is the following:

Repeat a predetermined number of times

1. Randomly generate Virgin Cells

2. Calculate DTF's VC

3. Evaluate VC with its own DTF

4. Insert a percentage of Virgin Cells into Effector Cells population

5. Calculate DTF's EC's

6. Repeat 50 times 6.1. Make the Effector Cells React 6.2. Evaluate EC's with its own DTF End repeat.

7. Insert a percentage of Effectors Cells into Memory Cells population

8. Repeat 100 times 8.1. Make the Memory Cells React 8.2. Evaluate MC

End repeat.

End repeat.

The most relevant aspects of our proposed model are the following:

* All equality constraints are converted into inequality constraints, [absolute value of h([??])] - [delta] [less than or equal to] 0, using a tolerance factor.

* VC's cells and MC's cell are sorted using the following criterion: the feasible cell whose objective function are the best are placed first. Then, we place the infeasible cells that have the lowest sum of constraint violation.

* EC_f's cells are sorted in ascending order on their objective function.

* EC_f's cells are sorted in ascending order on their sum of constraint violation.

5 DIFFERENCES BETWEEN MODELS

The immune system models described in 2 are based on different immunological theories. Clonal Selection is based on the replication of antibodies according to their affinity. The Immune Network Model is a probabilistic approach to idiotypic networks. Negative Selection is based on the principles of self-nonself discrimination that take place in the immune system. Aditionally, Negative Selection and T-Cell Model are both based on the mechanisms of the T-Cell. However, these models give a completely different treatment to the cells (in T-Cell Model) and detectors (in Negative Selection). The Negative Selection Model tries to detect some change, whereas T-Cell Models categorizes the T-cell and it uses their phases in order to achieve different goals.

6 EXPERIMENTAL SETUP

In order to validate our proposed model we tested it with two benchmarks, the first is composed by 19 test functions taken from the specialized literature [5] and the second one has five problems taken from [8]. The functions g02, g03, g08 and g12, from the first benchmark and problem 2 from the second one, are maximization problems (for simplicity, these problems were converted into minimization problems using - f (x)) and the rest are minimization problems.

For the first benchmark, our results are compared with respect to Stochastic Ranking, we take its result from [18], which is a constraint handling technique representative of the state-of-the-art in the area, and with respect to the AIS approach reported in [7]. For equation 1, we used m = [10.sup.7] for all functions except for g02, here we used m = [10.sup.2]. 25 independents runs were performed for each problem, each consisting of 350,000 fitness function evaluations. We experimented with different population sizes, the best results were obtained using: 1) for VC 100 cells for all functions, except for g19 here we used 10 cells and for g10 and g15 we used 20 cells, 2) for EC_f, EC_inf and MC we used 20 cells for all functions, except for g10 and g19, here we used 10 cells.

For the second benchmark, our results are compared with respect to a self-organizing migrating genetic algorithm for constrained optimization (C-SOMGA) [8]. For equation 1, we used m = [10.sup.7] for all problems. 100 independents runs were performed for each problem, each consisting of 150,000 fitness function evaluations. We experimented with different population sizes, the best results were obtained using: 1) for VC 100 cells for problem 1 [13] and problem 3 [24], for the others (problems 2 [12], 4 [3] and 5 [3]) is 10 cells, 2) for EC_f, EC_inf and MC we used 20 cells for problems 1 and 3, 10 cells for problem 2 and 50 cells for problems 4 and 5.

For both benchmarks, we adopted a 100% and 50% replacement for the cells in EC's and MC, respectively. All the statistical measures reported are taken only with respect to the runs in which a feasible solution was reached at the end.

7 DISCUSSION OF RESULTS

7.1 Results for the First Benchmark

Tables 1, 2 and 3 show the results obtained with the AIS proposed in [7], Stochastic Ranking and our T-Cell Model, respectively.

From Table 3, we can see that our model was able to reach the global optimum in 8 test functions (g01, g04, g06, g08, g11, g12, g15 and g16). Additionally, our model reached feasible solutions close to the global optimum in 7 more test functions (g02, g03, g07, g09, g13, g14 and g18) and it found acceptable (but not too close from the global optimum) feasible solutions for the rest of the test functions.

Comparing T-Cell Model with respect to Stochastic Ranking (see Tables 2 and 3), T-Cell Model obtained better results in 9 test functions (g03, g04, g06, g11, g14, g15, g16, g17 and g18). Both approaches found similar solutions for g01, g08 and g12. Our model was outperformed in 7 functions (g02, g05, g07, g09, g10, g13 and g19) . With respect to the mean and worst found solutions, our model was outperformed all functions except g03, g04, g06, g11, g14 and g16.

Comparing T-Cell Model with the AIS proposed in [7] (see Tables 1 and 3), T-Cell Model obtained better results in 8 test functions (g01, g02, g03, g05, g06, g07, g10 and g11). Both approaches found similar solutions for g04, g08 and g12. Finally, our model was outperformed in g09 and g13. With respect to the mean and worst found solutions, our model was outperformed only in g02, g07, g09 and g13.

We conducted an analysis of variance of the results obtained by our T-Cell Model and of the results obtained by Stochastic Ranking [18]. Due to, for some functions, the results do not follow a normal distribution, we used the Kruskal Wallis test [9] and then Turkey method [9]. The first test indicates if the means between the results of the algorithms had significant differences and the second one indicates in which experimental conditions the means had significant differences. Table 4 shows the values obtained for these tests. The first column represents to the function, the second column shows the values for Kruskal Wallis test (the means had significant differences if this value p is lower than 0.05), the third and fourth column indicate the lower and upper limits (if the values contained inside this interval does not contain the zero then the means had significant differences). After the analysis of Table 4, we observed that for all function the means have significant differences except for g11. Note that we do not apply these tests to g01, g08 and 12 because, for these functions both algorithms found the optimum solution in all runs.

We argue that the model is capable of performing an efficient local search over each cell, which allows the model to improve on the feasible solutions found. In cases in which no feasible solutions are found in the initial population, the mutation applied is capable of reaching the feasible region even when dealing with very small feasible search spaces.

Although there is clearly room for improving our proposed model, we have empirically shown that this approach is able of dealing with a variety of constrained optimization problems (i.e., with both linear and nonlinear constraints and objective function, and with both equality and inequality constraints). The benchmark adopted includes test functions with both small and large feasible regions, as well as a disjoint feasible region.

7.2 Results for the Second Benchmark

Tables 5 and 6 show the results obtained with our T-Cell Model and the approach in [8], respectively.

From Table 5, we can see that our model was able to reach the global optimum in four problems (problem 2, 3, 4 and 5). Additionally, our model reached feasible solutions close to the global optimum in problem 1. All runs found feasible solutions.

Comparing T-Cell Model with respect to CSOGMA (see Tables 5 and 6), T-Cell Model obtained better results in all problems.

8 CONCLUSIONS AND FUTURE WORK

This paper has presented a new AIS model for solving constrained optimization problems in which novel mutation operators are adopted. One of the operators incorporates knowledge of the problem, by modifying the decision variables involve in the most violated constraint. For some functions, the feasible region is very small, which makes it difficult to find good solutions. For this reason, we were motivated to develop a dynamic tolerance factor. It allows to explore regions of the search space that, otherwise, would be unreachable, if we use a tolerance factor very restrictive.

The proposed model was found to be competitive in a well-known benchmark commonly adopted in the specialized literature on constrained evolutionary optimization. The approach was also found to be robust and able to converge to feasible solutions in most cases.

Our analysis of the benchmarks adopted made us realize that these test functions require small step sizes, except for g02, due to this function has a feasible region bigger than the other functions. A lot of work remains to be done in order to improve the quality of some solutions found, so that the approach can be competitive with respect to the algorithms representative of the state-of-the-art in the area. For example, we plan to improve the mutation operators in order to find the frontier and feasible zone faster. Nevertheless, it is important to emphasize that there is very little work regarding the use of artificial immune systems for constrained numerical optimization, and in that context, this approach provides a viable alternative.

Acknowledgements

The first two authors acknowledge support from the Universidad Nacional de San Luis and the ANPCYT. The third author acknowledges support from the Consejo Nacional de Ciencia y Tecnologia (CONACyT) through project number 42435-Y.

Received: Aug. 2008. Accepted: Sep. 2008.

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Victoria S. Aragon, Susana C. Esquivel

Laboratorio de Investigacion y Desarrollo en Inteligencia Computacional *

Universidad Nacional de San Luis

Ejercito de los Andes 950

(5700) San Luis, Argentina

{vsaragon, esquivel}@Quns1.edu.ar

Carlos A. Coello Coello

CINVESTAV-IPN (Evolutionary Computation Group)

Electrical Eng. Department, Computer Science Dept.

Av. IPN No. 2508, Col. San Pedro Zacatenco

Mexico D.F. 07300, MEXICO

ccoello@cs.cinvestav.mx

* LIDIC is financed by Universidad Nacional de San Luis and ANPCyT (Agencia Nacional para promover la Ciencia y Tecnologia).
Table 1: Results obtained with AIS proposed
in [7]. The asterisk (*) indicates a
case in which only 90% of the runs converged
to a feasible solution

Fun Opt Best Mean Std. Dev

g01 -15 -14.9874 -14.7264 0.6070
g02 -0.803619 -0.8017 -0.7434 0.0414
g03 -1.0005 -1.0 -1.0 0.0000
g04 -30665.5386 -30665.5387 -30665.5386 0.0000
g05 * 5126.4967 5126.999 5436.1278 300.88
g06 -6961.81387 -6961.8105 -6961.8065 0.0027
g07 24.306 24.5059 25.4167 0.4637
g08 -0.095825 -0.095825 -0.095825 0.0000
g09 680.63 680.6309 680.6521 0.0176
g10 7049.24 7127.9502 8453.7902 1231.37
g11 0.7499 0.75 0.75 0.0000
g12 -1.0 -1.0 -1.0 0.0000
g13 0.05395 0.05466 0.45782 0.3790

Table 2: Results obtained with Stochastic
Ranking [18]

Fun Opt Best Mean

g01 -15 -15.0 -15.0
g02 -0.803619 -0.803 -0.784
g03 -1.0005 -1.0 -1.0
g04 -30665.539 -30665.539 -30665.480
g05 5126.4967 5126.497 5130.752
g06 -6961.81387 -6961.814 -6863.645
g07 24.306 24.310 24.417
g08 -0.095825 -0.095825 -0.095825
g09 680.63 680.63 680.646
g10 7049.24 7050.194 7423.434
g11 0.7499 0.750 0.750
g12 -1.0 -1.0 -1.0
g13 0.05395 0.053 0.061
g14 -47.7648 -41.551 -41.551
g15 961.71502 961.715 961.731
g16 -1.905155 -1.905 -1.703
g17 8853.539 8811.692 8805.99
g18 -0.86602 -0.866 -0.786
g19 32.655 33.147 34.337

Table 3: Results obtained with our proposed
T-Cell Model. The asterisk (*) indicates
a case in which only 96% of the runs
converged to a feasible solution

Fun Opt Best Mean Std. Dev

g01 -15.0 -15.0 -15.0 0.0
g02 -0.803619 -0.802914 -0.546031 0.1683
g03 -1.0005 -1.000499 -1.000499 0.00
g04 -30665.5386 -30665.5386 -30665.5386 0.0
g05 * 5126.4967 5126.6595 5307.1073 230.24
g06 -6961.81387 -6961.81387 -6961.81387 0.0
g07 24.306 24.3118 25.8927 1.1297
g08 -0.095825 -0.095825 -0.095825 0.0
g09 680.63 680.6312 680.6730 0.0305
g10 7049.24 7061.67 7451.88 218.39
g11 0.7499 0.7499 0.7499 0.0
g12 -1.0 -1.0 -1.0 0.0
g13 0.05395 0.054879 0.64231 0.5346
g14 -47.7648 -46.2546 -43.6876 1.5383
g15 961.71502 961.71502 965.02171 3.10
g16 -1.905155 -1.905155 -1.905155 0.0
g17 8853.539 8862.383 8984.399 117.59
g18 -0.86602 -0.866019 -0.78805 0.0928
g19 32.655 34.649 52.617 10.100

Table 4: Analysis of Variance

Function p lower limit upper limit

g02 2.54392e-09 16.0252 31.7348
g03 4.53296e-11 -35.0356 -18.9644
g04 8.98673e-11 17.4421 32.5579
g05 2.17934e-09 16.2556 32.0911
g06 9.06124e-11 17.4406 32.5594
g07 2.93747e-09 15.4582 30.7018
g09 1.08889e-08 14.9037 30.4563
g10 1.74435e-08 14.5837 30.1363
g11 0.1298 -1.7753 13.8553
g13 3.35698e-10 17.1443 32.6957
g14 0.0009 -21.3391 -5.4609
g15 2.01142e-08 14.966 31.0340
g16 3.97653e-11 -32.4185 -17.5815
g17 3.60989e-10 17.1853 32.8147
g18 7.25903e-10 16.6646 32.2157
g19 3.15542e-10 16.867 32.1330

Table 5: Results obtained with our proposed
T-Cell Model for Problems 1 to 5

Prob Opt Best Mean Std.Dev

1 0.0156 0.01561957 0.015620 0.0
2 310.0 310.0 310.0 0.0
3 13.59085 13.59085 13.59085 0.0
4 0.75 0.75 0.75 0.0
5 1.0 1.0 0.999985 0.000645

Table 6: Mean and standard deviation of
objective function value obtained by C-SOMGA
for Problems 1 to 5

Prob Opt Mean Std. Dev

1 0.0156 0.01531 0.00026
2 310.0 309.15 3.08
3 13.59085 13.59610 0.00315
4 0.75 0.825 0.097
5 1.0 0.887 0.212
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Author:Aragon, Victoria S.; Esquivel, Susana C.; Coello, Carlos A. Coello
Publication:Journal of Computer Science & Technology
Geographic Code:1USA
Date:Oct 1, 2008
Words:5370
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