Feasibility Preserving ConstraintHandling Strategies for Real Parameter Evolutionary Optimization
Abstract
Evolutionary Algorithms (EAs) are being routinely applied for a variety of optimization tasks, and realparameter optimization in the presence of constraints is one such important area. During constrained optimization EAs often create solutions that fall outside the feasible region; hence a viable constrainthandling strategy is needed. This paper focuses on the class of constrainthandling strategies that repair infeasible solutions by bringing them back into the search space and explicitly preserve feasibility of the solutions. Several existing constrainthandling strategies are studied, and two new single parameter constrainthandling methodologies based on parentcentric and inverse parabolic probability (IP) distribution are proposed. The existing and newly proposed constrainthandling methods are first studied with PSO, DE, GAs, and simulation results on four scalable testproblems under different location settings of the optimum are presented. The newly proposed constrainthandling methods exhibit robustness in terms of performance and also succeed on search spaces comprising upto variables while locating the optimum within an error of . The working principle of the IP based methods is also demonstrated on (i) some generic constrained optimization problems, and (ii) a classic ‘Weld’ problem from structural design and mechanics. The successful performance of the proposed methods clearly exhibits their efficacy as a generic constrainedhandling strategy for a wide range of applications.
Keywords constrainthandling, nonlinear and constrained optimization, particle swarm optimization, realparameter genetic algorithms, differential evolution.
1 Introduction
Optimization problems are widespread in several domains of science and engineering. The usual goal is to minimize or maximize some predefined objective(s). Most of the realworld scenarios place certain restrictions on the variables of the problem i.e. the variables need to satisfy certain predefined constraints to realize an acceptable solution.
The most general form of a constrained optimization problem (with inequality constraints, equality constraints and variable bounds) can be written as a nonlinear programming (NLP) problem:
(1)  
The NLP problem defined above contains decision variables (i.e. is a vector of size ), greaterthanequalto type inequality constraints (lessthanequalto can be expressed in this form by multiplying both sides by ), and equalitytype constraints. The problem variables are bounded by the lower () and upper () limits. When only the variable bounds are specified then the constrainthandling strategies are often termed as the boundaryhandling methods. ^{1}^{1}1For the rest of the paper, by constrainthandling we imply tackling all of the following: variable bounds, inequality constraints and equality constraints. And, by a feasible solution it is implied that the solution satisfies all the variable bounds, inequality constraints, and equality constraints. The main contribution of the paper is to propose an efficient constrainthandling method that operates and generates only feasible solutions during optimization.
In classical optimization, the task of constrainthandling has been addressed in a variety of ways: (i) using penalty approach developed by Fiacoo and McCormick [13], which degrades the function value in the regions outside the feasible domain, (ii) using barrier methods which operate in a similar fashion but strongly degrade the function values as the solution approaches a constraint boundary from inside the feasible space, (iii) performing search in the feasible directions using methods such gradient projection, reduced gradient and Zoutendijk’s approach [28] (iv) using the augmented Lagrangian formulation of the problem, as commonly done in linear programming and sequential quadratic programming (SQP). For a detailed account on these methods along with their implementation and convergence characteristics the reader is referred to [22, 5, 14]. The classical optimization methods reliably and effectively solve convex constrained optimization problems while ensuring convergence and therefore widely used in such scenarios. However, same is not true in the presence of nonconvexity. The goal of this paper is to address the issue of constrainthandling for evolutionary algorithms in realparameter optimization, without any limitations to convexity or a special form of constraints or objective functions.
In context to the evolutionary algorithms the constrainthandling has been addressed by a variety of methods; including borrowing of the ideas from the classical techniques. These include (i) use of penalty functions to degrade the fitness values of infeasible solutions such that the degraded solutions are given less emphasis during the evolutionary search. A common challenge in employing such penalty methods arises from choosing an appropriate penalty parameter () that strikes the right balance between the objective function value, the amount of constraint violation and the associated penalty. Usually, in EA studies, a trialanderror method is employed to estimate . A study [7] in 2000 suggested a parameterless approach of implementing the penalty function concepts for populationbased optimization method. A recent biobjective method [4] was reported to find the appropriate values adaptively during the optimization process. Other studies [27, 26] have employed the concepts of multiobjective optimization by simultaneously considering the minimization of the constraint violation and optimization of the objective function, (ii) use of feasibility preserving operators, for example, in [15] specialized operators in the presence of linear constraints were proposed to create new and feasibleonly individuals from the feasible parents. In another example, generation of feasible child solutions within the variable bounds was achieved through Simulated Binary Crossover (SBX) [6] and polynomial mutation operators [8]. The explicit feasibility of child solutions was ensured by redistributing the probability distribution function in such a manner that the infeasible regions were assigned a zero probability for childcreation [7]. Although explicit creation of feasibleonly solutions during an EA search is an attractive proposition, but it may not be possible always since generic crossover or mutation operators or other standard EAs do not gaurantee creation of feasibleonly solutions, (iii) deployment of repair strategies that bring an infeasible solution back into the feasible domain. Recent studies [19, 11, 10, 3] investigated the issue of constrainthandling through repair techniques in context to PSO and DE, and showed that the repair mechanisms can introduce a bias in the search and hinder exploration. Several repair methods proposed in context PSO [16, 12, 1] exploit the information about location of the optimum and fail to perform when the location of optimum changes [17]. These issues are universal and often encountered with all EAs (as shown in later sections). Furthermore, the choice of the evolutionary optimizer, the constrainthandling strategy, and the location of the optima with respect to the search space, all play an important role in the optimization task. To this end, authors have realized a need for a reliable and effective repairstrategy that explicitly preserves feasibility. An ideal evolutionary optimizer (evolutionary algorithm and its constrainedhandling strategy) should be robust in terms of finding the optimum, irrespective of the location of the optimal location in the search space. In rest of the paper, the term constrainthandling strategy refers to explicit feasibility preserving repair techniques.
First we review the existing constrainthandling strategies and then propose two new constrainthandling schemes, namely, Inverse Parabolic Methods (IPMs). Several existing and newly proposed constrainedhandling strategies are first tested on a class of benchmark unimodal problems with variable bound constraints. Studying the performance of constrainthandling strategies on problems with variable bounds allows us to gain better understanding into the operating principles in a simplistic manner. Particle Swarm Optimization, Differential Evolution and realcoded Genetic Algorithms are chosen as evolutionary optimizers to study the performance of different constrainthandling strategies. By choosing different evolutionary optimizers, better understanding on the functioning of constrainthandlers embedded in the evolutionary framework can be gained. Both, the search algorithm and constrainthandling strategy must operate efficiently and synergistically in order to successfully carry out the optimization task. It is shown that the constrainthandling methods possessing inherent predisposition; in terms of bringing infeasible solutions back into the specific regions of the feasible domain, perform poorly. Deterministic constrainthandling strategies such as those setting the solutions on the constraint boundaries result in the loss of population diversity. On the other hand, random methods of bringing the solutions back into the search space arbitrarily; lead to complete loss of all useful information carried by the solutions. A balanced approach that utilizes the useful information from the solutions and brings them back into the search space in a meaningful way is desired. The newly proposed IPMs are motivated by these considerations. The stochastic and adaptive components of IPMs (utilizing the information of the solution’s feasible and infeasible locations), and a userdefined parameter () render them quite effective.
The rest of the paper is organized as follows: Section 2 reviews existing constrainthandling techniques commonly employed for problems with variable bounds. Section 3 provides a detailed description on two newly proposed IPMs. Section 4 provides a description on the benchmark test problems and several simulations performed on PSO, GAs and DE with different constrainthandling techniques. Section 5 considers optimization problems with larger number of variables. Section 6 shows the extension and applicability of proposed IPMs for generic constrained problems. Finally, conclusions and scope for future work are discussed in Section 7.
2 Feasibility Preserving ConstraintHandling Approaches for Optimization Problems with Variable Bounds
Several constrainthandling strategies have been proposed to bring solutions back into the feasible region when constraints manifest as variable bounds. Some of these strategies can also be extended in presence of general constraints. An exhaustive recollection and comparison of all the constrainthandling techniques is beyond the scope of this study. Rather, we focus our discussions on the popular and representative constrainthandling techniques.
The existing constrainthandling methods for problems with variable bounds can be broadly categorized into two groups: Group techniques that perform feasibility check variable wise, and Group techniques that perform feasibility check vectorwise. According to Group techniques, for every solution, each variable is tested for its feasibility with respect to its supplied bounds and made feasible if the corresponding bound is violated. Here, only the variables violating their corresponding bounds are altered, independently, and other variables are kept unchanged. According to Group techniques, if a solution (represented as a vector) is found to violate any of the variable bounds, it is brought back into the search space along a vector direction into the feasible space. In such cases, the variables that explicitly do not violate their own bounds may also get modified.
It is speculated that for variablewise separable problems, that is, problems where variables are not linked to one another, techniques belonging to Group are likely to perform well. However, for the problems with high correlation amongst the variables (usually referred to as linkedproblems), Group techniques are likely to be more useful. Next, we provide description of these constrainthandling methods in detail ^{2}^{2}2The implementation of several strategies as C codes can be obtained by emailing npdhye@gmail.com or pulkitm.iitk@gmail.com.
2.1 Random Approach
This is one of the simplest and commonly used approaches for handling boundary violations in EAs [3]. This approach belongs to Group . Each variable is checked for a boundary violation and if the variable bound is violated by the current position, say , then is replaced with a randomly chosen value in the range , as follows:
(2) 
Figure 1 illustrates this approach. Due to the random choice of the feasible location, this approach explicitly maintains diversity in the EA population.
2.2 Periodic Approach
This strategy assumes a periodic repetition of the objective function and constraints with a period . This is carried out by mapping a violated variable in the range to , as follows:
(3) 
In the above equation, % refers to the modulo operator. Figure 2 describes the periodic approach. The above operation brings back an infeasible solution in a structured manner to the feasible region. In contrast to the random method, the periodic approach is too methodical and it is unclear whether such a repair mechanism is supportive of preserving any meaningful information of the solutions that have created the infeasible solution. This approach belongs to Group .
2.3 SetOnBoundary Approach
As the name suggests, according to this strategy a violated variable is reset on the bound of the variable which it violates.
(4) 
Clearly this approach forces all violated solutions to lie on the lower or on the upper boundaries, as the case may be. Intuitively, this approach will work well on the problems when the optimum of the problem lies exactly on one of the variable boundaries. This approach belongs to Group .
2.4 Exponentially Confined (ExpC) Approach
This method was proposed in [1]. According to this approach, a particle is brought back inside the feasible search space variablewise in the region between its old position and the violated bound. The new location is created in such a manner that higher sampling probabilities are assigned to the regions near the violated boundary. The developers suggested the use of an exponential probability distribution, shown in Figure 4. The motivation of this approach is based on the hypothesis that a newly created infeasible point violates a particular variable boundary because the optimum solution lies closer to that variable boundary. Thus, this method will probabilistically create more solutions closer to the boundaries, unless the optimum lies well inside the restricted search space. This approach belongs to Group .
Assuming that the exponential distribution is , the value of can be obtained by integrating the probability from to (where or , as the case may be). Thus, the probability distribution is given as . For any random number within , the feasible solution is calculated as follows:
(5) 
2.5 Exponential Spread (ExpS) Approach
This is a variation of the above approach, in which, instead of confining the probability to lie between and the violated boundary, the exponential probability is spread over the entire feasible region, that is, the probability is distributed from lower boundary to the upper boundary with an increasing probability towards the violated boundary. This requires replacing with (when the lower boundary is violated) or (when the upper boundary is violated) in the Equation 5 as follows:
(6) 
The probability distribution is shown in Figure 4. This approach also belongs to Group .
2.6 Shrink Approach
This is a vectorwise approach and belongs to Group in which the violated solution is set on the intersection point of the line joining the parent point (), child point (, and the violated boundary. Mathematically, the mapped vector is created as follows:
(7) 
where is computed as the minimum of all positive values of intercept
for a violated boundary
and for a violated boundary
.
This operation is shown in Figure 5. In the case shown,
needs to be
computed for variable bound only.
3 Proposed Inverse Parabolic (IP) ConstraintHandling Methods
The exponential probability distribution function described in the previous section brings violated solutions back into the allowed range variablewise, but ignores the distance of the violated solution with respect to the violated boundary. The distance from the violated boundary carries useful information for remapping the violated solution into the feasible region. One way to utilize this distance information is to bring solutions back into the allowed range with a higher probability closer to the boundary, when the fallenout distance (, as shown in Figure 6) is small. In situations, when points are too far outside the allowable range, that is, the fallenout distance is large, particles are brought back more uniformly inside the feasible range. Importantly, when the fallenout distance is small (meaning that the violated child solution is close to the variable boundary), the repaired point is also close to the violated boundary but in the feasible side. Therefore, the nature of the exponential distribution should become more and more like a uniform distribution as the fallenout distance becomes large.
Let us consider Figure 6 which shows a violated solution and its parent solution . Let denote the distance between the violated solution and the parent solution. Let and be the intersection points of the line joining and with the violated boundary and the nonviolated boundary, respectively. The corresponding distances of these two points from are and , respectively. Clearly, the violated distance is . We now define an inverse parabolic probability distribution function from along the direction as:
(8) 
where is the upper bound of allowed by the constrainthandling scheme (we define later) and is a predefined parameter. By calculating and equating the cumulative probability equal to one, we find:
The probability is maximum at (at the violated boundary) and reduces as the solution enters the allowable range. Although this characteristic was also present in the exponential distribution, the above probability distribution is also a function of violated distance , which acts like a variance to the probability distribution. If is small, then the variance of the distribution is small, thereby resulting in a localized effect of creating a mapped solution. For a random number , the distance of the mapped solution from in the allowable range is given as follows:
(9) 
The corresponding mapped solution is as follows:
(10) 
Note that the IP method makes a vectorwise operation and is sensitive to the relative locations of the infeasible solution, the parent solution, and the violated boundary.
The parameter has a direct external effect of inducing small or large variance to the above probability distribution. If is large, the variance is large, thereby having uniformlike distribution. Later we shall study the effect of the parameter . A value 1.2 is found to work well in most of the problems and is recommended. Next, we describe two particular constrainthandling schemes employing this probability distribution.
3.1 Inverse Parabolic Confined (IPC) Method
In this approach, the probability distribution is confined between , thereby making . Here, a mapped solution lies strictly between violated boundary location () and the parent ().
3.2 Inverse Parabolic Spread (IPS) Method
4 Results and Discussions
In this study, first we choose four standard scalable unimodal test functions (in presence of variable bounds): Ellipsoidal (), Schwefel (), Ackley (), and Rosenbrock (), described as follows:
(11)  
(12)  
(13)  
(14) 
In the unconstrained space, , and have a minimum at , whereas has a minimum at . All functions have minimum value . is the only variable separable problem. is a challenging test problem that has a ridge which poses difficulty for several optimizers. In all the cases the number of variables is chosen to be .
For each test problem three different scenarios corresponding to the relative location of the optimum with respect to the allowable search range are considered. This is done by selecting different variable bounds, as follows:
 On the Boundary:

Optimum is exactly on one of the variable boundaries (for , and , and for , ),
 At the Center:

Optimum is at the center of the allowable range (for , and , and for , ), and
 Close to Boundary:

Optimum is near the variable boundary, but not exactly on the boundary (for , and , and for , ).
These three scenarios are shown in the Figure 7 for a twovariable problem having variable bounds: and . Although in practice, the optimum can lie anywhere in the allowable range, the above three scenarios pose adequate representation of different possibilities that may exist in practice.
For each test problem, the population is initialized uniformly in the allowable range. We count the number of function evaluations needed for the algorithm to find a solution close to the known optimum solution and we call this our evaluation criterion . Choosing a high accuracy (i.e. small value of ) as the termination criteria minimizes the chances of locating the optimum due to random effects, and provides a better insight into the behavior of a constrainthandling mechanism.
To eliminate the random effects and gather results of statistical importance, each algorithm is tested on a problem times (each run starting with a different initial population). A particular run is terminated if the evaluation criterion is met (noted as a successful run), or the number of function evaluations exceeds one million (noted as an unsuccessful run). If only a few out of the runs are successful, then we report the number of successful runs in the bracket. In this case, the best, median and worst number of function evaluations are computed from the successful runs only. If none of the runs are successful, we denote this by marking (DNC) (Did Not Converge). In such cases, we report the best, median and worst attained function values of the best solution at the end of each run. To distinguish the unsuccessful results from successful ones, we present the fitness value information of the unsuccessful runs in italics.
An indepth study on the constrainthandling techniques is carried out in this paper. Different locations of the optimum are selected and systematic comparisons are carried out for PSO, DE and GAs in Sections 4.1, 4.3 and 4.4 , respectively.
4.1 Results with Particle Swarm Optimization (PSO)
In PSO, decision variable and the velocity terms are updated independently. Let us say, that the initial position is , the newly created position is infeasible and represented by , and the repaired solution is denoted by .
If the velocity update is based on the infeasible solution as:
(15) 
then, we refer to this as “Velocity Unchanged”. However, if the velocity update is based on the repaired location as:
(16) 
then, we refer to this as “Velocity Recomputed”. This terminology is used for rest of the paper. For inverse parabolic (IP) and exponential (Exp) approaches, we use “Velocity Recomputed” strategy only. We have performed “Velocity Unchanged” strategy with IP and exponential approaches, but the results were not as good as compared to “Velocity Recomputed” strategy. For the SetOnBoundary approach, we use the “Velocity Recomputed” strategy and two other strategies discussed as follows.
Another strategy named “Velocity Reflection” is used, which simply implies that if a particle is set on the th boundary, then is changed to . The goal of the velocity reflection is to explicitly allow particles to move back into the search space. In the “Velocity Set to Zero” strategy, if a particle is set on the th boundary, then the corresponding velocity component is set to zero i.e. . For the shrink approach, both “Velocity Recomputed” and “Velocity Set to Zero” strategies are used.
For PSO, a recently proposed hyperbolic [11] constrainthandling approach is also included in this study. This strategy operates by first calculating velocity according to the standard mechanism 15, and in the case of violation a linear normalization is performed on the velocity to restrict the solution from jumping out of the constrained boundary as follows:
(17) 
Essentially, the closer the particle gets to the boundary (e.g., only slightly smaller than ), the more difficult it becomes to reach the boundary. In fact, the particle is never completely allowed to reach the boundary as the velocity tends to zero. We emphasize again that this strategy is only applicable to PSO. A standard PSO is employed in this study with a population size of 100. The results for all the above scenarios with PSO are presented in Tables 1 to 4.
Strategy  Velocity  Best  Median  Worst 

in [0,10]: On the Boundary  
IP Spread  Recomputed  67,200  257,800  970,400 
IP Confined  Recomputed  112,400 (6)  126,500  145,900 
Exp. Spread  Recomputed  3.79e+00 (DNC)  8.37e+00  1.49e+01 
Exp. Confined  Recomputed  4,900  6,100  13,500 
Periodic  Recomputed  4.85e+03 (DNC)  7.82e+03  1.34e+04 
Periodic  Unchanged  7.69e+03 (DNC)  1.11e+04  1.51e+04 
Random  Recomputed  2.61e+02 (DNC)  5.44e+02  1.05e+03 
Random  Unchanged  5.30e+03 (DNC)  7.60e+03  1.22e+04 
SetOnBoundary  Recomputed  800 (30)  1,100  3,900 
SetOnBoundary  Reflected  171,500  241,700  434,200 
SetOnBoundary  Set to Zero  1,000 (40)  1,600  5,300 
Shrink  Recomputed  6,900  9,100  11,600 
Shrink  Set to Zero  17,900  31,900  49,800 
Hyperbolic  Modified (Eq. (17))  36,400  41,700  48,700 
in [10,10]: At the Center  
IP Spread  Recomputed  106,700  127,500  144,300 
IP Confined  Recomputed  111,500  130,100  149,900 
Exp. Spread  Recomputed  112,300  131,400  149,000 
Exp. Confined  Recomputed  116,400  131,300  148,200 
Periodic  Recomputed  113,400  130,900  150,600 
Periodic  Unchanged  121,200  137,800  159,100 
Random  Recomputed  112,900  129,800  151,100 
Random  Unchanged  117,000  130,600  148,100 
SetOnBoundary  Recomputed  118,500 (49)  132,300  161,100 
SetOnBoundary  Reflected  3.30e06 (DNC)  8.32e+01(DNC)  2.95e+02 (DNC) 
SetOnBoundary  Set to Zero  111,900  132,200  149,700 
Shrink.  Recomputed  111,800 (49)  131,800  183,500 
Shrink.  Set to Zero  108,400  125,100  143,600 
Hyperbolic  Modified (Eq. (17))  101,300  117,700  129,700 
in [1,10]: Close to Boundary  
IP Spread  Recomputed  107,200  130,400  272,400 
IP Confined  Recomputed  120,100 (44)  171,200  301,200 
Exp. Spread  Recomputed  92,800  109,200  126,400 
Exp. Confined  Recomputed  110,200  127,400  256,100 
Periodic  Recomputed  8.09e+02 (DNC)  2.01e+03 (DNC)  5.53e+03(DNC) 
Periodic  Unchanged  2.16e+03 (DNC)  4.36e+03 (DNC)  6.87e+03 (DNC) 
Random  Recomputed  123,300  165,600  280,000 
Random  Unchanged  8.17e+02 (DNC)  1.96e+03 (DNC)  2.68e+03 (DNC) 
SetOnBoundary  Recomputed  2.50e+00 (DNC)  1.25e+01 (DNC)  5.75e+02 (DNC) 
SetOnBoundary  Reflected  1.86e+00 (DNC)  7.76e+00 (DNC)  5.18e+01 (DNC) 
SetOnBoundary  Set to Zero  1.00e+00 (DNC)  5.00e+00 (DNC)  4.21e+02 (DNC) 
Shrink  Recomputed  5.00e01 (DNC)  3.00e+00 (DNC)  1.60e+01 (DNC) 
Shrink  Set to Zero  108,300 (8)  130,300  143,000 
Hyperbolic  Modified (Eq. (17))  93,100  108,300  119,000 
Strategy  Best  Median  Worst  

in [0,10]: On the Boundary  
IP Spread  Recomputed  150,600 (49)  220,900  328,000 
IP Confined  Recomputed  4.17e+00 (DNC)  6.53e+00  8.79e+00 
Exp. Spread  Recomputed  2.76e01 (DNC)  9.62e01  2.50e+00 
Exp. Confined  Recomputed  7,800  9,600  11,100 
Periodic  Recomputed  6.17e+00 (DNC)  6.89e+00  9.22e+00 
Periodic  Unchanged  8.23e+00 (DNC)  9.10e+00  9.68e+00 
Random  Recomputed  3.29e+00 (DNC)  3.40e+00  4.19e+00 
Random  Unchanged  6.70e+00 (DNC)  7.46e+00  8.57e+00 
SetOnBoundary  Recomputed  800  1,100  2,100 
SetOnBoundary  Reflected  420,600  598,600  917,400 
SetOnBoundary  Set to Zero  1,100  1,800  3,100 
Shrink.  Recomputed  33,800 (5)  263,100  690,400 
Shrink.  Set Zero  3.65e+00 (DNC)  6.28e+00  8.35e+00 
Hyperbolic  Modified (Eq. (17))  24,600 (25)  26,100  28,000 
in [10,10]: At the Center  
IP Spread  Recomputed  53,900 (46)  58,600  66,500 
IP Confined  Recomputed  54,800 (49)  59,200  64,700 
Exp. Spread  Recomputed  55,100  59,300  63,600 
Exp. Confined  Recomputed  56,800  59,600  65,000 
Periodic  Recomputed  55,700 (48)  59,900  64,700 
Periodic  Unchanged  57,900 (49)  62,100  66,700 
Random  Recomputed  55,100 (47)  59,400  65,100 
Random  Unchanged  56,300  59,700  65,500 
SetOnBoundary  Recomputed  55,100 (49)  58,900  65,400 
SetOnBoundary  Reflected  86,900 (4)  136,400  927,600 
SetOnBoundary  Set to Zero  53,900 (49)  59,600  67,700 
Shrink  Recomputed  55,800 (47)  58,700  65,800 
Shrink  Set to Zero  55,700 (49)  58,900  62,000 
Hyperbolic  Modified (Eq. (17))  52,900 (49)  56,200  64,400 
in [1,10]: Close to Boundary  
IP Spread  Recomputed  54,600 (5)  55,100  56,600 
IP Confined  Recomputed  63,200 (1)  63,200  63,200 
Exp. Spread  Recomputed  51,300  55,200  58,600 
Exp. Confined  Recomputed  1.42e+00 (DNC)  2.17e+00  2.92e+00 
Periodic  Recomputed  2.88e+00 (DNC)  4.03e+00  5.40e+00 
Periodic  Unchanged  6.61e+00 (DNC)  7.46e+00  8.37e+00 
Random  Recomputed  60,300 (45)  66,200  72,200 
Random  Unchanged  4.21e+00 (DNC)  4.93e+00  6.11e+00 
SetOnBoundary  Recomputed  2.74e+00 (DNC)  3.16e+00  3.36e+00 
SetOnBoundary  Reflected  824,700 (1)  824,700  824,700 
SetOnBoundary  Set to Zero  1.70e+00 (DNC)  2.63e+00  3.26e+00 
Shrink  Recomputed  1.45e+00 (DNC)  2.34e+00  2.73e+00 
Shrink  Set to Zero  2.01e+00 (DNC)  3.96e+00  6.76e+00 
Hyperbolic  Modified (Eq. (17))  50,000 (39)  53,500  58,100 
The extensive simulation results are summarized using the following method. For each (say ) of the 14 approaches, the corresponding number of the successful applications () are recorded. Here, an application is considered to be successful if more than 45 runs out of 50 runs are able to find the optimum within the specified accuracy. It is observed that IPS is successful in 10 out of 12 problem instances. Exponential confined approach (ExpC) is successful in 9 cases. To investigate the required number of function evaluations (FE) needed to find the optimum, by an approach (say ), we compute the average number of needed to solve a particular problem () and construct the following objective for th approach:
(18) 
where FE is the FEs needed by the th approach to solve the th problem. Figure 8 shows the performance of each (th) of the 14 approaches on the twoaxes plot ( and ).
The best approaches should have large values and small values. This results in a tradeoff between three best approaches which are marked in filled circles. All other 11 approaches are dominated by these three approaches. The SetBound (SetOnBoundary) with velocity set to zero performs in only six out of 12 problem instances. Thus, we ignore this approach. There is a clear tradeoff between IPS and ExpC approaches. IPS solves one problem more, but requires more FEs in comparison to ExpC. Hence, we recommend the use of both of these methods visavis all other methods used in this study.
Other conclusions of this extensive study of PSO with different constrainthandling methods are summarized as follows:

The constrainthandling methods show a large variation in the performance depending on the choice of test problem and location of the optimum in the allowable variable range.

When the optimum is on the variable boundary, periodic and random allocation methods perform poorly. This is expected intuitively.

When the optimum is on the variable boundary, methods that set infeasible solutions on the violated boundary (SetOnBoundary methods) work very well for obvious reasons, but these methods do not perform well for other cases.

When the optimum lies near the center of the allowable range, most constrainthandling approaches work almost equally well. This can be understood intuitively from the fact that tendency of particles to fly out of the search space is small when the optimum is in the center of the allowable range. For example, the periodic approaches fail in all the cases but are able to demonstrate some convergence characteristics for all test problems, when the optimum is at the center. When the optimum is on the boundary or close to the boundary, then the effect of the chosen constrainthandling method becomes critical.

The shrink method (with “Velocity Recomputed” and “Velocity Set Zero” strategies) succeeded in 10 of the 12 cases.
4.2 Parametric Study of
The proposed IP approaches involve a parameter affecting the variance of the probability distribution for the mapped variable. In this section, we perform a parametric study of to determine its effect on the performance of the IPS approach.
Following values are chosen: , , , and . To have an idea of the effect of , we plot the probability distribution of mapped values in the allowable range () for in Figure 9. It can be seen that for and 1,000, the distribution is almost uniform.
Figure 10 shows the effect of on problem. For the same termination criterion we find that and perform better compared to other values. With larger values of the IPS method does not even find the desired solution in all 50 runs.
4.3 Results with Differential Evolution (DE)
Differential evolution, originally proposed in [24], has gained popularity as an efficient evolutionary optimization algorithm. The developers of DE proposed a total of ten different strategies [20]. In [2] it was shown that performance of DE largely depended upon the choice of constrainthandling mechanism. We use Strategy 1 (where the offspring is created around the populationbest solution), which is most suited for solving unimodal problems [18]. A population size of was chosen with parameter values of and . Other parameters are set the same as before. We use as our termination criterion. Results are tabulated in Tables 5 to 8. Following two observations can be drawn:

For problems having optimum at one of the boundaries, SetOnBoundary approach performs the best. This is not a surprising result.

However, for problems having the optimum near the center of the allowable range, almost all eight algorithms perform in a similar manner.

For problems having their optimum close to one of the boundaries, the proposed IP and existing exponential approaches perform better than the rest of the approaches with DE.
Despite the differences, somewhat similar performances of different constrainthandling approaches with DE indicates that the DE is an efficient optimization algorithm and its performance is somewhat less dependent on the choice of constrainthandling scheme compared to the PSO algorithm.
Strategy  Best  Median  Worst 

in [0,10]: On the Boundary  
IP Spread  25,600  26,850  27,650 
IP Confined  22,400  23,550  24,200 
Exp. Spread  38,350  39,800  41,500 
Exp. Confined  19,200  20,700  21,900 
Periodic  42,400  43,700  45,050 
Random  40,650  43,050  44,250 
SetOnBoundary  2,850  3,350  3,900 
Shrink  4,050  4,900  5,850 
in [10,10]: At the Center  
IP Spread  29,950  31,200  32,500 
IP Confined  29,600  31,200  32,400 
Exp. Spread  29,950  31,300  32,400 
Exp. Confined  30,500  31,400  32,250 
Periodic  29,650  31,300  32,400 
Random  30,000  31,200  31,250 
SetOnBoundary  29,850  31,200  32,700 
Shrink  30,300  31,250  32,750 
in [1,10]: Close to Boundary  
IP Spread  28,550  29,600  30,550 
IP Confined  28,500  29,500  30,650 
Exp. Spread  28,050  28,900  29,850 
Exp. Confined  28,150  29,050  29,850 
Periodic  29,850  30,850  32,100 
Random  28,900  30,200  31,000 
SetOnBoundary  28,650  29,600  30,500 
Shrink  28,800  29,900  31,200 
Strategy  Best  Median  Worst 

in [0,10]: On the Boundary  
IP Spread  26,600  27,400  28,000 
IP Confined  22,450  23,350  24,300 
Exp. Spread  40,500  42,050  43,200 
Exp. Confined  19,650  20,350  22,050 
Periodic  44,700  46,300  48,250 
Random  43,850  45,150  47,000 
SetOnBoundary  2,100  3,100  3,750 
Shrink  3,450  4,400  5,100 
in [10,10]: At the Center  
IP Spread  258,750  281,650  296,300 
IP Confined  268,150  283,050  300,450 
Exp. Spread  266,850  283,950  304,500 
Exp. Confined  266,450  283,700  305,550 
Periodic  269,700  284,100  310,100 
Random  263,300  282,600  306,250 
SetOnBoundary  267,750  284,550  298,850 
Shrink  263,600  282,750  304,350 
in [1,10]: Close to Boundary  
IP Spread  228,950  242,300  255,700 
IP Confined  232,200  243,900  263,400 
Exp. Spread  227,550  243,000  261,950 
Exp. Confined  228,750  243,800  262,500 
Periodic  231,950  247,150  260,700 
Random  228,550  244,850  261,900 
SetOnBoundary  237,100  255,750  266,400 
Shrink  234,000  253,250  275,550 
Strategy  Best  Median  Worst 

in [0,10]: On the Boundary  
IP Spread  43,400  44,950  45,950 
IP Confined  37,300  38,700  40,350 
Exp. Spread Dist.  66,300  69,250  71,300 
Exp. Confined Dist  32,750  34,600  36,200 
Periodic  72,500  74,250  75,900 
Random  70,650  73,000  74,750 
SetOnBoundary  2,550  3,250  3,950 
Shrink  3,500  4,700  5,300 
in [10,10]: At the Center  
IP Spread  50,650  52,050  53,450 
IP Confined  51,050  52,200  53,800 
Exp. Spread  51,200  52,150  53,400 
Exp. Confined  51,100  52,300  53,850 
Periodic  51,250  52,250  53,500 
Random  50,950  52,200  53,450 
SetOnBoundary  50,950  52,300  53,450 
Shrink  50,450  52,300  53,550 
in [1,10]: Close to Boundary  
IP Spread  49,100  50,650  51,650 
IP Confined  48,650  50,400  52,100 
Exp. Spread  48,300  49,900  51,750 
Exp. Confined  48,900  50,000  51,250 
Periodic  50,400  51,950  53,300 
Random  50,250  51,200  52,150 
SetOnBoundary  49,900 (33)  51,100  53,150 
Shrink  50,200  51,400  52,750 
Strategy  Best  Median  Worst 

in [1,10]: On the Boundary  
IP Spread  38,850  62,000  89,700 
IP Confined  24,850  45,700  73,400 
Exp. Spread  57,100  86,800  118,600 
Exp. Confined  16,600  21,400  79,550 
Periodic  69,550  93,500  18,1150 
Random  65,850  92,950  157,600 
SetOnBoundary  2,950  4,700  30,450 
Shrink  5,450  8,150  55,550 
in [8,10]: At the Center  
IP Spread  133,350 (41)  887,250  995,700 
IP Confined  712,500 (44)  854,800  991,400 
Exp. Spread  390,700 (48)  866,150  998,950 
Exp. Confined  138,550 (40)  883,500  994,350 
Periodic  764,650 (39)  874,700  999,650 
Random  699,400 (49)  885,450  999,600 
SetOnBoundary  743,600 (38)  865,450  995,500 
Shrink  509,900 (40)  873,450  998,450 
in [0,10]: Close to Boundary  
IP Spread  36,850  78,700  949,700 
IP Confined  46,400 (46)  95,900  891,450 
Exp. Spread  49,550 (49)  85,900  968,200 
Exp. Confined  87,300 (43)  829,200  973,350 
Periodic  38,750  62,200  94,750 
Random  41,200  61,300  461,500 
SetOnBoundary  8.23E+00 (DNC)  1.62E+01  1.89E+01 
Shrink  252,650 (9)  837,700  985,750 
4.4 Results with RealParameter Genetic Algorithms (RGAs)
We have used two realparameter GAs in our study here:

StandardRGA using the simulated binary crossover (SBX) [9] operator and the polynomial mutation operator [8]. In this approach, variables are expressed as real numbers initialized within the allowable range of each variable. The SBX and polynomial mutation operators can create infeasible solutions. Violated boundary, if any, is handled using one of the approaches studied in this paper. Later we shall investigate a rigid boundary implementation of these operators which ensures creation of feasible solutions in every recombination and mutation operations.

ElitistRGA in which two newly created offsprings are compared against the two parents, and the best two out of these four solutions are retained as parents (thereby introducing elitism). Here, the offspring solutions are created using nonrigid versions of SBX and polynomial mutation operators. As before, we test eight different constrainthandling approaches and, later explore a rigid boundary implementation of the operators in presence of elite preservation.
Parameters for RGAs are chosen as follows: population size of 100, crossover probability =0.9, mutation probability =0.05, distribution index for crossover =2, distribution index for mutation =100. The results for the StandardRGA are shown in Tables 9 to 12 for four different test problems. Tables 13 to 16 show results using the ElitistRGA. Following key observations can be made:

For all the four test problems, StandardRGA shows convergence only in the situation when optima is on the boundary.

ElitistRGA shows good convergence on when the optimum is on the boundary and, only some convergence is noted when the optima is at the other locations. For other three problems, convergence is only obtained when optimum is present on the boundary.

Overall, the performance of EliteRGA is comparable or slightly better compared to StandardRGA.
The Did Not Converge cases can be explained on the fact that the SBX operator has the property of creating solutions around the parents; if parents are close to each other. This property is a likely cause of premature convergence as the population gets closer to the optima. Furthermore the results suggest that the elitism implemented in this study (parentchild comparison) is not quite effective.
Although RGAs are able to locate the optima, however, they are unable to finetune the optima due to undesired properties of the generation scheme. This emphasizes the fact that generation scheme is primarily responsible for creating good solutions, and the constrainthandling methods cannot act as surrogate for generating efficient solutions. Each step of the evolutionary search should be designed effectively in order to achieve overall success. On the other hand one could argue that strategies such as increasing the mutation rate (in order to promote diversity so as to avoid premature convergence) should be tried, however, creation of good and meaningful solutions in the generation scheme is rather an important and a desired fundamentalfeature.
As expected, when the optima is on the boundary SetOnBoundary finds the optima most efficiently within a minimum number of function evaluations. Like in PSO the performance of Exp. Confined is better than Exp. Spread. Periodic and Random show comparable or slightly worse performances (these mechanisms don’t have any preference of creating solutions close to the boundary and actually promote spread of the population).
Strategy  Best  Median  Worst 

in [0,10]  
IP Spread  9,200  10,500  12,900 
IP Confined  7,900  9,400  10,900 
Exp. Spread  103,100 (6)  718,900  931,200 
Exp. Confined  4,500  5,700  7,000 
Periodic  15,200 (1)  15,200  15,200 
Random  68,300 (12)  314,700  939,800 
SetOnBoundary  1,800  2,400  2,800 
Shrink  3,700  5,100  6,600 
in [10,10]  
2.60e02 (DNC)  
in [1,10]  
1.02e02 (DNC) 
Strategy  Best  Median  Worst 

in [0,10]  
IP Spread  6,800  9,800  11,800 
IP Confined  6,400  8,200  10,300 
Exp. Spread  21,200 (47)  180,000  772,200 
Exp. Confined  4,300  5,500  6,300 
Periodic  14,800 (26)  143,500  499,400 
Random  8,700 (43)  195,200  979,300 
SetOnBoundary  1,800  2,300  2,900 
Shrink.  3,600  4,600  5,500 
in [10,10]  
1.20e01 (DNC)  
in [1,10]  
8.54e02 (DNC) 
Strategy  Best  Median  Worst 

in [0,10]  
IP Spread  12,100  22,600  43,400 
IP Confined  9,800  13,200  16,400 
Exp. Spread  58,100 (29)  355,900  994,000 
Exp. Confined  6,300  9,100  11,900 
Periodic  19,600 (46)  122,300  870,200 
Random  35,700 (38)  229,200  989,500 
SetOnBoundary  1,800  2,500  3,100 
Shrink  4,200  5,700  8,600 
in [10,10]  
7.76e02(DNC)  
in [1,10] 

4.00e02 (DNC)  

Strategy  Best  Median  Worst 

in [1,10]: On the Boundary  
IP Spread  12,400 (39)  15,800  20,000 
IP Confined  9,400 (39)  11,800  13,600 
Exp. Spread  9.73e+00 (DNC)  1.83e+00  2.43e+01 
Exp. Confined  6,000  6,900  8,200 
Periodic  6.30E+01 (DNC)  4.92e+02  5.27e+04 
Random  3.97e+02 (DNC)  9.28e+02  1.50e+03 
SetOnBoundary  1,900  2,700  3,400 
Shrink  4,100  5,200  6,500 
in [8,10]  
3.64e+00 (DNC)  
in [1,10]: On the Boundary  
1.04e+01(DNC) 
Strategy  Best  Median  Worst 

in [0,10]  
IP Spread  6,600  8,000  9,600 
IP Confined  6,300  8,100  9,800 
Exp. Spread  4,800  6,900  8,300 
Exp. Confined  4,600  5,800  6,700 
Periodic  6,500  8,800  11,500 
Random  6,400  7,900  10,300 
SetOnBoundary  2,200  2,600  3,500 
Shrink  4,000  5,200  6,700 
in [10,10]  
IP Spread  980,200 (1)  980,200  980,200 
IP Confined  479,000 (1)  479,000  479,000 
Exp. Spread  2.06e01 (DNC)  4.53e01  4.86e01 
Exp. Confined  954,400 (1)  954,400  954,400 
Periodic  1.55E01 (DNC)  2.48E01  2.36E01 
Random  1.92E01 (DNC)  2.00E01  2.46E01 
SetOnBoundary  2.11E01 (DNC)  2.95E01  1.94E01 
Shrink  530,900 (3)  654,000  779,000 
in [1,10]  
IP Spread  803,400 (5)  886,100  947,600 
IP Confined  643,300 (2)  643,300  963,000 
Exp. Spread  593,300 (3)  628,900  863,500 
Exp. Confined  655,400 (3)  940,500  946,700 
Periodic  653,800 (3)  842,900  843,100 
Random  498,500 (2)  498,500  815,500 
SetOnBoundary  593,800 (5)  870,500  993,500 
Shrink  781,000 (2)  781,000  928,300 
Strategy  Best  Median  Worst 

in [0,10]  
IP Spread  5,000  6,500  7,900 
IP Confined  4,900  6,500  7,900 
Exp. Spread  4,300  5,800  7,800 
Exp. Confined  4,300  4,900  5,600 
Periodic  5,400  7,000  11,300 
Random  5,300  6,600  8,500 
SetOnBoundary  1,600  2,200  2,600 
Shrink  3,100  4,200  5,400 
in [10,10]  
8.12e05(DNC)  
in [1,10]  
1.61e01(DNC) 
Strategy  Best  Median  Worst 

in [0,10]  
IP Spread  6,300  8,700  46,500 
IP Confined  6,800  9,200  32,000 
Exp. Spread  5,600  6,800  8,700 
Exp. Confined  5,200  7,800  9,900 
Periodic  6,300  9,300  12,200 
Random  6,200  8,300  53,700 
SetOnBoundary  1,900  2,500  4,000 
Shrink  3,900  5,100  7,700 
in [10,10]  
1.03e01(DNC)  
in [1,10]  
1.15e00 (DNC) 
Strategy  Best  Median  Worst 

in [0,10]  
IP Spread  9,900 (13)  12,500  14,000 
IP Confined  10,100 (12)  12,100  14,400 
Exp. Spread  8,500 (10)  11,000  15,400 
Exp. Confined  6,600 (30)  7,800  8,900 
Periodic  9,500 (10)  13,300  16,800 
Random  14,000 (3)  15,300  16,100 
SetOnBoundary  2,300 (44)  3,200  4,500 
Shrink  4,500 (32)  6,100  8,100 
in [8,10]  
1.27e00 (DNC)  
in [1,10]: On the Boundary 

1.49e00 (DNC) 
4.4.1 RGAs with Rigid Boundary
Optimum on the boundary  

Strategy  Best  Median  Worst 
8,100  8,500  8,800  
7,800  8,100  8,300  
9,500  10,100  10,800  
10,100 (39)  10,900  143,600  
Optimum in the center  
3.88e02 (DNC)  
Optimum close to the edge of the boundary  
9.44e03 (DNC) 
We also tested RGA (standard and its elite version) with a rigid bound consideration in its operators. In this case, the probability distributions of both SBX and polynomial mutation operator are changed in a way so as to always create a feasible solution. It is found that the StandardRGA with rigid bounds shows convergence only when optimum is on the boundary (Table 17). The performance of EliteRGA with rigid bounds is slightly better (Table 18). Overall, SBX operating within the rigid bounds is found to perform slightly better compared to the RGAs employing boundaryhandling mechanisms. However, as mentioned earlier, in the scenarios where the generation scheme cannot guarantee creation of feasible only solutions there is a necessary need for constrainthandling strategies.
Strategy  Best  Median  Worst 

Optimum on the boundary  
7,300  7,900  8,400  
6,500  6,900  7,500  
9,400  10,400  12,200  
11000 (10)  12700  16400  
Optimum in the center  
1.24e01(DNC)  
Optimum close to the boundary edge  
579,800 (3)  885,900  908,600  
2.73E00 DNC  6.18E00  1.34E00  
1.75E01 DNC  8.38E01  2.93E00  
3.29E00 DNC  4.91E+00  5.44E+00 
5 HigherDimensional Problems
As the dimensionality of the search space increases it becomes difficult for a search algorithm to locate the optimum. Constrainthandling methods play even a more critical role in such cases. So far in this study, DE has been found to be the best algorithm. Next, we consider all four unimodal test problems with an increasing problem size: , , , , , and . For all problems the variable bounds were chosen such that optima occured near the center of the search space. No population scaling is used for DE. The DE parameters are chosen as and . For we were able to achieve a high degree of convergence and results are shown in Figure 11. As seen from the figure, although it is expected that the required number of function evaluations would increase with the number of variables, the increase is subquadratic. Each case is run times and the termination criteria is set as . All 20 runs are found to be successful in each case, demonstrating the robustness of the method in terms of finding the optimum with a high precision. Particularly problems with large variables, complex search spaces and highly nonlinear constraints, such a methodology should be useful in terms of applying the method to different realworld problems.
It is worth mentioning that authors independently tested other scenarios for large scale problems with corresponding optimum on the boundary, and in order to achieve convergence with IP methods we had to significantly reduce the values of . Without lowering , particularly, PSO did not show any convergence. As expected in larger dimensions the probability to sample a point closer to the boundary decreases and hence a steeper distribution is needed. However, this highlights the usefulness of the parameter to modify the behavior of the IPMs so as to yield the desired performance.
6 General Purpose ConstraintHandling
So far we have carried out simulations on problems where constraints have manifested as the variable bounds. The IP methods proposed in this paper can be easily extended and employed for solving nonlinear constrained optimization problems (inclusive of variable bounds).^{3}^{3}3By General Purpose ConstraintHandling we imply tackling of all variable bounds, inequality constraints and equality constraints.
As an example, let us consider a generic inequality constraint function: – the th constraint in a set of inequality constraints. In an optimization algorithm, every created (offspring) solution at an iteration must be checked for its feasibility. If satisfies all inequality constraints, the solution is feasible and the algorithm can proceed with the created solution. But if does not satisfy one or more of constraints, the solution is infeasible and should be repaired before proceeding further.
Let us illustrate the procedure using the inverse parabolic (IP) approach described in Section 3; though other constrainthandling methods discussed before may also be used. The IP approaches require us to locate intersection points and : two bounds in the direction of (), where is one of the parent solutions that created the offspring solution (see Figure 12). The critical intersection point can be found by finding multiple roots of the direction vector with each constraint and then choosing the smallest root.
We define a parameter as the extent of a point from , as follows:
(19) 
Substituting above expression for ^{4}^{4}4Note that here for calculating points should not be confused with parameter introduced in the proposed constrainthandling methods.in the th constraint function, we have the following rootfinding problem for the th constraint:
(20) 
Let us say the roots of the above equation are for . The above procedure can now be repeated for all inequality constraints and corresponding roots can be found one at a time. Since the extent of to reach from is given as
we are now ready to compute the two closest bounds (lower and upper bounds) on for our consideration, as follows:
(21)  
(22) 
IPS and IPC approaches presented in Section 3 now can be used to map the violated variable value into the feasible region using (the lower bound), (the upper bound) and (location of parent). It is clear that the only difficult aspect of this method is to find multiple intersection points in presence of nonlinear constraints.
For the sake of completeness, we show here that the two bounds for each variable: and used in previous sections can be also be treated uniformly using the above described approach. The two bounds can be written together as follows:
(23) 
Note that a simultaneous nonpositive value of each of the bracketed terms is not possible, thus only way to satisfy the above left side is to make each bracketed term nonnegative. The above inequality can be considered as a quadratic constraint function, instead of two independent variable bounds and treated as a single combined nonlinear constraint and by finding both roots of the resulting quadratic rootfinding equation.
Finally, the above procedure can also be extended to handle equality constraints () with a relaxation as follows: . Again, they can be combined together as follows:
(24) 
Alternatively, the above can also be written as and may be useful for nongradient based optimization methods, such as evolutionary algorithms. We now show the working of the above constraint handling procedure on a number of constrained optimization problems.
6.1 Illustrations on Nonlinear Constrained Optimization
First, we consider the three unconstrained problems used in previous sections as , but now add an inequality constraint by imposing a quadratic constraint that makes the solutions fall within a radius of oneunit from a chosen point :
(25) 
There are no explicit variable bounds in the above problem. By choosing different locations of the center of the hypersphere (), we can have different scenarios of the resulting constrained optimum. If the minimum of the objective function (without constraints) lies at the origin, then setting the unconstrained minimum is also the solution to the constrained problem, and this case is similar to the “Optimum at the Center” (but in the context of constrained optimization now). The optimal solution is at with . DE with IPS and previous parameter settings is applied to this new constrained problem, and the results from 50 different runs for this case are shown in Table 19.
Strategy  Best  Median  Worst 

22,800  23,750  24,950  
183,750  206,000  229,150  
42,800  44,250  45,500 
As a next case, we consider . The constrained minimum is now different from that of the unconstrained problem, as the original unconstrained minimum is no more feasible. This case is equivalent to “Optima on the Constraint Boundary”. Since the optimum value is not zero as before, instead of choosing a termination criterion based on value, we allocate a maximum of one million function evaluations for a run and record the obtained optimized solution. The best fitness values for as , and are shown in Table 20. For each function, we verified that the obtained optimized solution satisfies the KKT optimality conditions [22, 5] suggesting that a truly optimum solution has been found by this procedure.
640.93 0.00  8871.06 0.39  6.56 0.00 
Next, we consider two additional nonlinear constrained optimization
problems (TP5 and TP8) from [7] and a wellstudied structural design and mechanics problem (‘Weld’) taken from [21].
The details on the mechanics of the welded structure and the beam deformation can be found in [23, 25].
These problems have
multiple nonlinear inequality constraints and our goal is to demonstrate the performance of our proposed constrainthandling
methods. We used DE with
IPS method with the following
parameter settings: , , , and for all three problems.
A maximum of 200,000 function evaluations were allowed and a termination criteria of
from the known optima is chosen.
The problem definitions of TP5, TP8 and ‘Weld’ are as follows:
TP5:
(26) 
TP8: