Pseudo-Adaptive Penalization to Handle Constraints in Particle Swarm Optimizers
PPreprint submitted to Proceedings of the tenth International Conference on Computational Structures Technology. doi:10.4203/ccp.93.123 Abstract
The penalization method is a popular technique to provide particle swarm optimizers with the ability to handle constraints. The downside is the need of penalization coefficients whose settings are problem-specific. While adaptive coefficients can be found in the literature, a different adaptive scheme is proposed in this paper, where coefficients are kept constant. A pseudo-adaptive relaxation of the tolerances for constraint violations while penalizing only violations beyond such tolerances results in a pseudo-adaptive penalization. A particle swarm optimizer is tested on a suite of benchmark problems for three types of tolerance relaxation: no relaxation; self-tuned initial relaxation with deterministic decrease; and self-tuned initial relaxation with pseudo-adaptive decrease. Other authors’ results are offered as frames of reference.
Keywords: particle swarm optimization, constrained problems, penalization with constant coefficients, pseudo-adaptive tolerance relaxation. Introduction
The particle swarm optimization paradigm is a population-based and gradient-free optimization method suitable for unconstrained problems. In order to be able to deal with constrained problems, some external mechanism needs to be incorporated. One of the most straightforward and popular techniques is the penalization method, where infeasible solutions are penalized by increasing their objective function value (minimization problems). The key issue is in the amount of penalization, which is typically linked to the amount of constraint violations. By turning the constrained problem into an unconstrained one, these methods are especially well suited for Particle Swarm Optimizers (PSOs) because they do not disrupt the normal dynamics of the swarm as, for instance, a repair algorithm would do. The drawback is that the penalization coefficients involved typically require problem-specific tuning. An excessive penalization might lead to premature convergence, whereas too mild a
Pseudo-Adaptive Penalization to Handle Constraints in Particle Swarm Optimizers
M. S. Innocente and J. Sienz ADOPT Research Group, School of Engineering Swansea University, Swansea, UK reprint submitted to Proceedings of the tenth International Conference on Computational Structures Technology. doi:10.4203/ccp.93.123 Constrained optimization
Different families of optimization problems require different approaches to be dealt with. The first important difference is in the type of solution sought. Thus, problems whose optimum solution is a scalar or vector of scalars are called parameter optimization problems whereas problems whose optimum solution is a function or vector of functions are called variational problems. Another critical difference is in the type of variables, where a problem is referred to as continuous ‒or real-valued‒ optimization if the variables can take on real values and discrete optimization if they can only take on values from a given discrete set. Only real-valued, parameter optimization problems are considered within this paper. The constraints bound the regions of the search-space where solutions are admissible. Thus, they may be due to geometric, technological, and/or physical restrictions, to name a few. While their linearity or nonlinearity does not make a difference in particle swarm optimization, their formulation as inequality or equality constraints does. The general, constrained, real-valued optimization problem may be formulated as follows: = ++== = niuxl r, ... ,qqjg , ... ,qjgf iii jj , ... ,1 ; 1 ; 0)( 1 ; 0)( subject to )( Minimize xxx (1) reprint submitted to Proceedings of the tenth International Conference on Computational Structures Technology. doi:10.4203/ccp.93.123 ineq
Tol here both for generality and to allow its relaxation. ( ) ( ) ( ) =+−+− ++= = nilxux r, ... ,qqjTolg , ... ,qjTolgf iiii eqj ineqj , ... ,1 ; 0 ,0max,0max 1 ; )(abs 1 ; )( subject to )( Minimize xxx (2) Particle swarm optimization
First introduced by James Kennedy and Russell C. Eberhart in 1995 [1], the particle swarm optimization (PSO) method has strong roots on different disciplines such as social psychology, artificial intelligence, and mathematical optimization. From the ‘optimization’ perspective, it is a gradient-free search method suitable for optimization problems whose solutions can be represented as points in an n -dimensional space. While variables need to be real-valued in its original version, binary and other discrete versions of the method have also been proposed. Refer, for instance, to [2, pp. 289-299]; [3]; [4]; and [5]. Since the method is not designed to optimize but to carry out procedures that are not directly related to the optimization problem, it is frequently referred to as a modern heuristics. Optimization occurs, nevertheless, without obvious links between the implemented technique and the resulting optimization process. Influenced by Evolutionary Algorithms (EAs), the function to be minimized is commonly called ‘fitness function’. It is more appropriately referred to as ‘conflict function’ hereafter due to the social-psychology metaphor that inspired the method. Gradient information is not required, which enables the method to deal with non-differentiable and even discontinuous problems. Therefore there is no restriction to the characteristics of the objective function for the approach to be applicable. In fact, the function does not even need to be explicit. The PSO algorithm is not designed to optimize but to perform a sort of simulation of a social milieu, where the ability of the population (swarm) to optimize its performance emerges from the cooperation among individuals (particles). While this makes it difficult to understand how optimization is actually performed, it shows astonishing robustness in handling different kinds of complex problems that it was not specifically designed for. It has the disadvantage that its theoretical bases are very difficult to be understood deterministically. Nevertheless, considerable theoretical work has been carried out on simplified versions of the algorithm (e.g. [6], [7], [2], [8], [9], [10], and [11]. For a comprehensive review of the method, refer to [12] and [13]. reprint submitted to Proceedings of the tenth International Conference on Computational Structures Technology. doi:10.4203/ccp.93.123 Basic algorithm
While the emergent optimization properties of the PSO algorithm result from local interactions among particles in a swarm, the behaviour of a single particle can be summarized in three sequential processes: evaluation ; comparison ; and imitation . Thus, the performance of a particle in its current position is evaluated in terms of the conflict function. In order to decide upon its next position, the particle compares its current conflict to those associated with its own and with its neighbours’ best experiences. Finally, the particle imitates its own best experience and the best experience in its neighbourhood to some extent. The basic update equations are: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) += −+−+= − −−−−− tijtijtij tijtij,tijtij,tijtij vxx xlbestUswxpbestUiwvwv (3) where: ( ) tij v : j th coordinate of the velocity of particle i at time-step t . ( ) tij x : j th coordinate of the position of particle i at time-step t . U (0,1) : Random number from a uniform distribution in the range [0,1] resampled anew every time it is referenced. w , iw , sw : Inertia, individuality, and sociality weights, respectively. ( ) tij pbest : j th coordinate of the best position found by particle i by time-step t . ( ) tij lbest : j th coordinate of the best position found by any particle in the neighbourhood of particle i by time-step t . As it can be observed, there are three coefficients that govern the dynamics of the swarm: the inertia ( w ), the individuality ( iw ), and the sociality ( sw ) weights. The settings of these coefficients greatly affect the behaviour of the swarm. Equation (3) can also be formulated in terms of ϕ i and ϕ s as shown in Equation (4): ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) += +=+=+ −+−+= − −−−−− tijtijtij ,,si tijtijstijtijitijtij vxx swiwawUswUiw xlbestxpbestvwv (4) Two alternative formulations
Two alternative formulations were proposed by Innocente [11], consisting of defining a desired average behaviour and then adding randomness as a type of noise. Thus the strength of the attraction towards the weighted average of the best experiences is no longer a random value between zero and the acceleration weight. Instead the random value is within the range maxmin , , and the acceleration weight is the centre of the interval rather than its upper limit. Furthermore, the individuality reprint submitted to Proceedings of the tenth International Conference on Computational Structures Technology. doi:10.4203/ccp.93.123 ) ip and ( sp , where their aggregation =+ spip . The procedure starts with the user selecting the desired strength awarded to the individuality and the sociality by choosing ) ip . The common formulation for both proposed alternatives is shown in Equation (5). ( ) ( ) ( ) ( ) ) += −= −+= −+= −+−+= − −−−−− )()1()( )1,0(minmaxmin )1,0(minmaxmin )1()1()1()1()1()(
1 ; 1,0 tijtijtijsi tijtijstijtijitijtij vxx ipspip Usp Uip xlbestxpbestvwv (5)
PSO-RRR1 : The first alternative consists of selecting ( ) aw , and ( )( ) += += −=
121 123 1 minmax wwaww (6) PSO-RRR2 : The second alternative consists of selecting ( aw , and ( ) −= += +−= maxminmax aww awaww (7) It is advisable not to take values of aw too close to the limits. For further details on these two formulations, refer to [11], chapter 6. Pseudo-adaptive penalization
Static penalization
Penalization methods can be viewed as optimizing two objectives: minimizing the objective function on one hand and minimizing constraint violations on the other, where the second objective is already an aggregation of objectives (minimizing each constraint function violation). These methods combine all these objectives into a single function to be optimized, thus turning the constrained problem into an reprint submitted to Proceedings of the tenth International Conference on Computational Structures Technology. doi:10.4203/ccp.93.123 ( ) = += mj jjp j fkff )()()( xxx (8) ( ) ; )(abs 1 ; )(,0max)( = mjqg qjgf j jj x xx (9) where )( x f is the conflict function; )( x p f is the penalized conflict function; )( x j f is the amount of violation of j th constraint; and j k and j α are penalization coefficients. These coefficients may be constant, time-varying or adaptive, and they can be the same or different for different constraints. Typically, k j is set to high and α j to small values. It is not recommendable to use different penalization coefficients for different constraints because that makes the coefficients’ tuning more difficult for every problem. An alternative to account for the different sensitivity of the penalized objective function to the different constraint violations which may be of different orders of magnitude consists of normalizing the constraint violations. Even further, the original conflict function may also be normalized so that the latter and the overall measure of constraint violations are of the same order of magnitude. These normalizations are not considered here and therefore left for future work. Proposed pseudo-adaptive penalization
As previously mentioned, arbitrarily set ‒not tuned‒ constant coefficients are used in the experiments hereafter. The study of their convenient setting is left for future research, where the hope is that the normalization of constraints and perhaps of the conflict function would allow for their removal from Equation (8). Thus, the reprint submitted to Proceedings of the tenth International Conference on Computational Structures Technology. doi:10.4203/ccp.93.123 ( ) = += mj jp fkff )()()( xxx (10) ( ) ( )( ) ; )(abs,0max 1 ; )(,0max)( − −= mjqTolg qjTolgf eqj ineqjj xxx (11) == xx jj ffk (12) Self-tuned initial tolerances relaxation
The use of tolerances for equality constraint violations in population-based methods is of common practice. It is also not uncommon to relax such tolerances, where the decrease of such relaxations is typically deterministic. The aim of these relaxations is to temporarily increase the feasible region of the search-space. However, the impact of a given relaxation on the feasibility ratio (FR) of the search-space is problem-dependent, and can vary greatly. For instance, to obtain a %25%,20FR , a tolerance for equality constraint violations of around 0.26 is required for the problem g11, whereas a tolerance of around 6.63 is required for problem g13 (both problems involving equality constraints only). In addition, since there are problems involving only inequality constraints that present very small FRs, the same concept can be applied. That is, the tolerance for inequality constraint violations can also be dynamically relaxed. Note, for instance, that the tolerance required for problem g10 to present a %25%,20FR is around 10.83, whereas it is around 2790 for problem g06 (both involving inequality constraints only). These examples clearly illustrate how problem-dependent the effect of a given tolerance may be. Hence an initial self-tuned tolerance relaxation is proposed aiming for a desired FR. Thus, the self-tuning procedure consists of starting with a small, minimum value for the tolerance and evaluating the constraint functions of 1000 randomly selected solutions. The FR is evaluated, and the tolerance is adequately increased or decreased. For problems involving inequality and equality constraints, the tolerance for the violations of equality constraints are arbitrarily kept 10 times greater than that of inequality constraint violations. For further details on the implementation of this procedure, refer to [11], chapter 8. Pseudo-adaptive decrease of tolerances relaxation
The aim is to make the tolerance update adaptive so that updates are performed when they would have a less disruptive effect in maintaining potentially good reprint submitted to Proceedings of the tenth International Conference on Computational Structures Technology. doi:10.4203/ccp.93.123 per ( t ) is the current percentage of feasible PBESTs. The exponential update of the tolerances is as posed in Equation (14). Thus )( = t ktol for min)( perper t = ; min)( ktolktol t = for )( = t per ; and the variation in between is linear. Therefore the greater the percentage above a minimum established the smaller the value of ktol ( t ) , and hence the greater the tolerance decrease. Obviously, there is no update for min)( perper t . ( ) min)(minmin)( ktolperperktolktol tt +−−−= (13) )1()()( − = ttt TolktolTol (14)
Since the tolerance for inequality constraint violations is typically set to zero, whenever it goes below 10 –5 it is automatically reset to zero. Aiming to avoid too many time-steps without a tolerance update, a safety mechanism is implemented by enforcing a tolerance update if tes tol. updan t o (15) where t is the number of time-step and n o tol. updates is the number of tolerance updates performed so far. When the update is enforced by Equation (15), the coefficient used in Equation (14) equals )( = t ktol . In order to give some time for the particles to find feasible solutions once the tolerances have reached their desired value, it is arbitrarily set that such values are reached by the time 80% of the search has been carried out ( t min ). If the desired tolerance was not reached by min tt = , an update is performed at every time-step (i.e. from min += tt to min tt = ) using the coefficient in Equation (16). Thus final tolerances are ensured to be reached by min tt = . ( )( ) minminmin ttt TolTolktol = (16) where ktol is calculated independently for inequality and equality constraints. Since ( ) min = tineq Tol , the latter is replaced by 10 –5 for the calculation of ktol in Equation (16) and Tol ineq is set to zero as soon as it reaches a value below or equal to 10 –5 . Of course t min , min t , and min t are rounded-off to integer values if necessary. reprint submitted to Proceedings of the tenth International Conference on Computational Structures Technology. doi:10.4203/ccp.93.123 Experimental results
The details of the implementation for the experiments are as follows: 50 particles; 10000 time-steps, constant penalization coefficients = k and = in Equation (10); the best of 1000 Latin Hypercube samplings for the particles’ initialization according to the maximum minimum distance criterion; velocities initialized to zero while the best individual experiences are initialized in the same manner as the initial positions; synchronous update of the best experiences; the forward topology with three sub-neighbourhoods as proposed by Innocente [11]; the coefficients and formulations for the first sub-neighbourhood are those of the PSO-RRR2 with = aw as in Equations (5) and (7); the coefficients and formulations for the second sub-neighbourhood are those of the PSO-RRR1 with = aw as in Equations (5) and (6); the coefficients and formulations for the third sub-neighbourhood are those of the classical PSO with = w and = aw as in Equation (3) ‒also coinciding with Clerc et al.’s constriction factor type 1” [9] with = cf and = aw ‒; initial self-tuned tolerance relaxation such that %2520FR −= . If the FR of the problem is already greater than that, it is increased in around 5%. If there are inequality and equality constraints, the tolerance for the violations of the latter are arbitrarily set 10 times greater than that of the inequality constraints. The results obtained with no relaxation are also offered for comparison. For relaxed tolerances, two types of decrease of the initial ones are considered: the proposed pseudo-adaptive scheme, and an exponential decrease with a constant coefficient )( = t ktol . For the pseudo-adaptive decrease, min = ktol and %80 min = per in Equation (13), while maxmin == tt in Equation (16). The proposed pseudo-adaptive penalization method is tested on a well known benchmark suite composed of 13 constrained problems. Their main features are offered in Table 1 together with their approximate FRs without tolerances, with final tolerances, and with mean initial tolerances. 25 runs are performed for the statistics. The values of the mean initial tolerances that result in the desired initial FRs are also provided in the table. The results obtained for the three different types of tolerance relaxations are presented in Table 2, including the exact number of objective and constraint function evaluations. A solution is considered successful if the error is not greater than 10 ‒4 . The ‘mean [%] feasible PBESTs’ is the mean percentage –among 25 runs– of individual best experiences of the particles that are feasible in the final time-step. For comparison, the results obtained for the same suite of benchmark problems by two other PSO algorithms in the literature ‒namely those proposed by Toscano Pulido et al. [16] and by Muñoz Zavala et al. [17]‒ are offered in Table 3 together with the results obtained by the pseudo-adaptive approach proposed here. Since the values of the tolerances are pseudo-adaptive, it is interesting to observe the form of the curves of their evolution throughout the search. Due to the stochastic nature of the paradigm, those curves vary from one run to the next for a given problem. The curves showing the evolution of the tolerances corresponding to four selected problems are offered in Figure 1 to Figure 4. Thus, Figure 1 corresponds to problem g01 involving 9 inequality constraints; Figure 2 corresponds to problem reprint submitted to Proceedings of the tenth International Conference on Computational Structures Technology. doi:10.4203/ccp.93.123
10 g03 involving 1 equality constraint only; Figure 3 corresponds to problem g05 involving 3 equality constraints and 1 inequality constraint (in the form of an interval); and Figure 4 corresponds to problem g13 involving 3 equality constraints. In those figures, the plain tolerance curves correspond to one single arbitrary run whereas the average curves correspond to the average tolerances among all 25 runs.
Problem Optimum Dim. IC EC FR [%] FR [%] for desired tolerance FR [%] for initial tolerance Mean initial inequality tolerance Mean initial equality tolerance g01 -15.000000 13 9 0 0.0003 0.0003 23.4617 89.92 N/A g02 -0.803619 20 2 0 99.9971 99.9971 99.9971 0.01 N/A g03 -1.000500 10 0 1 < 0.0001 0.0002 24.5335 N/A 1.66 g04 -30665.538672 5 3 (
0 26.9887 26.9887 30.2026 0.11 N/A g05 (*)
3 < 0.0001 < 0.0001 23.3053 68.88 688.79 g06 -6961.813876 2 2 0 0.0074 0.0074 24.3050 2790.51 N/A g07 g08 -0.095825 2 2 0 0.8610 0.8610 23.4371 9.88 N/A g09 g10 g11 g12 -1.000000 3 1 (@)
0 4.7713 4.7713 22.0256 0.11 N/A g13 (
Other authors claim there are 6 inequality constraints, but each one defines an interval, therefore no more than 3 constraints can be violated simultaneously. (*)
Other authors claim there are two inequality constraints, but they define an interval, so that no more than 1 constraint can be violated simultaneously. (@)
Most authors claim there are 9 inequality constraints, but in reality it is one constraint that splits the feasible space in 9 (729) disjointed spheres. The solution needs to be inside one sphere to be feasible, so that membership to all 729 spheres is not possible. If the constraint is viewed as 729 constraints, then at least 728 of them will be always violated. Table 1. Features of the problems in the test suite: number of dimensions, inequality and equality constraints; feasibility ratios (FRs) of the problem with no tolerance, desired tolerance, and initial tolerance; and the mean self-tuned initial inequality and equality tolerances. FRs are calculated by randomly generating 10 solutions, where final (desired) equality constraint violations tolerance equals 10 ‒4 Due to the tolerance relaxations, intermediate solutions that are temporarily regarded as feasible smaller than the actual feasible minimum solution of the problem can be found by the optimizer. In these cases, the best solution might increase rather than decrease as the search progresses. An example of this is shown in Figure 5 for problem g05. Conclusions and future research
The results in Table 2 show that the pseudo-adaptive scheme obtains the best results overall, in terms of the best, median and mean solutions found. The technique results in remarkable improvement compared to the same optimizer without relaxation, especially in problems with equality constraints such as g03, g05, g11 and g13. The exception in this test suite is in problem g10, for which the solution is decreased by the pseudo-adaptive scheme. The reason for this is still to be studied. reprint submitted to Proceedings of the tenth International Conference on Computational Structures Technology. doi:10.4203/ccp.93.123 P r ob l e m Optimum Type of tolerance relaxation BEST MEDIAN MEAN WORST [%] Feasible Solutions [%] Successful Solutions Mean FEs Mean CEs Mean [%] Feasible PBESTs g01 -15.000000 NONE -15.000000 -15.000000 -15.000000 -15.000000 100.00 100.00 5.00E+05 5.00E+05 99.92 EXP. -15.000000 -15.000000 -15.000000 -15.000000 100.00 100.00 5.00E+05 5.86E+05 99.76 ADAPTIVE -15.000000 -15.000000 -15.000000 -15.000000 100.00 100.00 5.00E+05 5.75E+05 99.36 g02 -0.803619 NONE -0.803618 -0.794896 -0.792568 -0.687854 100.00 40.00 5.00E+05 5.00E+05 100.00 EXP. -0.803618 -0.803429 -0.794852 -0.758093 100.00 48.00 5.00E+05 5.35E+05 100.00 ADAPTIVE -0.803618 -0.803429 -0.794852 -0.758093 100.00 48.00 5.00E+05 5.08E+05 100.00 g03 -1.000500 NONE -0.999213 -0.983214 -0.972554 -0.894905 100.00 0.00 5.00E+05 5.00E+05 98.80 EXP. -1.000493 -1.000477 -1.000467 -1.000326 100.00 96.00 5.00E+05 5.77E+05 100.00 ADAPTIVE -1.000499 -1.000496 -1.000493 -1.000459 100.00 100.00 5.00E+05 5.84E+05 99.92 g04 -30665.538672 NONE -30665.538672 -30665.538672 -30665.538672 -30665.538672 100.00 100.00 5.00E+05 5.00E+05 100.00 EXP. -30665.538672 -30665.538672 -30665.538672 -30665.538672 100.00 100.00 5.00E+05 5.50E+05 100.00 ADAPTIVE -30665.538672 -30665.538672 -30665.538672 -30665.538672 100.00 100.00 5.00E+05 5.14E+05 100.00 g05 5126.496714 NONE 5126.498381 5158.465976 5242.672049 5708.280940 100.00 0.00 5.00E+05 5.00E+05 25.20 EXP. 5126.516036 5155.072539 5235.566090 5885.912510 100.00 0.00 5.00E+05 5.87E+05 24.80 ADAPTIVE 5126.593807 5130.122719 5142.265330 5318.299822 100.00 0.00 5.00E+05 6.31E+05 57.52 g06 -6961.813876 NONE -6961.813876 6961.813876 -6961.813876 -6961.813876 100.00 100.00 5.00E+05 5.00E+05 100.00 EXP. -6961.813876 -6961.813876 -6961.813876 -6961.813876 100.00 100.00 5.00E+05 6.18E+05 100.00 ADAPTIVE -6961.813876 -6961.813876 -6961.813876 -6961.813876 100.00 100.00 5.00E+05 6.38E+05 100.00 g07 24.306209 NONE 24.308588 24.397361 24.447352 25.185085 100.00 0.00 5.00E+05 5.00E+05 100.00 EXP. 24.322332 24.438153 24.447324 24.796222 100.00 0.00 5.00E+05 6.03E+05 100.00 ADAPTIVE 24.322181 24.483518 24.515330 24.948167 100.00 0.00 5.00E+05 5.58E+05 100.00 g08 -0.095825 NONE -0.095825 -0.095825 -0.095825 -0.095825 100.00 100.00 5.00E+05 5.00E+05 100.00 EXP. -0.095825 -0.095825 -0.095825 -0.095825 100.00 100.00 5.00E+05 5.72E+05 100.00 ADAPTIVE -0.095825 -0.095825 -0.095825 -0.095825 100.00 100.00 5.00E+05 5.20E+05 100.00 g09 680.630057 NONE 680.630477 680.632574 680.633039 680.638835 100.00 0.00 5.00E+05 5.00E+05 100.00 EXP. 680.630753 680.632033 680.632550 680.636595 100.00 0.00 5.00E+05 6.03E+05 100.00 ADAPTIVE 680.630093 680.632446 680.632900 680.637958 100.00 8.00 5.00E+05 5.36E+05 100.00 g10 7049.248021 NONE 7059.928996 7154.036097 7169.147324 7440.039736 100.00 0.00 5.00E+05 5.00E+05 98.72 EXP. 7049.728978 7105.765355 7140.793308 7348.447274 100.00 0.00 5.00E+05 5.73E+05 97.04 ADAPTIVE 7118.872990 7489.944970 7570.781098 8155.654975 96.00 0.00 5.00E+05 6.20E+05 82.88 g11 0.749900 NONE 0.749900 0.749900 0.749901 0.749915 100.00 100.00 5.00E+05 5.00E+05 99.20 EXP. 0.749900 0.749900 0.749907 0.749975 100.00 100.00 5.00E+05 5.19E+05 99.68 ADAPTIVE 0.749900 0.749900 0.749900 0.749903 100.00 100.00 5.00E+05 5.95E+05 90.24 g12 -1.000000 NONE -1.000000 -1.000000 -1.000000 -1.000000 100.00 100.00 5.00E+05 5.00E+05 100.00 EXP. -1.000000 -1.000000 -1.000000 -1.000000 100.00 100.00 5.00E+05 5.48E+05 100.00 ADAPTIVE -1.000000 -1.000000 -1.000000 -1.000000 100.00 100.00 5.00E+05 5.16E+05 100.00 g13 0.053942 NONE 0.170131 0.600263 0.632353 0.983725 100.00 0.00 5.00E+05 5.00E+05 38.80 EXP. 0.277980 0.720584 0.706277 0.996941 100.00 0.00 5.00E+05 5.08E+05 40.96 ADAPTIVE 0.053943 0.054119 0.131239 0.439679 100.00 36.00 5.00E+05 6.29E+05 79.92
Table 2. Statistical results obtained for the 13 problems in the test suite for three types of tolerance relaxation: none, initially self-tuned with exponential decrease, and initially self-tuned with (pseudo) adaptive decrease. The percentages of feasible solutions, successful solutions (error not greater than 10 ‒4 ), the mean numbers of FEs and CEs, and the mean percentage of feasible PBESTs at the end of the search are also provided. reprint submitted to Proceedings of the tenth International Conference on Computational Structures Technology. doi:10.4203/ccp.93.123 P r ob l e m Optimum Type of tolerance relaxation BEST MEDIAN MEAN WORST [%] Feasible Solutions [%] Successful Solutions Mean FEs Mean CEs Mean [%] Feasible PBESTs g01 -15.000000 ADAPTIVE -15.000000 -15.000000 -15.000000 -15.000000 100.00 100.00 5.00E+05 5.75E+05 25 Toscano -15.000000 - -15.000000 -15.000000 - - 3.40E+05 - 30 PESO -15.000000 -15.000000 -15.000000 -15.000000 - - 3.40E+05 - 30 g02 -0.803619 ADAPTIVE -0.803618 -0.803429 -0.794852 -0.758093 100.00 48.00 5.00E+05 5.08E+05 25 Toscano -0.803432 - -0.790400 -0.750393 - - 3.40E+05 - 30 PESO -0.792608 -0.731693 -0.721749 0.614135 - - 3.40E+05 - 30 g03 -1.000500 ADAPTIVE -1.000499 -1.000496 -1.000493 -1.000459 100.00 100.00 5.00E+05 5.84E+05 25 Toscano -1.004720 - -1.003814 -1.002490 - - 3.40E+05 - 30 PESO -1.005010 -1.005008 -1.005006 -1.004989 - - 3.40E+05 - 30 g04 -30665.538672 ADAPTIVE -30665.538672 -30665.538672 -30665.538672 -30665.538672 100.00 100.00 5.00E+05 5.14E+05 25 Toscano -30665.500000 -30665.500000 -30665.500000 -30665.500000 - - 3.40E+05 - 30 PESO -30665.538672 -30665.538672 -30665.538672 -30665.538672 - - 3.40E+05 - 30 g05 5126.496714 ADAPTIVE 5126.593807 5130.122719 5142.265330 5318.299822 100.00 0.00 5.00E+05 6.31E+05 25 Toscano 5126.640000 - 5461.081333 6104.750000 - - 3.40E+05 - 30 PESO 5126.484154 5126.538302 5129.178298 5148.859414 - - 3.40E+05 - 30 g06 -6961.813876 ADAPTIVE -6961.813876 -6961.813876 -6961.813876 -6961.813876 100.00 100.00 5.00E+05 6.38E+05 25 Toscano -6961.810000 - -6961.810000 -6961.810000 - - 3.40E+05 - 30 PESO -6961.813876 -6961.813876 -6961.813876 -6961.813876 - - 3.40E+05 - 30 g07 24.306209 ADAPTIVE 24.322181 24.483518 24.515330 24.948167 100.00 0.00 5.00E+05 5.58E+05 25 Toscano 24.351100 - 25.355771 27.316800 - - 3.40E+05 - 30 PESO 24.306921 24.371253 24.371253 24.593504 - - 3.40E+05 - 30 g08 -0.095825 ADAPTIVE -0.095825 -0.095825 -0.095825 -0.095825 100.00 100.00 5.00E+05 5.20E+05 25 Toscano -0.095825 - -0.095825 -0.095825 - - 3.40E+05 - 30 PESO -0.095825 -0.095825 0.095825 -0.095825 - - 3.40E+05 - 30 g09 680.630057 ADAPTIVE 680.630093 680.632446 680.632900 680.637958 100.00 8.00 5.00E+05 5.36E+05 25 Toscano 680.638000 - 680.852393 681.553000 - - 3.40E+05 - 30 PESO 680.630057 680.630057 680.630057 680.630058 - - 3.40E+05 - 30 g10 7049.248021 ADAPTIVE 7118.872990 7489.944970 7570.781098 8155.654975 96.00 0.00 5.00E+05 6.20E+05 25 Toscano 7057.590000 - 7560.047857 8104.310000 - - 3.40E+05 - 30 PESO 7049.459452 7069.926219 7099.101385 7251.396244 - - 3.40E+05 - 30 g11 0.749900 ADAPTIVE 0.749900 0.749900 0.749900 0.749903 100.00 100.00 5.00E+05 5.95E+05 25 Toscano 0.749999 - 0.750107 0.752885 - - 3.40E+05 - 30 PESO 0.749000 0.749000 0.749000 0.749000 - - 3.40E+05 - 30 g12 -1.000000 ADAPTIVE -1.000000 -1.000000 -1.000000 -1.000000 100.00 100.00 5.00E+05 5.16E+05 25 Toscano -1.000000 - -1.000000 -1.000000 - - 3.40E+05 - 30 PESO -1.000000 -1.000000 -1.000000 -1.000000 - - 3.40E+05 - 30 g13 0.053942 ADAPTIVE 0.053943 0.054119 0.131239 0.439679 100.00 36.00 5.00E+05 6.29E+05 25 Toscano 0.068665 - 1.716426 13.669500 - - 3.40E+05 - 30 PESO 0.081498 0.631946 0.626881 0.997586 - - 3.40E+05 - 30
Table 3. Statistical results obtained for the 13 problems in the test suite for the (pseudo) adaptive penalization scheme proposed, together with those reported by Toscano Pulido et al. [16] and by Muñoz Zavala et al. [17] as references. It is quite surprising to observe the improvement in the quality of the solutions resulting from the pseudo-adaptive scheme on problem g02, which presents a very reprint submitted to Proceedings of the tenth International Conference on Computational Structures Technology. doi:10.4203/ccp.93.123
13 high FR. The reason for this is probably that the solution lies near the boundary of feasible space, and the relaxation of the tolerance allows particles to approach the solution from every direction rather than from feasible space only. The self-tuned initial relaxation with deterministic, exponential decrease presents competitive or better performance than no relaxation on all problems in terms of the best, median, and mean solutions found, except for g13. The surprising decrease in the quality of the solutions for problem g13 ‒while the pseudo-adaptive decrease leads to remarkable improvement‒ seems to suggest that updating the tolerance ‘too soon’ may turn the relaxation of tolerances into a harmful mechanism, as potentially good solutions may be lost during the updates.
Figure 1: Pseudo-adaptive tolerance for inequality constraint violations in problem g01. The average is among the 25 runs. The figure on the right is just a zoom of the one on the left.
Figure 2: Pseudo-adaptive tolerance for equality constraint violations in problem g03. The average is among the 25 runs. The figure on the right is just a zoom of the one on the left. The results obtained with the proposed pseudo-adaptive penalization are compared to those obtained by two other PSO algorithms in the literature for reference. Results obtained here are better than those reported by Toscano Pulido et reprint submitted to Proceedings of the tenth International Conference on Computational Structures Technology. doi:10.4203/ccp.93.123
14 al. [16] for problems g02, g05, g07, g09, g11 and g13, while they are worse for problem g10. They are competitive for the other problems, namely g01, g03, g04, g06, g08, and g12. It is fair to note that Toscano Pulido et al.’s experiments used a smaller number of FEs. Also note that the optimum in Table 2 corresponds to a tolerance for equality constraint violations of 10 ‒4 , while Toscano Pulido et al. used a tolerance of 10 ‒3 . Hence smaller results than the actual minimum feasible solution can be observed in the table. Figure 3: Pseudo-adaptive tolerance for inequality and equality constraint violations in problem g05. The average is among the 25 runs. The figure on the right is just a zoom of the one on the left.
Figure 4: Pseudo-adaptive tolerance for equality constraint violations in problem g13. The average is among the 25 runs. The figure on the right is just a zoom of the one on the left. Results obtained here are better than those reported by Muñoz Zavala et al. [17] for problems g02 and g13, while they are worse for problems g05, g07, g09 and g10. They are competitive for the remaining problems (g01, g03, g04, g06, g08, g11 and g12). The number of FEs carried out in [17] are reported to be smaller (it is not clear whether they count or not the FEs performed by their mutation operators). reprint submitted to Proceedings of the tenth International Conference on Computational Structures Technology. doi:10.4203/ccp.93.123
15 Figure 5: Evolution of the mean best and average conflict values among 25 runs for problem g05. Recall that this is a minimization problem. The shape of the curve is due to the tolerance relaxations. There are several aspects of the proposed pseudo-adaptive penalization scheme which remain to be studied more thoroughly. Namely, the study of the possible improvement of simply normalizing the constraint violations so that more sensitive constraints do not overtake the search; the initial FR (20‒25% here); the percentage of feasible PBESTs that triggers the tolerance update (80% here); the setting of ktol min ; the calculation of the pseudo-adaptive coefficient ktol (see Equation (13)); and the scheme used to force an update if the desired percentage of feasible PBESTs is not being achieved (see Equation (15)). Preliminary tests showed that the initial FR is not critical within a range such as [1‒100%], as the greater the initial FR the more updates of the tolerances are carried out early in the search, as achieving 80% of feasible PBESTs is not difficult. It seems that the results are more sensitive to the setting of ktol min ; the pseudo-adaptive scheme for the update of ktol ; and the percentage of feasible PBESTs required to triggering the updates. A more extensive and systematic study of these aspects still needs to be carried out.
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