Solving Constrained Optimization Problems with a Self-Adaptive Differential Evolution Algorithm

Chukiat Worasucheep

Applied Computer Science, Department of Mathematics Faculty of Science, King Mongkut's University of Technology Thonburi

Bangkok 10140, Thailand.

Abstract- Solving constrained optimization problems has been

challenging for many decades. Although a number of evolutionary algorithms have been proposed recently for solving these problems, most of them require a careful tuning of several essential algorithmic parameters. This paper proposes a constrained self-adaptive differential evolution algorithm, named cwDE. This algorithm is an enhancement of a previous work that requires no parameter settings at all. The constraints-handling mechanism equipped into cwDE requires no additional parameters. The algorithm is described and its performance is evaluated using three real-world constrained engineering optimization problems that are widely tested in literature. The results show that the proposed algorithm is as highly competitive as other state-of-the-art algorithms for constrained optimization.

I. INTRODUCTION A large class of engineering optimization problems is

subjected to nonlinear objective functions and (non)linear constraints. Solving such constrained optimization problems is very important in engineering fields; but it is challenging. Differential Evolution (DE) is a stochastic population-based search algorithm for the global optimization in continuous space [1]. Its simplicity and effectiveness have been successfully demonstrated in many application areas. Like all other evolutionary-type algorithms, the standard DE has a few parameters that requires hand-tuning prior to an actual optimization process for a successful run. However proper tuning of essential parameters takes time and requires user’s expertise. In addition, the standard DE lacks an explicit mechanism for handling the constraints that are often found in many real-world optimization problems. A few developments of parameterless versions of DE have been proposed by using some forms of self-adaptation of parameters [2]-[3]. Also, many DE versions have been enhanced to handle constraints [3]-[5]. However, only a few works have both the self-adaptation of parameters and the mechanism for handling constraints [6]-[7].

In this work, a variant of DE algorithm for constrained optimization problems called cwDE is proposed. It is based on a previous work of self-adaptive DE called wDE [3] with an enhancement of a mechanism for handling constraints with no additional parameters. The rest of this paper is organized as follows. Section II reviews the related works on DE algorithm, constrained optimization problems, and techniques for handling constraints in evolutionary algorithms and DE in particular. Section III describes the self-adaptation and the

constraint handling mechanism of the proposed algorithm. Section IV reports the application of the proposed algorithm for solving nonlinear constrained engineering problems that are widely-tested in literature. Section V concludes this paper.

II. LITERATURE REVIEW

A. Differential Evolution Standard Differential Evolution algorithm [1] starts with an

initial population of vectors that are randomly generated. Successive populations are generated by adding the weighted difference between two population vectors to a third vector. Then the crossover operation shuffles the competing vectors to create a trial vector. If the objective function value of the trial vector is better than that of the target vector, then the target vector is replaced with the trial vector for next generation. This process will be iterated until some stopping criteria such as a maximum number of generations or the number of objective function calls allowed.

There exist many mutation/crossover strategies in DE algorithms resulting in different variants. The performance of DE algorithm depends on a selection of strategy and three critical parameters: NP, F, and CR. The population size NP is a user-specified parameter to tackle problems with different complexity. The mutation scaling factor F controls the amplification of differential variation. The crossover rate CR controls the influence of parent in the generation of offspring. Proper setting of these parameters is crucial to the success of algorithm [8].

B. Constrained Optimization Problems The general constrained optimization problem (P) with

inequality, equality, upper bound and lower bound constraints is defined as follows [9]:

(P) minimize f ( x ); subject to gj ( x ) ≤ 0, j = 1,…,q,

hj ( x ) = 0, j = q + 1,…,m, li ≤ xi ≤ ui, i = 1,…,D,

where x = (x1, x2, ..., xD) is a D dimensional vector of decision variables, f( x ) is an objective function, gj( x ) ≤ 0 are q inequality constraints and hj( x ) are m – q equality constraints. Usually equality constraints are transformed into inequalities of the form

| hi ( x ) | - ε ≤ 0, j = q + 1,…,m (1)

978-1-4244-3388-9/09/$25.00 ©2009 IEEE

where ε is the allowed tolerance and set to a very small value. The functions f, gj, and hj are linear or nonlinear real-valued functions. The values li and ui are lower and upper bounds defining the search space S. Inequality and equality constraints define the feasible region F, and F ⊆ S. The problem P is said to be feasible if there exist at least one feasible point. If x is feasible and gj ( x ) = 0, the constraint gj ( x ) ≤ 0 is said to be active at x . Equality constraints are also active at all feasible points. By using equation (1), equality constraints can then be transformed into inequality constraints, which can be treated more easily.

There are many studies on solving constrained optimization problems using evolutionary algorithms. The constraint handling techniques can be grouped into several categories, according to how the constraints are treated as follows [9]: preserving feasibility of solutions, penalty functions, transforming into multi-objective functions, repair algorithms, and separation of objectives and constraints, etc.

There have been a few attempts to handle constraints based on Differential Evolution. Storn proposed constraint adaptation [11], in which all the constraints are relaxed so that the individuals become feasible at each generation. Lampinen [12] proposed some rules for the replacement made during a selection process, in which the feasible vectors are preferable to over the infeasible vectors concerning the violation for all constraints. Huang et al. [13] recently proposed a two-population DE that decouples constraints from objective values. One population is evolved by DE according to the objectives, while the other stores feasible solutions which are used to repair some infeasible solutions in the former population. Huang, Qin, and Suganthan [6] proposed a self-adaptive DE in which the constraints mechanism prefers the feasible vectors over the infeasible ones. When comparing both infeasible vectors, the algorithm prefers the one with less overall constrain violation that is calculated from a weighted mean value of all constraints.

III. THE PROPOSED CONSTRAINED ALGORITHM This section describes the proposed cwDE (Constrained

wDE) algorithm for the constrained optimization problems. cwDE algorithm is an enhanced wDE [3] for handling the constrained optimization problems with few minor modifications particularly on the handling of boundary. Similar to the case of wDE, the learning strategies as well as parameters CR, F, and NP in cwDE are self-adapted (as will be described in subsection A). In addition, cwDE incorporates a mechanism for handling constraints (in subsection B).

A. Adaptation of Learning Strategies and DE Parameters For each individual in the current population, cwDE

probabilistically selects one out of three learning strategies. During a learning period of 30 generations, the probabilities of applying three different strategies to each individual in the current population are updated. The update is based on the rate of the trial vectors, using each strategy, that successfully pass

to the next generation. The suitable learning strategy can gradually be adapted for different evolutionary stage of the problem. Furthermore, each individual i in the population is extended with parameters CRi (randomly initialized in [0, 1]), and Fi (randomly initialized in [0, 2] for global search ability at the beginning). The adaptation of these two control parameters happens at the same period of strategy adaptation. The average values of CR and F of the vectors that successfully pass to the next generation are calculated. The algorithm then regenerates new values of CR and F for those individuals whose probability of pass to next generation is below the corresponding average value.

Unless specified by user, the initial number of individuals, NP0, is set to 5D, where D is the problem’s dimensionality. The adaptation of NP occurs less frequently than that of other control parameters. For every 200 generation period, NP is decreased by five percents. The minimum population size at max(NP0/2, 15) is maintained to allow enough individuals for the mutation. As an option, to satisfy the bound-constrained problems, any variable of the trial vectors that violates boundary constraints must be reset to a new uniformly random value. If the variable is over/under boundary, the randomized value will be in the upper/lower 5% of valid range. The rounding of discrete variables is then performed, if necessary, right after boundary checking.

B. Constraints Handling Mechanism The constraint handling mechanism based on feasibility

proposed by Deb [14] is employed in cwDE when comparing between trial vector and target vector. The rules are as follows. Between 2 feasible vectors, the one with the better fitness value wins. If one vector is feasible and the other one is infeasible, the feasible vector wins. If both vectors are infeasible, the one with the lowest sum of constraint violation is preferred. By using this mechanism, no additional parameters are introduced into the algorithm.

IV. CONSTRAINED ENGINEERING OPTIMIZATION PROBLEMS This section reports the applications of cwDE in optimizing

three well-known constrained engineering problems that are widely studied in optimization literature for performance comparisons. The 3 tested problems are as follows: welded beam design problem, pressure vessel design problem, and tension/compression spring problem. These problems have linear and non-linear constraints and have been solved by various methods and compared in [9][18]-[23]. Their definitions are briefly described as follows. 1) Welded Beam Design

A welded beam is designed for the minimum cost subject to constraints on shear stress (τ), bending stress in the beam (σ), buckling load on the bar (Pc), end deflection of the beam (δ) and side constraints [15]. This problem has 4 design variables: thickness of weld h (x1), length of weld l (x2), width of beam t (x3), and the thickness of beam b (x4). The problem can be stated as follows.

Minimize 2( , , , ) 1.10471 0.04811 (14.0 )C h l t b h l tb l= + +

Subject to 1 2

3 4

( ) 13,600 0 ( ) 30,000 0( , ) 0 ( ) 6,000 0

5( ) 0.25 0

g gg b h b h g Pc Pcg

τ τ σ σ

δ δ

= − ≥ = − ≥= − ≥ = − ≥

= − ≥

The expressions for τ, σ, Pc and δ are given by: 2 2

2 2 2

3

3

2 2

( ) ( ) /

0.25( ) ) 504000 /( )

64746.022(1 0.0282346)

2.1952 /( ) 6000 /( 2 )6000(14 0.5 )

2(0.707 ( /12 0.25( ) ))

l

l h t t b

Pc tb

t b hll

hl l h t

τ τ τ τ τ α

α σ

δ τατ

′ ′′ ′ ′′= + +

= + + =

= −

′= =+′′ =+ +

2) Pressure Vessel Design A pressure vessel is a cylindrical vessel that is capped at

both ends by the hemispherical heads. The vessel is designed to minimize total cost including the cost of material, forming and welding [16]. There are 4 design variables: thickness of the shell Ts (x1), thickness of the head Th (x2), the inner radius R (x3), and length of the cylindrical section of the vessel not including the head L (x4). Ts and Th are integer multiples of 0.0625 inches, which are the available thickness of rolled steel plates, whereas R and L are continuous. The design variables are given in inches and the weight is written as:

Minimize 2( , , , ) 0.6224 1.7781s h s h hW T T R L T T R T R= + 2 23.1661 19.84s sT L T R+ +

Subject to 1

22 3

3

4

( , ) 0.0193 0( , ) 0.00954 0

( , ) 4 / 3 1,296,000 0( ) 240 0

s s

h h

g T R T Rg T R T R

g R L R L Rg L L

π π

= − ≥= − ≥

= + − ≥= − + ≥

where 0.0625 , 5 (in constant steps of 0.0625)Ts Th≤ ≤ , 10 , and 200R L≤ ≤

3) The Tension/Compression Spring This problem aims to minimize the volume V of a

tension/compression spring subject to constraints on minimum deflection, shear stress, surge frequency, limits on outside diameter and on design variables [17]. The design variables are the mean coil diameter D, the wire diameter d, and the number of active coils N. The problem has 4 inequality constraints. The volume of the coil to be minimized is:

Minimize: 2( , , ) ( 2)V N D d N Dd= + Subject to:

3 41

2

2 3 4 2

( ) 1 /(71785 ) 0

4 1( ) 1 012566( ) 5108

g x D N d

D dDg xDd d d

= − ≤

−= + − ≤−

3 2

4

140.45( ) 1 0

( ) ( ) /1.5 1 0

dg xD N

g x D d

= − ≤

= + − ≤

where 0.05 2, 0.25 1.3, 2 15d D N≤ ≤ ≤ ≤ ≤ ≤

B. Experimental Setups In this study, the population size is set at 20, and the

maximum number of objective function calls (MAXNFC) is set at 10,000 for all three problems. In each problem, 30 runs are performed and the statistical values are calculated.

The results of the first two problems will be compared to those from ε-constrained Differential Evolution (εDE) [18], ε

constrained hybrid algorithm of PSO and GA [19], MGA (Multiobjective genetic algorithm) [20] and Co-evolutionary penalty approach [22], as shown in Tables 1 and 2.

For the last problem (Tension/Compression Spring), the results are compared with those from a hybrid PSO with a feasibility-based rule (HPSO) [21] by He and Wang, Co-evolutionary penalty approach [22] by Coello, and a hybrid Genetic Algorithm and Artificial Immune System (AIS-GA) by Bernardino et al. [23], as shown in Table 3.

The referred papers have recently reported the best results in such problems and widely considered as state of the art.

C. Results and Discussions Table 1 to 3 report the comparative results for those three

engineering problems tested. The results are as follows. 1. For the first two problems, εDE achieves the best results

with consistency (low S.D.), whereas cwDE achieves a very close performance as εDE with respect to all statistical values. Note that both algorithms use a lower number of function calls than other algorithms.

2. For Tension/Compression String problem, cwDE achieves the better results than both Co-evolutionary and AIS-GA with respect to all statistical values. HPSO outperforms cwDE only at a margin; but HPSO uses far more number of function calls than cwDE does.

The εDE by Takahama [18] enhances the robust and efficient DE with the ε-level comparison, in which the search points are compared based on the constraint violation of them. One additional parameter (the ε level) is added into the standard DE which already has 3 parameters (NP, F, and CR), while cwDE is based on a similar robust DE engine but requires no parameter settings (although setting of NP is optional to the user). HPSO by He and Wang [21] hybridize simulated annealing (SA) method into the particle swarm optimization (PSO) to avoid premature convergence and, like cwDE, uses a feasibility-based rule [14] to handle constraints. However, as reported in [21], HPSO has 9 parameters (6 from PSO and 3 from SA) that require proper tuning by the users.

V. CONCLUSIONS This paper proposes cwDE algorithm for solving

constrained optimization problems. The learning strategies and three critical parameters of differential evolution are self-adapted in cwDE. The employed constraint handling mechanism introduced no additional parameters. Therefore cwDE requires no fine-tuning of essential parameters at all. The algorithm is tested with three widely-tested real-world engineering constrained optimization problems. The results show a highly competitive performance to recent state-of-the-art constrained evolutionary algorithms.

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heuristic for global optimization over continuous spaces, J Global Optimization, vol. 11, 1997, pp. 341–359.

[2] A. K. Qin and P. N. Suganthan, Self-adaptive differential evolution algorithm for numerical optimization, Proc. of the Congress on Evolutionary Computation 2005, pp. 630–636.

TABLE 1 RESULTS OF WELDED BEAM DESIGN PROBLEM

Algorithm Max. NFC Best Average Worst S.D.

cwDE 10,000 1.724852 1.724853 1.724856 0.000001 εDE [18] 10,000 1.724852 1.724852 1.724852 0.000000 Hybrid εPSO-GA [19] 50,000 1.724852 1.725100 1.727011 0.000565 Co-evolutionary [22] 900,000 1.7483 1.772 1.7858 0.0112 MGA [20] 50,000 1.8245 1.919 1.995 0.0538

TABLE 2 RESULTS OF PRESSURE VESSEL DESIGN PROBLEM

Algorithm Max. NFC Best Average Worst S.D.

cwDE 10,000 6059.714335 6067.043323 6090.528380 14.733305εDE[18] 10,000 6059.714335 6065.876708 6090.526202 12.324747Hybrid εPSO-GA [19] 50,000 6059.714335 6112.675030 6410.086760 91.493420Co-evolutionary [22] 900,000 6288.7445 6293.8432 6308.1497 7.4133MGA [20] 50,000 6069.3267 6263.7925 6403.4500 97.9445

TABLE 3 RESULTS OF TENSION/COMPRESSION STRING PROBLEM

Algorithm Max. NFC Best Average Worst S.D.

cwDE 10,000 0.0126653 0.0127174 0.013087 8.70e-05 HPSO [21] 81,000 0.0126652 0.0127072 0.012719 1.58e-05 Co-evolutionary [22] n.a. 0.0127048 0.012756* 0.012822 n.a. AIS-GA [23] ** 36,000 0.012666 0.012974 0.013880 1.68e-04

Notes: * median value reported instead of the mean value. N.a. means ‘not available’ in the referred paper. ** MaxNFC and results are obtained from personal email communication.

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