Una metodología eficiente para manejo de restricciones en algoritmos evolutivos multi-objetivo

Mauricio Granada Echeverri¹, Jesús María López Lezama², Ruben Romero³

¹Departamento de Ingeniería Eléctrica, Universidad Tecnológica de Pereira, Vereda la Julita, Pereira, Risaralda, Colombia

²Grupo Gimel. Facultad de Ingeniería, Universidad de Antioquia, Calle 67 N^o53-108, Medellín, Colombia

³Departmento de Ingeniería Eléctrica, Feis-Unesp-Ilha Solteira-Brasil. Avenida Brasil, 56 - Centro, 15385-000, Ilha Solteira - SP, Brasil

Abstract

This paper presents a new approach for solving constraint optimization problems (COP) based on the philosophy of lexicographical goal programming. A two-phase methodology for solving COP using a multi-objective strategy is used. In the first phase, the objective function is completely disregarded and the entire search effort is directed towards finding a single feasible solution. In the second phase, the problem is treated as a bi-objective optimization problem, turning the constraint optimization into a two-objective optimization. The two resulting objectives are the original objective function and the constraint violation degree. In the first phase a methodology based on progressive hardening of soft constraints is proposed in order to find feasible solutions. The performance of the proposed methodology was tested on 11 well-known benchmark functions.

Keywords: Evolutionary algorithms, multi-objective algorithms, constraint optimization.

Resumen

Este artículo presenta un nuevo enfoque para resolver problemas de optimización restrictos (POR) basado en la filosofía de programación lexicografita de objetivos. En este caso se utiliza una metodología de dos fases usando una estrategia multi-objetivo. En la primera fase se concentra el esfuerzo en encontrar por lo menos una solución factible, descartando completamente la función objetivo. En la segunda fase se aborda el problema como bi-objetivo, convirtiendo el problema de optimización restricta a un problema de optimización irrestricto de dos objetivos. Los dos objetivos resultantes son la función objetivo original y el grado de violación de las restricciones. En la primera fase se propone una metodología basada en el endurecimiento progresivo de restricciones blandas para encontrar soluciones factibles. El desempeño de la metodología propuesta es validado a través de 11 casos de prueba bastante conocidos en la literatura especializada.

Palabras clave: Algoritmos evolutivos, algoritmos multi-objetivo, optimización restricta.

Introduction

Evolutionary algorithms (EA) have been widely used in the solution of optimization problems. These techniques, compared with the traditional nonlinear programming methods, handle a smaller amount of information (gradients, and Hessians, among others), are of easy implementation, and constitute useful tools for global search. Additionally, they have a smaller probability of converging to a local optimal solution, and are able to obtain good quality results in problems of great size [1]. Many researchers have developed a great amount of EA to solve constraint optimization problems (COP). The different methodologies found in the literature to handle with COP can be classified in four main groups: 1) methods based on penalty functions, 2) methods based on the preference of feasible solutions instead of the non feasible ones, 3) hybrid methods, and 4) methods based on multi-objective optimization. This last group is currently of great scientific interest and becomes the state-of-the-art of the constraint optimization algorithms. A detail description of these methodologies is out of the scope of this paper. For a more in dept reading, the interested reader is referred to [2, 3] and [4].

Most of the real world problems involve equality and inequality constraints. The general problem formulation with continuous parameters and constraints is defined in [5] as shown in (1):

The objective function f is defined on the search space S ⁿ, and the set F S defines the feasible region. The feasible region F is restricted by a set of m constraints (m ≥ 0) with q inequality constraints g_j (X) , and m -q equality constraints h_j (X) .

- Constraint problems can be handled in a natural fashion. That is, it is not necessary to formulate artificially penalized objective functions, and additionally penalty parameters are not needed. These parameters introduce a subjective component to the problem solution.

- In real world problems, it is unusual to find rigid or hard constraints. Therefore, a constraint violation margin (soft constraints) is permitted as long as an important improvement in the objective function is obtained.

- MOEA allow obtaining a set of solutions, denominated Pareto-Optimal-Front (POF), with the best commitments between the objective functions involved in the problem. Thus, it is possible to find solutions that violate constraints marginally. Figure 1 shows a non-dominated set of solutions in the ƒ -v space, where ƒ is the original objective function value, and v is the constraint violation index. In this scheme, one objective is the violation degree of constraints and the other is the original objective function value. The minimum feasible solution (point A), the minimum solution considering soft constraints within a violation margin ε(point B), and the original feasible solutions of the single-objective problem are also shown in figure 1. All the solutions of the POF that are between the points A and B are of great interest.

The philosophy of the proposed methodology is inspired by the goal programming methods [7], where the main idea is to find solutions that reach a reference (predefined objective) for one or more objective functions. If these solutions do not exist, the task will be to find solutions where the difference with the reference is minimum. On the other hand, if there is a solution with the same value as the reference objective function, the task will be to identify this solution. The lexicographical method is among the goal programming methods. In this case the different goals are categorized within many levels of priority. In This way, the problem is first solved considering only one goal with the corresponding constraints of the first priority level.

If there are multiple solutions in the previous step, another goal programming problem is formulated considering the second level of priority. The goals of the first level of priority are used as equality constraints to assure that the second problem solution does not violate the first level constraints. The procedure is repeated sequentially for other priority levels.

Figure 2 illustrates the operation principle of the lexicographical goal programming method for a minimization problem with two objective functions f₁ and f₂. If it is considered that f₁ is more important that f₂ , the procedure consists of first minimizing the problem only considering f₁ and ignoring f₂. In this way, the set of solutions is represented by the segments AB and CD for the first priority level. The solution of the second priority level will be the one that minimizes f₂ along the segments AB and CD. In this case, the solution of the second priority level will be D which is the global solution of the problem. If f₂ is more important than f₁ the problem solution changes to point E.

Figure 1 Search space of a two-objective problem with hard and soft constraints

Proposed methodology for COP

The proposed methodology for COP is based on the philosophy of lexicographical goal programming. It consists of turning a COP into a bi-objective problem, where one objective function is the original one ƒ(X) and the other, is the constrained violation degree v (X) . In other words, one objective function considers optimality and the other one considers feasibility. The algorithm is composed by two phases. In the first stage, the original objective function is completely discarded and the optimization problem is concentrated on minimizing the constrained violation degree of the solutions. Thus, the algorithm might find a feasible solution because the search is concentrated only on the minimization of the constrained violation degree. The second phase consists of optimizing simultaneously the original objective function and the constrained violation degree using a multi-objective strategy.

Phase I: Constraint enforcement algorithm

In this phase, the objective function is completely discarded and all the algorithm effort is directed towards finding at least one feasible solution. For each alternative i of the population, a fitness function, according to v_i(X) , is assigned. Then, an elitist strategy is used to assure that the solution with smaller v (X) is included in the following generation. This phase allows obtaining a solution that satisfies all the constraints (usable solution in the real world).

This technique is appropriate to solve highly constrained problems, where finding a feasible solution can be difficult. In order to find the constrained violation degree of an alternative X in the constraint j, the first step of the proposed strategy consists of turning the equality constraints into soft constraints using a tolerance δ. Thus, the constraint violation degree of the alternative will be given by (2)

Where | | denotes the absolute value.

In order to give the same importance degree to all constraints, each alternative violated must be normalized dividing it by the greatest violation value of the population. In this case the greatest violation value for each constraint j is calculated using (3).

Obtaining the fitness function for phase one

In order to illustrate the calculus of the fitness function, the constraint optimization problem presented in [8] and defined by the set of equations (5) is considered.

Where r₁, r₂, …, r₇ Kare the problem constraints. Table 1 shows a population of 5 alternatives randomly generated. The small letters (x₁, x₂, …, x₅) stand for the variables and the capital letters (X₁, X₂, …, X₅) stand for the solution alternatives. During the generation of the population it is guaranteed that constraints r₆ and r₇ are satisfied (limits of the decision variables). Thus, the problem is only limited by the first 5 constraints (r₁, r₂, r₅).

Evaluating each alternative of the population for each of the constraints in problem (5) and discarding the objective function completely, the data registered in table 2 are obtained. Applying (2) and assuming a tolerance δ = 0.0001 the data presented in table 3 are obtained, and the term c_max (j) is calculated using (3).

Table 1 Randomly generated population

Table 2 Violation values for each alternative and each constraint

Then, when a new alternative is generated, a comparison between the constraint violations for each alternative, and the maximum violations calculated in (3) allows keeping the values of the vector c_max(j) updated. It is advisable to generate an additional vector i_max(j) containing the index of the alternatives that produce each c_max(j). Thus, for example, the maximum violation of constraint 1 is caused by individual 5 (X₅) as shown table 3.

Table 3 Constraint violations considering δ = 0.0001

Finally, applying expression (4) to the data shown in table 3, a scalar vector v (X) that quantifies the infeasible degree of each individual of the population is obtained as shown in (6).

The vector v (X) corresponds to the fitness function of phase one, which will be used in the selection process. For the feasible solution search, a traditional genetic algorithm (GA) with real codification is used incorporating progressive hardening of soft constraints.

Progressive hardening of soft constraints (PHSC) - Phase one

Figure 1 shows the soft constraints handled through a violation margin ε. The technique used to find the feasible solutions consists of considering an interval of the violation margin (ε_min ≤ ε ≤ ε_max). In this way, the initial objective of the GA is to minimize the parameter v(X) of each alternative, calculated with expression (4) considering ε_max. The algorithm is initially run with a high violation margin, and therefore, the GA reaches its objective with a low computational effort. Next, the violation margin is reduced every time the GA reaches a partial objective, until a constraint violation margin smaller or equal to ε_min is finally reached. In this point, the GA has found a feasible solution.

The expression ε = (1 - τ) is used as a reduction strategy of the violation margin. ε_min corresponds to the tolerance used to evaluate the equality constraints fulfillment (a typical value is ε_min = δ = 0.0001). ε_max is a "bait" value that allows the GA to easily find a population with a reasonable infeasibility margin. From this population, the optimization process guides the search towards feasible regions of better quality until finding a feasible solution with the desired precision degree. Figure 3 shows the search process of feasible solutions for problem (5), starting with a random population with ε_max = 0.4 and τ = 0.5. The white circles correspond to the initial population, the asterisks indicate the evolution of the population after several generations, and the vertical dashed lines indicate the current violation margin.

Figure 3 Search process of feasible solutions. Evolution of the population alternatives for different ε

Phase II: Optimization algorithm for constraint problems

hase II is activated when at least one feasible solution has been found by phase one. In phase one the fitness function corresponds to the constraint violation degree, and the evolution of the alternatives considers the quality of each non-dominated set to which each alternative belongs. In phase II the constraint violation and the original objective function must be minimized simultaneously within a modified objective space as shown in figure 1 (space f-v). The feasible alternative with the best objective function will be the current incumbent of the space search.

A GA and an elitist operator based on a non-dominated sorting (NSGA-II [9]) are used in this paper for solving the bi-objective problem. In addition, to preserve diversity in the alternatives belonging to the non-dominated solutions set, a niches scheme is used, taking into consideration the normalized Euclidean distance between two objective vectors. This distance is known as crowding distance metric. A detailed description of the calculus of this distance is presented in [7], and [10]. The multi-objective theory introduces the concept of dominance, which defines that a solution X₁ dominates another solution X₂ if both conditions 1) and 2) are true:

1) The solution X₁ is not worse than X₂ in all objectives.

2) The solution X₁ is strictly better than X₂ in at least one objective.

If any of the above conditions is violated, the solution X₁ does not dominate the solution X₂ . It is possible to apply this definition in an iterative way to any set of solutions belonging to a multi-objective optimization problem to establish the dominant and non-dominant sets of alternatives.

The set of dominant solutions through all the objective space is called the Pareto-optimal front. Therefore, the GA (or any other evolutionary approach) aims to move the current front in each iteration towards regions of better quality. Figure 4 shows the optimal front evolution for problem (5) using NSGA-II approach.

Figure 4 Pareto-Optimal-Front and target search region

Where a_ij represents the improvement in the j-th objective function for a one-unit loss in the i-th objective function. Then, the new dominance concept is:

A solution X₁ dominates another solution X₂ if Ω_i(f(x₁)) ≤ Ω_i(f(x₂)) for all i = 1,2, …,M and the strict inequality is satisfied at least for one objective.

In this problem, there are two objective functions (M = 2). The two weighted functions are shown in (8) and (9).

Thus, the modified definition of dominance allows a larger region to become dominated by any solution than the one allowed by the traditional definition.

Besides, by choosing appropriate values of the coefficients a₁₂ and a₂₁, a section of the Pareto-optimal region can be emphasized (see Figure 5). In this paper, in order to intensify the exploration of an interesting region of the Pareto-optimal front (as shown in Figure 4) the following coefficients were used: a₁₂ = 0 and a₂₁ = 1.33.

Figure 5 The non-dominated portion of the Pareto-optimal region

Genetic Algorithm

The multi-objective technique (NSGA-II) requires the incorporation of a GA that improves the Pareto-optimal front quality during the iterative process. A GA with the following characteristics is used:

Real codification: binary chain codification is not used, which implies a modification in the recombination and mutation operators.

Linear crossover: the implemented crossover operator creates three solutions (offspring) in each generation t from two parent solutions X_i^1,t and X_i^{2, t} as shown in expressions (10), (11) and (12).

Out of these three solutions, one is eliminated by tournament.

Random mutation: the mutation scheme consists of creating a random alternative Y_i^{(1, t +1)} considering all search space Y_i^{(1, t + 1)} = r_i (X_i^(U) - X_i^(L)). Where ri is a random number between [0,1] and superscripts U and L indicate the superior and inferior search space limit, respectively.

Test cases and results

In general terms, the use of PHSC showed a better performance of the algorithm in phase I. Particularly, the G5 case is highly constrained and to find a feasible solution is a difficult task. Venkatraman reports, for the G5 case, an average number of generations of 1807.82 to find the first feasible solution on 50 runs with ε = 0.001. Using PHSC an average of 405.3 generations is obtained on 30 runs with ε = 0.0001. The use of a smaller tolerance allows obtaining a greater number of non-dominated solutions in the target search region. Another strongly constraint case is G10, for this case the average number of generations reported by Venkatraman is 99.86. However, the average number obtained applying PHSC was 38.7.

Table 4 Summary of eleven test cases. (for G2 and G3 it is assumed k =30)

The algorithm performance in phase II is similar to the one reported by Venkatraman in [11]. Nevertheless, for the G5 case the proposed method was able to find, in 30 runs, 3 solutions with a better objective function than the one reported in [11]. These solutions belong to the dominated front and have an acceptable constraint violation degree (see table 6). Comparing the best alternative reported with the 3 alternatives found, it can be noticed that the alternatives 1 and 3 exactly satisfy constraints R3 = 0 and R4 =0. All alternatives satisfy the inequality constraints R1 and R2.

Table 5 Comparison of best results. ANG = average number of generations when the first feasible solution is found.

Table 6 Variables and constraint values for the best alternatives of problem G

Conclusions

The multi-objective NSGA-II method along with the proposed methodology was implemented on 11 test cases widely studied in the specialized literature. Results showed that the proposed methodology is competitive with the state-of-the-art constraint optimization algorithms. In particular, for test case G5, three different alternatives better than the one reported in the literature were found. For the other test cases the algorithm found the best solution already reported. However, in some cases, a considerable reduction of the number of generations was achieved.

Future work will consider other recombination and mutation strategies using real codification, such as blend crossover, simulated binary crossover, simplex crossover, non-uniform mutation and polynomial mutation, among others. The use of algorithms that incorporate the lateral diversity concept, such as NSGAII-controlled, allows a search of greater quality in the target search region and can be implemented with the purpose of improving some results. This philosophy can be applied to highly constrained problems.

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(Recibido el 29 de octubre de 2008. Aceptado el 26 de mayo de 2009)

^*Autor de correspondencia: teléfono: + 57 + 4 + 219 55 55, fax: + 57 + 4 + 219 05 07, correo electrónico: lezama@udea.edu.co (J. López).