Solving Constrained Global Optimization Problems by Using Hybrid Evolutionary Computing and Artificial Life Approaches

Abstract

This work presents a hybrid real-coded genetic algorithm with a particle swarm optimization (RGA-PSO) algorithm and a hybrid artificial immune algorithm with a PSO (AIA-PSO) algorithm for solving 13 constrained global optimization (CGO) problems, including six nonlinear programming and seven generalized polynomial programming optimization problems. External RGA and AIA approaches are used to optimize the constriction coefficient, cognitive parameter, social parameter, penalty parameter, and mutation probability of an internal PSO algorithm. CGO problems are then solved using the internal PSO algorithm. The performances of the proposed RGA-PSO and AIA-PSO algorithms are evaluated using 13 CGO problems. Moreover, numerical results obtained using the proposed RGA-PSO and AIA-PSO algorithms are compared with those obtained using published individual GA and AIA approaches. Experimental results indicate that the proposed RGA-PSO and AIA-PSO algorithms converge to a global optimum solution to a CGO problem. Furthermore, the optimum parameter settings of the internal PSO algorithm can be obtained using the external RGA and AIA approaches. Also, the proposed RGA-PSO and AIA-PSO algorithms outperform some published individual GA and AIA approaches. Therefore, the proposed RGA-PSO and AIA-PSO algorithms are highly promising stochastic global optimization methods for solving CGO problems.

1. Introduction

Many scientific, engineering, and management problems can be expressed as constrained global optimization (CGO) problems, as follows: Ms.ti.nimize𝑔𝑓_𝑚(𝐱),ℎ(𝐱)≤0,𝑚=1,2,…,𝑀,_𝑘𝑥(𝐱)=0,𝑘=1,2,…,𝐾,_𝑙𝑛≤𝑥_𝑛≤𝑥^𝑢𝑛,𝑛=1,2,…,𝑁,(1.1) where 𝑓(𝐱) denotes an objective function; 𝑔_𝑚(𝐱) represents a set of 𝑚 nonlinear inequality constraints; ℎ_𝑘(𝐱) refers to a set of 𝑘 nonlinear equality constraints; 𝐱 represents a vector of decision variables which take real values, and each decision variable 𝑥_𝑛 is constrained by its lower and upper boundaries [𝑥^𝑙𝑛,𝑥^𝑢𝑛]; 𝑁 is the total number of decision variables 𝑥_𝑛. For instance, generalized polynomial programming (GPP) belongs to the nonlinear programming (NLP) method. The formulation of GPP is a nonconvex objective function subject to nonconvex inequality constraints and possibly disjointed feasible region. The GPP approach has been successfully used to solve problems including alkylation process design, heat exchanger design, optimal reactor design [1], inventory decision problem (economic production quantity) [2], process synthesis and the design of separations, phase equilibrium, nonisothermal complex reactor networks, and molecular conformation [3].

Traditional local NLP optimization approaches based on a gradient algorithm are inefficient for solving CGO problems, while an objective function is nondifferentiable. Global optimization methods can be divided into deterministic or stochastic [4]. Often involving a sophisticated optimization process, deterministic global optimization methods typically make assumptions regarding the problem to be solved [5]. Stochastic global optimization methods that do not require gradient information and numerous assumptions have received considerable attention. For instance, Sun et al. [6] devised an improved vector particle swarm optimization (PSO) algorithm with a constraint-preserving method to solve CGO problems. Furthermore, Tsoulos [7] developed a real-coded genetic algorithm (RGA) with a penalty function approach for solving CGO problems. Additionally, Deep and Dipti [8] presented a self-organizing GA with a tournament selection method for solving CGO problems. Meanwhile, Wu and Chung [9] developed a RGA with a static penalty function approach for solving GPP optimization problems. Finally, Wu [10] introduced an artificial immune algorithm (AIA) with an adaptive penalty function method to solve CGO problems.

Zadeh [11] defined “soft computing” as the synergistic power of two or more fused computational intelligence (CI) schemes, which can be divided into several branches: granular computing (e.g., fuzzy sets, rough sets, and probabilistic reasoning), neurocomputing (e.g., supervised, unsupervised, and reinforcement neural learning algorithms), evolutionary computing (e.g., GAs, genetic programming, and PSO algorithms), and artificial life (e.g., artificial immune systems) [12]. Besides, outperforming individual algorithms in terms of solving certain problems, hybrid algorithms can solve general problems more efficiently [13]. Therefore, hybrid CI approaches have recently attracted considerable attention as a promising field of research. Various hybrid evolutionary computing (GA and PSO methods) and artificial life (such as AIA methods) approaches have been developed for solving optimization problems. These hybrid algorithms focus on developing diverse candidate solutions (such as chromosomes and particles) of population/swarm to solve optimization problems more efficiently. These hybrid algorithms use two different algorithms to create diverse candidate solutions using their specific operations and then merge these diverse candidate solutions to increase the diversity of the candidate population. For instance, Abd-El-Wahed et al. [14] developed an integrated PSO algorithm and GA to solve nonlinear optimization problems. Additionally, Kuo and Han [15] presented a hybrid GA and PSO algorithm for bilevel linear programming to solve a supply chain distribution problem. Furthermore, Shelokar et al. [16] presented a hybrid PSO method and ant colony optimization method for solving continuous optimization problems. Finally, Hu et al. [17] developed an immune cooperative PSO algorithm for solving the fault-tolerant routing problem.

Compared to the above hybrid CI algorithms, this work optimizes the parameter settings of an individual CI method by using another individual CI algorithm. A standard PSO algorithm has certain limitations [17, 18]. For instance, a PSO algorithm includes many parameters that must be set, such as the cognitive parameter, social parameter, and constriction coefficient. In practice, the optimal parameter settings of a PSO algorithm are tuned based on trial and error and prior knowledge is required to successfully manipulate the cognitive parameter, social parameter, and constriction coefficient. The exploration and exploitative capabilities of a PSO algorithm are limited to optimum parameter settings. Moreover, conventional PSO methods involve premature convergence that rapidly losses diversity during optimization.

Fortunately, optimization of parameter settings for a conventional PSO algorithm can be considered an unconstrained global optimization (UGO) problem, and the diversity of candidate solutions of the PSO method can be increased using a multi-nonuniform mutation operation [19]. Moreover, the parameter manipulation of a GA and AIA method is easy to implement without prior knowledge. Therefore, to overcome the limitations of a standard PSO algorithm, this work develops two hybrid CI algorithms to solve CGO problems efficiently. The first algorithm is a hybrid RGA and PSO (RGA-PSO) algorithm, while the second algorithm is a hybrid AIA and PSO (AIA-PSO) algorithm. The proposed RGA-PSO and AIA-PSO algorithms are considered to optimize two optimization problems simultaneously. The UGO problem (optimization of cognitive parameter, social parameter, constriction coefficient, penalty parameter, and mutation probability of an internal PSO algorithm based on a penalty function approach) is optimized using external RGA and AIA approaches, respectively. A CGO problem is then solved using the internal PSO algorithm. The performances of the proposed RGA-PSO and AIA-PSO algorithms are evaluated using a set of CGO problems (e.g., six benchmark NLP and seven GPP optimization problems).

The rest of this paper is organized as follows. Section 2 describes the RGA, PSO algorithm, AIA, and penalty function approaches. Section 3 then introduces the proposed RGA-PSO and AIA-PSO algorithms. Next, Section 4 compares the experimental results of the proposed RGA-PSO and AIA-PSO algorithms with those of various published individual GAs and AIAs [9, 10, 20–22] and hybrid algorithms [23, 24]. Finally, conclusions are drawn in Section 5.

2. Related Works

2.1. Real-Coded Genetic Algorithm

GAs are stochastic global optimization methods based on the concepts of natural selection and use three genetic operators, that is, selection, crossover, and mutation, to explore and exploit the solution space. RGA outperforms binary-coded GA in solving continuous function optimization problems [19]. This work thus describes operators of a RGA [25].

2.1.1. Selection

A selection operation selects strong individuals from a current population based on their fitness function values and then reproduces these individuals into a crossover pool. The several selection operations developed include the roulette wheel, ranking, and tournament methods [19, 25]. This work uses the normalized geometric ranking method, as follows: 𝑝_𝑗=𝑞^(1−𝑞)^𝑟−1,𝑗=1,2,…,ps_RGA,(2.1)𝑝_𝑗= probability of selecting individual 𝑗,𝑞= probability of choosing the best individual (here 𝑞=0.35) 𝑞^=𝑞1−(1−𝑞)^psRGA,(2.2)𝑟= individual ranking based on fitness value, where 1 represents the best, 𝑟=1,2,…,ps_RGA, ps_RGA = population size of the RGA.

2.1.2. Crossover

While exploring the solution space by creating new offspring, the crossover operation randomly selects two parents from the crossover pool and then uses these two parents to generate two new offspring. This operation is repeated until the ps_RGA/2 is satisfied. The whole arithmetic crossover is easily implemented, as follows: 𝐯𝐯^𝟏_𝟐=𝛽×𝐯_𝟏+(1−𝛽)×𝐯_𝟐,=(1−𝛽)𝐯_𝟏+𝛽×𝐯_𝟐,(2.3) where 𝐯_𝟏 and 𝐯_𝟐= parents, 𝐯^𝟏 and 𝐯^𝟐= offspring, 𝛽= uniform random number in the interval [0,1.5].

2.1.3. Mutation

Mutation operation can increase the diversity of individuals (candidate solutions). Multi-nonuniform mutation is described as follows: 𝑥_trial,𝑛=𝑥^current,𝑛𝑥+𝑥^𝑢𝑛−𝑥_current,𝑛pert𝑔_RGAif𝑈₁(0,1)<0.5,_current,𝑛−𝑥_current,𝑛−𝑥^𝑙𝑛pert𝑔_RGAif𝑈₁(0,1)≥0.5,(2.4) where pert(𝑔_RGA)=[𝑈₂(0,1)(1−𝑔_RGA/𝑔_max,RGA)]², perturbed factor, 𝑈₁(0,1) and 𝑈₂(0,1) = uniform random variable in the interval [0,1], 𝑔_max,RGA = maximum generation of the RGA, 𝑔_RGA = current generation of the RGA, 𝑥_current,𝑛 = current decision variable 𝑥_𝑛, 𝑥_trial,𝑛 = trial candidate solution 𝑥_𝑛.

2.2. Particle Swarm Optimization

Kennedy and Eberhart [26] first introduced a conventional PSO algorithm, which is inspired by the social behavior of bird flocks or fish schools. Like GAs, a PSO algorithm is a population-based algorithm. A population of candidate solutions is called a particle swarm. The particle velocities can be updated by (2.5), as follows: 𝑣_𝑗,𝑛𝑔_PSO+1=𝑣+_𝑗,𝑛𝑐₂𝑔_PSO+𝑐₁𝑟_1𝑗𝑔_PSO𝑝^{𝑙𝑏𝑗,𝑛}𝑔_PSO−𝑥_𝑗,𝑛𝑔_PSO𝑟_2𝑗𝑔_PSO𝑝^{𝑔𝑏𝑗,𝑛}𝑔_PSO−𝑥_𝑗,𝑛𝑔_PSO𝑗=1,2,…,ps_PSO,𝑛=1,2,…,𝑁,(2.5)𝑣_𝑗,𝑛(𝑔_PSO+1) = particle velocity of decision variable 𝑥_𝑛 of particle 𝑗 at generation 𝑔_PSO+1, 𝑣_𝑗,𝑛(𝑔_PSO)= particle velocity of decision variable 𝑥_𝑛 of particle 𝑗 at generation 𝑔_PSO, 𝑐₁= cognitive parameter, 𝑐₂= social parameter, 𝑥_𝑗,𝑛(𝑔_PSO)= particle position of decision variable 𝑥_𝑛 of particle 𝑗 at generation 𝑔_PSO, 𝑟_1,𝑗(𝑔_PSO),𝑟_2,𝑗(𝑔_PSO)= independent uniform random numbers in the interval [0,1] at generation 𝑔_PSO, 𝑝^{𝑙𝑏𝑗,𝑛}(𝑔_PSO)= best local solution at generation 𝑔_PSO,𝑝^{𝑔𝑏𝑗,𝑛}(𝑔_PSO)= best global solution at generation 𝑔_PSO, ps_PSO= population size of the PSO algorithm.

The particle positions can be computed using (2.6), as follows: 𝑥_𝑗,𝑛𝑔_PSO+1=𝑥_𝑗,𝑛𝑔_PSO+𝑣_𝑗,𝑛𝑔_PSO+1𝑗=1,2,…,ps_PSO,𝑛=1,2,…,𝑁.(2.6)

Shi and Eberhart [27] developed a modified PSO algorithm by incorporating an inertia weight (𝜔_in) into (2.7) to control the exploration and exploitation capabilities of a PSO algorithm, as follows: 𝑣_𝑗,𝑛𝑔_PSO+1=𝜔+_in𝑐𝑣_𝑗,𝑛2𝑔_PSO+𝑐₁𝑟_1𝑗𝑔_PSO𝑝^{𝑙𝑏𝑗,𝑛}𝑔_PSO−𝑥_𝑗,𝑛𝑔_PSO𝑟_2𝑗𝑔_PSO𝑝^{𝑔𝑏𝑗,𝑛}𝑔_PSO−𝑥_𝑗,𝑛𝑔_PSO𝑗=1,2,…,ps_PSO,𝑛=1,2,…,𝑁.(2.7)

A constriction coefficient (𝜒) was inserted into (2.8) to balance the exploration and exploitation tradeoff [28–30], as follows: 𝑣_𝑗,𝑛𝑔_PSO+1=𝜒+𝑣𝜌_𝑗,𝑛2𝑔_PSO+𝜌₁𝑔_PSO𝑝^{𝑙𝑏𝑗,𝑛}𝑔_PSO−𝑥_𝑗,𝑛𝑔_PSO𝑔_PSO𝑝^{𝑔𝑏𝑗,𝑛}𝑔_PSO−𝑥_𝑗,𝑛𝑔_PSO𝑗=1,2,…,ps_PSO,𝑛=1,2,…,𝑁,(2.8) where 𝜒=2𝑈³(0,1)|||2−𝜏−√𝜏(𝜏−4)|||,(2.9)𝑈₃(0,1)= uniform random variable in the interval [0,1],𝜏=𝜏₁+𝜏₂,𝜏₁=𝑐₁𝑟_1𝑗,𝜏₁=𝑐₂𝑟_2𝑗.

This work considers parameters 𝜔_in and 𝜒 to update the particle velocities, as follows: 𝑣_𝑗,𝑛𝑔_PSO+1=𝜒+𝜔𝑐_in2𝑣_𝑗,𝑛𝑔_PSO+𝑐₁𝑟_1𝑗𝑔_PSO𝑝^{𝑙𝑏𝑗,𝑛}𝑔_PSO−𝑥_𝑗,𝑛𝑔_PSO𝑟_2𝑗𝑔_PSO𝑝^{𝑔𝑏𝑗,𝑛}𝑔_PSO−𝑥_𝑗,𝑛𝑔_PSO𝑗=1,2,…,ps_PSO,𝑛=1,2,…,𝑁,(2.10) where 𝜔_in=((𝑔_max,PSO−𝑔_PSO)/𝑔_max,PSO), increased 𝑔_PSO value reduces the 𝜔_in, 𝑔_max,PSO= maximum generation of the PSO algorithm.

According to (2.10), the optimal values of parameters 𝑐₁, 𝑐₂, and 𝜒 are difficult to obtain through a trial and error. This work thus optimizes these parameter settings by using RGA and AIA approaches.

2.3. Artificial Immune Algorithm

Wu [10] presented an AIA based on clonal selection and immune network theories to solve CGO problems. The AIA approach comprises selection, hypermutation, receptor editing, and bone marrow operations. The selection operation is performed to reproduce strong antibodies (Abs). Also, diverse Abs are created using hypermutation, receptor editing, and bone marrow operations, as described in the following subsections.

2.3.1. Ab and Ag Representation

In the human immune system, an antigen (Ag) has multiple epitopes (antigenic determinants), which can be recognized by various Abs with paratopes (recognizers), on its surface. In the AIA approach, an Ag represents known parameters of a solved problem. The Abs are the candidate solutions (i.e., decision variables 𝑥_𝑛, 𝑛=1,2,…,𝑁) of the solved problem. The quality of a candidate solution is evaluated using an Ab-Ag affinity that is derived from the value of an objective function of the solved problem.

2.3.2. Selection Operation

The selection operation, which is based on the immune network principle [31], controls the number of antigen-specific Abs. This operation is defined according to Ab-Ag and Ab-Ab recognition information, as follows: 𝑝_𝑟𝑗=1𝑁^𝑁_𝑛=11𝑒^𝑑𝑛𝑗𝑑_𝑛𝑗,=||||𝑥^∗𝑛−𝑥_𝑛𝑗𝑥^∗𝑛||||,𝑗=1,2,…,rs,𝑛=1,2,…,𝑁,(2.11) where 𝑝_𝑟𝑗= probability that 𝐀𝐛𝑗 recognizes 𝐀𝐛^∗ (the best solution), 𝑥^∗𝑛= the best 𝐀𝐛^∗ with the highest Ab-Ag affinity, 𝑥_𝑛𝑗= decision variables 𝑥_𝑛 of 𝐀𝐛𝑗,rs= repertoire (population) size of the AIA.

The 𝐀𝐛^∗ is recognized by other Ab  𝑗 in a current Ab repertoire. Large 𝑝_𝑟𝑗 implies that Ab  𝑗 can effectively recognize 𝐀𝐛^∗. The Ab  𝑗 with 𝑝_𝑟𝑗 that is equivalent to or larger than the threshold degree 𝑝_𝑟𝑡 is reproduced to generate an intermediate Ab repertoire.

2.3.3. Hypermutation Operation

Multi-nonuniform mutation [19] is used as the somatic hypermutation operation, which can be expressed as follows: 𝑥_trial,𝑛=𝑥^current,𝑛𝑥+𝑥^𝑢𝑛−𝑥_current,𝑛pert𝑔_AIA,if𝑈₄(0,1)<0.5,_current,𝑛−𝑥_current,𝑛−𝑥^𝑙𝑛pert𝑔_AIA,if𝑈₄(0,1)≥0.5,(2.12) where pert(𝑔_AIA)={𝑈₅(0,1)(1−𝑔_AIA/𝑔_max,AIA)}²= perturbation factor, 𝑔_AIA= current generation of the AIA, 𝑔_max,AIA= maximum generation number of the AIA, 𝑈₄(0,1) and 𝑈₅(0,1)= uniform random number in the interval [0,1].

This operation has two tasks, that is, a uniform search and local fine-tuning.

2.3.4. Receptor Editing Operation

A receptor editing operation is developed using the standard Cauchy distribution 𝐶(0,1), in which the local parameter is zero and the scale parameter is one. Receptor editing is performed using Cauchy random variables that are generated from 𝐶(0,1), owing to their ability to provide a large jump in the Ab-Ag affinity landscape to increase the probability of escaping from the local Ab-Ag affinity landscape. Cauchy receptor editing can be defined by 𝐱_trial=𝐱_current+𝑈₅(0,1)²×𝝈,(2.13) where 𝝈=[𝜎₁,𝜎₂,…,𝜎_𝑁]^𝑇, vector of Cauchy random variables, 𝑈₅(0,1)= uniform random number in the interval [0,1].

This operation is employed in local fine-tuning and large perturbation.

2.3.5. Bone Marrow Operation

The paratope of an Ab can be generated by recombining gene segments 𝐕_𝐇𝐃_𝐇𝐉_𝐇 and 𝐕_𝐋𝐉_𝐋 [32]. Therefore, based on this metaphor, diverse Abs are synthesized using a bone marrow operation. This operation randomly chooses two Abs from the intermediate Ab repertoire and a recombination point from the gene segments of the paratope of the selected Abs. The selected gene segments (e.g., gene 𝑥₁ of Ab 1 and gene 𝑥₁ of the Ab 2) are reproduced to create a library of gene segments. The selected gene segments in the paratope are then deleted. The new Ab  1  is formed by inserting the gene segment, which is gene 𝑥₁ of the Ab 2 in the library plus a random variable created from standard normal distribution 𝑁(0,1), at the recombination point. The literature details the implementation of the bone marrow [10].

2.4. Penalty Function Methods

Stochastic global optimization approaches, including GAs, AIAs, and PSO, are naturally unconstrained optimization methods. Penalty function methods, which are constraint handling approaches, are commonly used to create feasible solutions to a CGO problem and transform it into an unconstrained optimization problem. Two popular penalty functions exist, namely, the exterior and interior functions. Exterior penalty functions use an infeasible solution as a starting point, and convergence is from the infeasible region to the feasible one. Interior penalty functions start from a feasible solution, then move from the feasible region to the constrained boundaries. Exterior penalty functions are favored over interior penalty functions, because they do not require a feasible starting point and are easily implemented. The exterior penalty functions developed to date include static, dynamic, adaptive, and death penalty functions [33]. This work uses the form of a static penalty function, as follows: Minimize𝑓_pseudo(𝐱,𝜌)=𝑓(𝐱)+𝜌⎧⎪⎨⎪⎩^𝑀_𝑚₌₁max0,𝑔^𝑚(𝐱)+^𝐾_𝑘=1ℎ_𝑘(𝐱)²⎫⎪⎬⎪⎭,(2.14) where 𝑓_pseudo(𝐱,𝜌)= pseudo-objective function obtained using an original objective function plus a penalty term, 𝜌= penalty parameter.

Unfortunately, the penalty function scheme is limited by the need to fine-tune the penalty parameter 𝜌 [8]. To overcome this limitation, this work attempts to find the optimum 𝜌 for each CGO problem using the RGA and AIA approaches. Additionally, to obtain high-quality RGA-PSO and AIA-PSO solutions accurate to at least five decimal places for the violation of each constraint to a specific CGO problem, the parameter 𝜌 is within the search space [1×10⁹,1×10¹¹].

3. Method

3.1. RGA-PSO Algorithm

Figure 1 shows the pseudocode of the proposed RGA-PSO algorithm. The external RGA approach is used to optimize the best parameter settings of the internal PSO algorithm, and the internal PSO algorithm is employed to solve CGO problems.

Figure 1 

The pseudocode of the proposed RGA-PSO algorithm.

External RGA

Step 1 (initialize the parameter settings). Parameter settings are given such as ps_RGA, crossover probability 𝑝_𝑐, mutation probability of the external RGA approach 𝑝_𝑚,RGA, the lower and upper boundaries of these parameters 𝑐₁, 𝑐₂, 𝜒, 𝜌, and the mutation probability of the internal PSO algorithm 𝑝_𝑚,PSO. The candidate solutions (individuals) of the external RGA represent the optimized parameters of the internal PSO algorithm. Finally, Figure 2 illustrates the candidate solution of the external RGA approach.

Figure 2 

Chromosome representation of the external RGA.

Step 2 (compute the fitness function value). The fitness function value fitness_𝑗 of the external RGA approach is the best objective function value 𝑓(𝐱^∗_PSO) obtained from the best solution 𝐱^∗_PSO of each internal PSO algorithm execution, as follows: fitness_𝑗=𝑓𝐱^∗_PSO,𝑗=1,2,…,ps_RGA.(3.1)

Candidate solution 𝑗 of the external RGA approach is incorporated into the internal PSO algorithm, and a CGO problem is then solved using the internal PSO algorithm, which is executed as follows.

Internal PSO Algorithm
Step (1) (create an initial particle swarm). An initial particle swarm is created based on the ps_PSO from [𝑥^𝑙𝑛, 𝑥^𝑢𝑛] of a CGO problem. A particle represents a candidate solution of a CGO problem, as shown in Figure 3. Step (2) (calculate the objective function value). According to (2.14), the pseudo-objective function value of the internal PSO algorithm is defined by 𝑓_pseudo,𝑗=𝑓𝐱_PSO,𝑗+𝜌×^𝑀_𝑚₌₁max0,𝑔_𝑚𝐱_PSO,𝑗²,𝑗=1,2,…,ps^PSO.(3.2)Step (3) (update the particle velocity and position). The particle position and velocity can be updated using (2.6) and (2.10), respectively.Step (4) (implement a mutation operation). The standard PSO algorithm lacks evolution operations of GAs such as crossover and mutation. To maintain the diversity of particles, this work uses the multi-nonuniform mutation operator defined by (2.4).Step (5) (perform an elitist strategy). A new particle swarm is created from internal step (3). Notably, 𝑓(𝐱_PSO,𝑗) of a candidate solution j (particle j) in the particle swarm is evaluated. Here, a pairwise comparison is made between the 𝑓(𝐱_PSO,𝑗) value of candidate solutions in the new and current particle swarms. A situation in which the candidate solution j (𝑗=1,2,…,ps_PSO) in the new particle swarm is superior to candidate solution 𝑗 in the current particle swarm implies that the strong candidate solution 𝑗 in the new particle swarm replaces the candidate solution 𝑗 in the current particle swarm. The elitist strategy guarantees that the best candidate solution is always preserved in the next generation. The current particle swarm is updated to the particle swarm of the next generation.

Figure 3 

Candidate solution of the internal PSO algorithm.

Internal steps (2) to (5) are repeated until the 𝑔_max,PSO value of the internal PSO algorithm is satisfied.

End

Step 3 (implement selection operation). The parents in a crossover pool are selected using (2.1).

Step 4 (perform crossover operation). In GAs, the crossover operation performs a global search. Thus, the crossover probability 𝑝_c usually exceeds 0.5. Additionally, candidate solutions are created using (2.3).

Step 5 (conduct mutation operation). In GAs, the mutation operation implements a local search. Additionally, a solution space is exploited using (2.4).

Step 6 (implement an elitist strategy). This work updates the population using an elitist strategy. A situation in which the fitness_𝑗 of candidate solution 𝑗 in the new population is larger than that in the current population suggests that the weak candidate solution 𝑗 is replaced. Additionally, a situation in which the fitness_𝑗 of candidate solution 𝑗 in the new population is equal to or worse than that in the current population implies that the candidate solution 𝑗 in the current population survives. In addition to maintaining the strong candidate solutions, this strategy eliminates weak candidate solutions.

External Steps 2 to 6 are repeated until the 𝑔_max,RGA value of the external RGA approach is met.

3.2. AIA-PSO Algorithm

Figure 4 shows the pseudocode of the proposed AIA-PSO algorithm, in which the external AIA approach is used to optimize the parameter settings of the internal PSO algorithm and the PSO algorithm is used to solve CGO problems.

Figure 4 

The pseudocode of the AIA-PSO algorithm.

External AIA

Step 1 (initialize the parameter settings). Several parameters must be predetermined. These include rs and the threshold for Ab-Ab recognition 𝑝_𝑟𝑡, as well as the lower and upper boundaries of these parameters 𝑐₁, 𝑐₂, 𝜒, 𝜌, and 𝑝_𝑚,PSO. Figure 5 shows the Ab and Ag representation.

Figure 5 

Ag and Ab representation.

Step 2 (evaluate the Ab-Ag affinity).   
Internal PSO Algorithm
The external AIA approach offers parameter settings 𝑐₁, 𝑐₂, 𝜒, 𝜌, and 𝑝_𝑚,PSO for the internal PSO algorithm, subsequently leading to the implementation of internal steps (1)–(5) of the PSO algorithm. The PSO algorithm returns the best fitness value of PSO 𝑓(𝐱^∗_PSO) to the external AIA approach. Step (1) (create an initial particle swarm). An initial particle swarm is created based on ps_PSO from [𝑥^𝑙𝑛, 𝑥^𝑢𝑛] of a CGO problem. A particle represents a candidate solution of a CGO problem.Step (2) (calculate the objective function value). Equation (3.2) is used as the pseudo-objective function value of the internal PSO algorithm. Step (3) (update the particle velocity and position). Equations (2.6) and (2.10) can be used to update the particle position and velocity. Step (4) (implement a mutation operation). The diversity of the particle swarm is increased using (2.4). Step (5) (perform an elitist strategy). A new particle swarm (population) is generated from internal step (3). Notably, 𝑓(𝐱_PSO,𝑗) of a candidate solution 𝑗 (particle 𝑗) in the particle swarm is evaluated. Here, a pairwise comparison is made between the 𝑓(𝐱_PSO,𝑗) value of candidate solutions in the new and current particle swarms. The elitist strategy guarantees that the best candidate solution is always preserved in the next generation. The current particle swarm is updated to the particle swarm of the next generation.

Internal steps (2) to (5) are repeated until the 𝑔_max,PSO value of the internal PSO algorithm is satisfied.

End
Consistent with the Ab-Ag affinity metaphor, an Ab-Ag affinity is determined using (3.3), as follows: maxaffinity_𝑗=−1×𝑓𝐱^∗_PSO𝑗=1,2,…,rs.(3.3) Following the evaluation of the Ab-Ag affinities of Abs in the current Ab repertoire, the Ab with the highest Ab-Ag affinity (𝐀𝐛^∗) is chosen to undergo clonal selection operation in external Step 3.

Step 3 (perform clonal selection operation). To control the number of antigen-specific Abs, (2.11) is used.

Step 4 (implement Ab-Ag affinity maturation). The intermediate Ab repertoire that is created in external Step 3 is divided into two subsets. These Abs undergo somatic hypermutation operation by using (2.12) when the random number is 0.5 or less. Notably, these Abs suffer receptor editing operation using (2.13) when the random number exceeds 0.5.

Step 5 (introduce diverse Abs). Based on the bone marrow operation, diverse Abs are created to recruit the Abs suppressed in external Step 3.

Step 6 (update an Ab repertoire). A new Ab repertoire is generated from external Steps 3–5. The Ab-Ag affinities of the Abs in the generated Ab repertoire are evaluated. This work presents a strategy for updating the Ab repertoire. A situation in which the Ab-Ag affinity of Ab  𝑗 in the new Ab repertoire exceeds that in the current Ab repertoire implies that a strong Ab in the new Ab repertoire replaces the weak Ab in the current Ab repertoire. Additionally, a situation in which the Ab-Ag affinity of Ab  𝑗 in the new Ab repertoire equals to or is worse than that in the current Ab repertoire implies that the Ab  𝑗 in the current Ab repertoire survives. In addition to maintaining the strong Abs, this strategy eliminates nonfunctional Abs.

External Steps 2–6 are repeated until the termination criterion 𝑔_max,AIA is satisfied.

4. Results

The 13 CGO problems were taken from other studies [1, 20, 21, 23, 34]. The set of CGO problems comprises six benchmark NLP problems (TPs 1–4 and 12–13), and seven GPP problems, in which TP 5 (alkylation process design in chemical engineering), TP 6 (optimal reactor design), TP 12 (a tension/compression string design problem), and TP 13 (a pressure vessel design problem) are constrained engineering problems, were used to evaluate the performances of the proposed RGA-PSO and AIA-PSO algorithms. In the appendix, the objective function, constraints, boundary conditions of decision variables, and known global optimum for TPs 1−11 are described and the problem characteristics of TPs 5, 12, and 13 are further detailed.

The proposed RGA-PSO and AIA-PSO algorithms were coded in MATLAB software and executed on a Pentium D 3.0 (GHz) personal computer. Fifty independent runs were conducted to solve each test problem (TP). Numerical results were summarized, including the best, median, mean, and worst results, as well as the standard deviation (S.D.) of objective function values obtained using RGA-PSO and AIA-PSO solutions, mean computational CPU times (MCCTs), and mean absolute percentage error MAPE, as defined by MAPE=∑^50𝑠=1||𝑓𝐱^∗−𝑓𝐱^𝑠_stochastic/𝑓𝐱^∗||50×100%,𝑠=1,2,…,50,(4.1) where 𝑓(𝐱^∗)= value of the known global solution, 𝑓(𝐱^𝑠_stochastic) = values obtained from solutions of stochastic global optimization approaches (e.g., RGA-PSO and AIA-PSO algorithms).

Table 1 lists the parameter settings for the RGA-PSO and AIA-PSO algorithms, as shown in Table 1.

Table 1 

The parameter settings for the RGA-PSO and AIA-PSO algorithms.

4.1. Comparison of the Results Obtained Using the RGA-PSO and AIA-PSO Algorithms

Table 2 summarizes the numerical results obtained using the proposed RGA-PSO and AIA-PSO algorithms for TPs 1–13. Numerical results indicate that the RGA-PSO and the AIA-PSO algorithms can obtain the global minimum solution to TPs 1–11, since each MAPE% is small. Moreover, the best, median, worst, and S.D. of objective function values obtained using the RGA-PSO and AIA-PSO solutions are identical for TPs 1, 2, 3, 4, 6, 7, 8, 9, and 11. Furthermore, the worst values obtained using the AIA-PSO algorithm for TPs 5 and 13 are smaller than those obtained using the RGA-PSO algorithm. Additionally, 𝑡-test is performed for each TP, indicating that the mean values obtained using the RGA-PSO and AIA-PSO algorithms are statistically significant for TPs 5, 10, 12, and 13, since 𝑃 value is smaller than a significant level 0.05. Based on the results of 𝑡-test, the AIA-PSO algorithm yields better mean values than the RGA-PSO algorithm for TPs 5, 12, and 13, and the AIA-PSO algorithm yields worse mean value than the RGA-PSO algorithm for TP 10.

Table 2 

Numerical results of the proposed RGA-PSO and AIA-PSO algorithms for TPs 1–13.

Tables 3 and 4 list the best solutions obtained using the RGA-PSO and AIA-PSO algorithms from TPs 1–13, respectively, indicating that each constraint is satisfied (i.e., the violation of each constraint is accurate to at least five decimal places) for every TP. Tables 5 and 6 list the best parameter settings of the internal PSO algorithm obtained using the external RGA and AIA approaches, respectively.

Table 3 

The best solutions obtained using the RGA-PSO algorithm from TPs 1–13.

Table 4 

The best solutions obtained using the AIA-PSO algorithm from TPs 1–13.

Table 5 

The best parameter settings of the best solution obtained using the RGA-PSO algorithm from TPs 1–13.

Table 6 

The best parameter settings of the best solution obtained using the AIA-PSO algorithm from TPs 1–13.

4.2. Comparison of the Results for the Proposed RGA-PSO and AIA-PSO Algorithms with Those Obtained Using the Published Individual GA and AIA Approaches and Hybrid Algorithms

Table 7 compares the numerical results of the proposed RGA-PSO and AIA-PSO algorithms with those obtained using published individual GA and AIA approaches for TPs 1–4. In this table, GA-1 is a GA with a penalty function methods, as used by Michalewicz [20]. Notably, GA-2 represents a GA with a penalty function, but without any penalty parameter, as used by Deb [21]. Also, GA-3 is an RGA with a static penalty function, as developed by Wu and Chung [9]. Notably, AIA-1 is an AIA method called CLONALG, as proposed by Cruz-Cortés et al. [22]. Finally, AIA-2 is an AIA approach based on an adaptive penalty function, as developed by Wu [10]. The numerical results of GA-1, GA-2, and AIA-1 methods for solving TPs 1–4 were collected from the published literature [20–22]. Furthermore, the GA-1, GA-2, and AIA-1 approaches were executed under 350,000 objective function evaluations. To fairly compare the performances of the proposed hybrid CI algorithms and the individual GA and AIA approaches, the GA-3, AIA-2, the internal PSO algorithm of RGA-PSO method, and the internal PSO algorithm of AIA-PSO method were independently executed 50 times under 350,000 objective function evaluations for solving TPs 1–4.

Table 7 

Comparison of the results of the proposed RGA-PSO and AIA-PSO algorithms and those of the published individual GA and AIA approaches for TPs 1–4.

For solving TP 1, the median values obtained using the RGA-PSO and AIA-PSO algorithms are smaller than those obtained using the GA-1, GA-3, and AIA-2 approaches, and the worst values obtained using RGA-PSO and AIA-PSO algorithms are smaller than those obtained using the GA-1, GA-2, GA-3, AIA-1, and AIA-2 approaches. For solving TP 2, the median and worst values obtained using the RGA-PSO and AIA-PSO algorithms are smaller than those obtained using the GA-3 method. For solving TP 3, the median and worst values obtained using the RGA-PSO and AIA-PSO algorithms are smaller than those obtained using the GA-1 and GA-3 approaches. For solving TP 4, the median and worst values obtained using the RGA-PSO and AIA-PSO algorithms are smaller than those obtained using the GA-3 method, and the worst values obtained using the RGA-PSO and AIA-PSO algorithms are smaller than those obtained using the AIA-1 approach. Moreover, the GA-3 method obtained the worst MAPE% for TP 1 and TP 4. Table 8 lists the results of the 𝑡-test for the GA-3, AIA-2, RGA-PSO, and AIA-PSO methods. This table indicates that the mean values of the RGA-PSO, and AIA-PSO algorithms are not statistically significant, since 𝑃 values are larger than a significant level 0.05, and the mean values between GA-3 versus AIA-2, GA-3 versus RGA-PSO, GA-3 versus AIA-PSO, AIA-2 versus RGA-PSO, and AIA-2 versus AIA-PSO are statistically significant. According to Tables 7 and 8, the mean values obtained using the RGA-PSO and AIA-PSO algorithms are better than those of obtained using the GA-3 and AIA-1 methods for TPs 1–4.

Table 8 

Results of the 𝑡-test for TPs 1–4.

Table 9 compares the numerical results obtained using the proposed RGA-PSO and AIA-PSO algorithms and those obtained using AIA-2 and GA-3 for solving TPs 5–13. The AIA-2, GA-3, the internal PSO algorithm of the RGA-PSO approach, and the internal PSO algorithm of AIA-PSO approach were independently executed 50 times under 300,000 objective function evaluations. Table 9 shows that MAPE% obtained using the proposed RGA-PSO and AIA-PSO algorithms is close to 1%, or smaller than 1% for TPs 5–11, indicating that the proposed RGA-PSO and AIA-PSO algorithms can converge to global optimum for TPs 5–11. Moreover, the worst values obtained using the RGA-PSO and AIA-PSO algorithms are significantly smaller than those obtained using the GA-3 method for TPs 5, 6, 11, and 13. Additionally, the worst values obtained using the RGA-PSO and AIA-PSO algorithms are smaller than those obtained using the AIA-2 method for TPs 5, 6, and 13.

Table 9 

Comparison of the numerical results of the proposed RGA-PSO and AIA-PSO algorithms and those of the published individual AIA and RGA for TPs 5–13.

Table 10 summarizes the results of the 𝑡-test for TPs 5–13. According to Tables 9 and 10, the mean values of the RGA-PSO and AIA-PSO algorithms are smaller than those of the GA-3 approach for TPs 5, 6, 7, 8, 9, 10, 11, and 13. Moreover, the mean values obtained using the RGA-PSO and AIA-PSO algorithms are smaller than those of the AIA-2 approach for TPs 6, 7, 8, 10, and 12. Totally, according to Tables 7−10, the performances of the hybrid CI methods are superior to those of individual GA and AIA methods.

Table 10 

Results of t-test for TPs 5–13.

The TPs 12 and 13 have been solved by many hybrid algorithms. For instance, Huang et al. [23] presented a coevolutionary differential evolution (CDE) that integrates a coevolution mechanism and a DE approach. Zahara and Kao [24] developed a hybrid Nelder-Mead simplex search method and a PSO algorithm (NM-PSO). Table 11 compares the numerical results of the CDE, NM-PSO, RGA-PSO, and AIA-PSO methods for solving TPs 12−13. The table indicates that the best, mean, and worst values obtained using the NM-PSO method are superior to those obtained using the CDE, RGA-PSO, and AIA-PSO approaches for TP 12. Moreover, the best, mean, and worst values obtained using the AIA-PSO algorithm are better than those of the CDE, NM-PSO, and RGA-PSO algorithms.

Table 11 

Comparison of the numerical results of the proposed RGA-PSO and AIA-PSO algorithms and those of the published hybrid algorithms for TPs 12-13.

According to the No Free Lunch theorem [35], if algorithm A outperforms algorithm B on average for one class of problems, then the average performance of the former must be worse than that of the latter over the remaining problems. Therefore, it is unlikely that any unique stochastic global optimization approach exists that performs best for all CGO problems.

4.3. Summary of Results

The proposed RGA-PSO and AIA-PSO algorithms with a penalty function method have the following benefits.(1)Parameter manipulation of the internal PSO algorithm is based on the solved CGO problems. Owing to their ability to efficiently solve an UGO problem, the external RGA and AIA approaches are substituted for trial and error to manipulate the parameters (𝜒, 𝑐₁, 𝑐₂, 𝜌, and 𝑝_𝑚,PSO). (2)Besides obtaining the optimum parameter settings of the internal PSO algorithm, the RGA-PSO and AIA-PSO algorithms can yield a global optimum for a CGO problem.(3)In addition to performing better than approaches of some published individual GA and AIA approaches, the proposed RGA-PSO and AIA-PSO algorithms reduce the parametrization for the internal PSO algorithm, despite the RGA-PSO and AIA-PSO algorithms being more complex than individual GA and AIA approaches.

The proposed RGA-PSO and AIA-PSO algorithms have the following limitations. (1)The proposed RGA-PSO and AIA-PSO algorithms increase the computational CPU time, as shown in Table 2.(2)The proposed RGA-PSO and AIA-PSO algorithms are designed to solve CGO problems with continuous decision variables 𝑥_𝑛. Therefore, the proposed algorithms cannot be applied to manufacturing problems such as job shop scheduling and quadratic assignment problems (combinatorial optimization problems).

5. Conclusions

This work presents novel RGA-PSO and AIA-PSO algorithms. The synergistic power of the RGA with PSO algorithm and the AIA with PSO algorithm is also demonstrated by using 13 CGO problems. Numerical results indicate that, in addition to converging to a global minimum for each test CGO problem, the proposed RGA-PSO and AIA-PSO algorithms obtain the optimum parameter settings of the internal PSO algorithm. Moreover, the numerical results obtained using the RGA-PSO and AIA-PSO algorithms are superior to those obtained using alternative stochastic global optimization methods such as individual GA and AIA approaches. The RGA-PSO and AIA-PSO algorithms are highly promising stochastic global optimization approaches for solving CGO problems.

Appendices

A. TP 1 [20, 21]

TP 1 has ten decision variables, eight inequality constraints, and 20 boundary conditions, as follows: Minimize𝑓(𝐱)=𝑥²¹+𝑥²²+𝑥₁𝑥₂−14𝑥₁−16𝑥₂+𝑥₃−10²+4𝑥₄−5²+𝑥₅−3²+2𝑥₆−1²Subjectto𝑔₁+5𝑥²⁷+7𝑥₈−11²+2𝑥₉−10²+𝑥₁₀−7²+45𝑔(𝐱)≡−105+4𝑥₁+5𝑥₂−3𝑥₇+9𝑥₈≤0,₂𝑔(𝐱)≡10𝑥₁−8𝑥₂−17𝑥₇+2𝑥₈≤0,₃𝑔(𝐱)≡−8𝑥₁+2𝑥₂+5𝑥₉−2𝑥₁₀−12≤0,₄𝑔(𝐱)≡3𝑥₁−2²+4𝑥₂−3²+2𝑥²³−7𝑥₄−120≤0,₅𝑔(𝐱)≡5𝑥²¹+8𝑥₂+𝑥₃−6²−2𝑥₄−40≤0,₆𝑔(𝐱)≡𝑥²¹+2𝑥₂−2²−2𝑥₁𝑥₂+14𝑥₅−6𝑥₆≤0,₇𝑔(𝐱)≡0.5𝑥₁−8²+2𝑥₂−4²+3𝑥²⁵−𝑥₆−30≤0,₈(𝐱)≡−3𝑥₁+−61𝑥0₂≤+1𝑥2_𝑛𝑥₉−8²−7𝑥₁₀≤0,≤10,𝑛=1,2,…,10.(A.1) The global solution to TP 1 is as follows: 𝐱^∗=(2.11.7413909567,42,.13.63𝑓326186𝐱34,48,.97.7832982762,65,.80.9258908049,20,.89.930765594287,),_∗=24.306.(A.2)

B. TP 2 [21]

TP 2 involves five decision variables, six inequality constraints, and ten boundary conditions, as follows: SMuibnjiemctizteo𝑔𝑓₁(𝐱)=5.3578547𝑥²³+0.8356891𝑥₁𝑥₅+37.293239𝑥₁−40792.141𝑔(𝐱)≡−85.334407−0.0056858𝑥₂𝑥₅−0.0006262𝑥₁𝑥₄+0.0022053𝑥₃𝑥₅≤0,₂𝑔(𝐱)≡−6.665593+0.0056858𝑥₂𝑥₅+0.0006262𝑥₁𝑥₄−0.0022053𝑥₃𝑥₅≤0,₃𝑔(𝐱)≡9.48751−0.0071371𝑥₂𝑥₅−0.0029955𝑥₁𝑥₂−0.0021813𝑥²³≤0,₄𝑔(𝐱)≡−29.48751+0.0071371𝑥₂𝑥₅+0.0029955𝑥₁𝑥₂+0.0021813𝑥²³≤0,₅𝑔(𝐱)≡10.669039−0.0047026𝑥₃𝑥₅−0.0012547𝑥₁𝑥₃−0.0019085𝑥₃𝑥₄≤0,₆(𝐱)≡−15.69970839≤+𝑥₁0.0047026𝑥₃𝑥₅+0.0012547𝑥₁𝑥₃+0.0019085𝑥₃𝑥₄≤0,≤102,33≤𝑥₂≤45,27≤𝑥_𝑛≤45,𝑛=3,4,5.(B.1)

The global solution to TP 2 is 𝐱^∗=(78.0,𝑓33.𝐱0,29.995256,45.0,36.775812),_∗=−30665.539.(B.2)

C. TP 3 [20, 21]

TP 3 has seven decision variables, four inequality constraints, and 14 boundary conditions, as follows: Minimize𝑓(𝐱)=𝑥₁−10²+5𝑥₂−12²+𝑥⁴³+3𝑥₄−11²+10𝑥⁶⁵+7𝑥²⁶+𝑥⁴⁷−4𝑥₆𝑥₇−10𝑥₆−8𝑥₇Subjectto𝑔𝑔₁(𝐱)≡−127+2𝑥²¹+3𝑥⁴²+𝑥₃+4𝑥²⁴+5𝑥₅≤0,₂𝑔(𝐱)≡−282+7𝑥₁+3𝑥₂+10𝑥²³+𝑥₄−𝑥₅≤0,₃𝑔(𝐱)≡−196+23𝑥₁+𝑥²²+6𝑥²⁶−8𝑥₇≤0,₄(𝐱)≡4𝑥²¹+−𝑥²²1−0≤3𝑥𝑥_1𝑛𝑥₂+2𝑥²³+5𝑥₆−11𝑥₇≤0,≤10,𝑛=1,2,…,7.(C.1) The global solution to TP 3 is 𝐱^∗=(2.330499,1.951372,−0.477𝑓54𝐱14,4.365726,−0.6244870,1.038131,1.594227),_∗=680.630.(C.2)

D. TP 4 [20, 21]

TP 4 involves 13 decision variables, nine inequality constraints, and 26 boundary conditions, as follows: Minimize𝑓(𝐱)=5⁴_𝑛=1𝑥_𝑛−5⁴_𝑛=1𝑥^2𝑛−¹³_𝑛=5𝑥_𝑛Subjectto𝑔𝑔₁(𝐱)≡2𝑥₁+2𝑥₂+𝑥₁₀+𝑥₁₁−10≤0,₂𝑔(𝐱)≡2𝑥₁+2𝑥₃+𝑥₁₀+𝑥₁₂−10≤0,₃𝑔(𝐱)≡2𝑥₂+2𝑥₃+𝑥₁₁+𝑥₁₂−10≤0,₄𝑔(𝐱)≡−8𝑥₁+𝑥₁₀≤0,₅𝑔(𝐱)≡−8𝑥₂+𝑥₁₁≤0,₆𝑔(𝐱)≡−8𝑥₃+𝑥₁₂≤0,₇𝑔(𝐱)≡−2𝑥₄−𝑥₅+𝑥₁₀≤0,₈𝑔(𝐱)≡−2𝑥₆−𝑥₇+𝑥₁₁≤0,₉(𝐱)≡−02≤𝑥𝑥_8𝑛−𝑥₉+𝑥₁₂≤0,0≤𝑥_𝑛≤1,𝑛=1,2,…,9,0≤𝑥₁₃≤100,𝑛=10,11,12,≤1.(D.1) The global solution to TP 4 is 𝐱^∗=(1,1,1,1,1,1,1,1,1,3,3,3,1),𝑓𝐱^∗=−15.(D.2)

E. TP 5 (Alkylation Process Design Problem in Chemical Engineering) [1]

TP 5 has seven decision variables subject to 12 nonconvex, two linear, and 14 boundary constraints. The objective function is to improve the octane number of some olefin feed by reacting it with isobutane in the presence of acid. The decision variables 𝑥_𝑛 are olefin feed rate (barrels/day) 𝑥₁, acid addition rate (thousands of pounds/day) 𝑥₂, alkylate yield (barrels/day) 𝑥₃, acid strength 𝑥₄, motor octane number 𝑥₅, external isobutane-to-olefin ration 𝑥₆, and F-4 performance number 𝑥₇: sM.ti.nimize𝑔𝑓₁(𝐱)=𝜔₁𝑥₁+𝜔₂𝑥₁𝑥₆+𝜔₃𝑥₃+𝜔₄𝑥₂+𝜔₅−𝜔₆𝑥₃𝑥₅𝑔(𝐱)=𝜔₇𝑥²⁶+𝜔₈𝑥¹⁻¹𝑥₃−𝜔₉𝑥₆≤1,₂𝑔(𝐱)=𝜔₁₀𝑥₁𝑥³⁻¹+𝜔₁₁𝑥₁𝑥³⁻¹𝑥₆−𝜔₁₂𝑥₁𝑥³⁻¹𝑥²⁶≤1,₃𝑔(𝐱)=𝜔₁₃𝑥²⁶+𝜔₁₄𝑥₅−𝜔₁₅𝑥₄−𝜔₁₆𝑥₆≤1,₄𝑔(𝐱)=𝜔₁₇𝑥⁵⁻¹+𝜔₁₈𝑥⁵⁻¹𝑥₆+𝜔₁₉𝑥₄𝑥⁵⁻¹−𝜔₂₀𝑥⁵⁻¹𝑥²⁶≤1,₅𝑔(𝐱)=𝜔₂₁𝑥₇+𝜔₂₂𝑥₂𝑥³⁻¹𝑥⁴⁻¹−𝜔₂₃𝑥₂𝑥³⁻¹≤1,₆𝑔(𝐱)=𝜔₂₄𝑥⁷⁻¹+𝜔₂₅𝑥₂𝑥³⁻¹𝑥⁷⁻¹−𝜔₂₆𝑥₂𝑥³⁻¹𝑥⁴⁻¹𝑥⁷⁻¹≤1,₇𝑔(𝐱)=𝜔₂₇𝑥⁵⁻¹+𝜔₂₈𝑥⁵⁻¹𝑥₇≤1,₈𝑔(𝐱)=𝜔₂₉𝑥₅−𝜔₃₀𝑥₇≤1,₉𝑔(𝐱)=𝜔₃₁𝑥₃−𝜔₃₂𝑥₁≤1,₁₀𝑔(𝐱)=𝜔₃₃𝑥₁𝑥³⁻¹+𝜔₃₄𝑥³⁻¹≤1,₁₁𝑔(𝐱)=𝜔₃₅𝑥₂𝑥³⁻¹𝑥⁴⁻¹−𝜔₃₆𝑥₂𝑥³⁻¹≤1,₁₂𝑔(𝐱)=𝜔₃₇𝑥₄+𝜔₃₈𝑥²⁻¹𝑥₃𝑥₄≤1,₁₃𝑔(𝐱)=𝜔₃₉𝑥₁𝑥₆+𝜔₄₀𝑥₁−𝜔₄₁𝑥₃≤1,₁₄(𝐱)=𝜔15₄₂0𝑥0≤¹⁻¹𝑥₃𝑥₁+𝜔₄₃𝑥¹⁻¹−𝜔₄₄𝑥₆≤1,85≤𝑥₄≤2000,1≤𝑥₂≤120,3000≤𝑥₃≤3500,≤93,90≤𝑥₅≤95,3≤𝑥₆≤12,145≤𝑥₇≤162,(E.1) where 𝜔_𝑙(𝑙=1,2,…,44) denotes positive parameters given in Table 12. The global solution to TP 5 is 𝐱^∗=(1698.18,53.66,3031.30,90.11,10.50,153.53),𝑓𝐱^∗=1227.1978.(E.2)

Table 12 

Coefficients for TP 5.

F. TP 6 (Optimal Reactor Design Problem) [1]

TP 6 contains eight decision variables subject to four nonconvex inequality constraints and 16 boundary conditions, as follows: Ms.ti.nimize𝑔𝑓₁(𝐱)=0.4𝑥^10.67𝑥^7−0.67+0.4𝑥^20.67𝑥^8−0.67+10−𝑥₁−𝑥₂𝑔(𝐱)=0.0588𝑥₅𝑥₇+0.1𝑥₁≤1,₂𝑔(𝐱)=0.0588𝑥₆𝑥₈+0.1𝑥₁+0.1𝑥₂≤1,₃𝑔(𝐱)=4𝑥₃𝑥⁵⁻¹+2𝑥^3−0.71𝑥⁵⁻¹+0.0588𝑥^3−1.3𝑥₇≤1.₄(𝐱)=4𝑥₄𝑥⁶⁻¹+2𝑥^4−0.71𝑥⁶⁻¹+0.0588𝑥^4−1.3𝑥₈≤1,0.1≤𝑥_𝑛≤10,𝑛=1,2,…,8.(F.1) The global solution to TP 6 is 𝐱^∗=(6.4747,2.2340,0.6671,0.5957,5.9310,5.5271,1.0108,0.4004),𝑓𝐱^∗=3.9511.(F.2)

G. TP 7 [1]

TP 7 has four decision variables, two nonconvex inequality constraints, and eight boundary conditions, as follows: sM.ti.nimize𝑔𝑓₁(𝐱)=−𝑥₁+0.4𝑥^10.67𝑥^3−0.67𝑔(𝐱)=0.05882𝑥₃𝑥₄+0.1𝑥₁≤1₂(𝐱)=4𝑥₂𝑥⁴⁻¹+2𝑥^2−0.71𝑥⁴⁻¹+0.05882𝑥^2−1.3𝑥₃≤1,0.1≤𝑥₁,𝑥₂,𝑥₃,𝑥₄≤10.(G.1)

The global solution to TP 7 is 𝐱^∗=(8.1267,0.6154,0.5650,5.6368),𝑓𝐱^∗=−5.7398.(G.2)

H. TP 8 [1]

TP 8 contains three decision variables subject to one nonconvex inequality constraint and six boundary conditions, as follows: Ms.ti.nimize𝑔𝑓₁(𝐱)=0.5𝑥₁𝑥²⁻¹−𝑥₁−5𝑥²⁻¹(𝐱)=0.01𝑥₂𝑥³⁻¹+0.01𝑥₁+0.0005𝑥₁𝑥₃≤1,1≤𝑥_𝑛≤100,𝑛=1,2,3.(H.1) The global solution to TP 8 is 𝐱^∗=(88.2890,7.7737,1.3120),𝑓𝐱^∗=−83.2540.(H.2)

I. TP 9 [1]

TP 9 contains eight decision variables subject to four nonconvex inequality constraints and 16 boundary conditions, as follows: Ms.ti.nimize𝑔𝑓₁(𝐱)=−𝑥₁−𝑥₅+0.4𝑥^10.67𝑥^3−0.67+0.4𝑥^50.67𝑥^7−0.67𝑔(𝐱)=0.05882𝑥₃𝑥₄+0.1𝑥₁≤1,₂𝑔(𝐱)=0.05882𝑥₇𝑥₈+0.1𝑥₁+0.1𝑥₅≤1,₃𝑔(𝐱)=4𝑥₂𝑥⁴⁻¹+2𝑥^2−0.71𝑥⁴⁻¹+0.05882𝑥^2−1.3𝑥₃≤1,₄(𝐱)=4𝑥₆𝑥⁸⁻¹+2𝑥^6−0.71𝑥⁸⁻¹+0.05882𝑥^6−1.3𝑥₇≤1,0.01≤𝑥_𝑛≤10,𝑛=1,2,…,8.(I.1) The global solution to TP 9 is 𝐱^∗=(6.4225,0.6686,1.0239,5.9399,2.2673,0.5960,0.4029,5.5288),𝑓𝐱^∗=−6.0482.(I.2)

J. TP 10 [34]

TP 10 contains three decision variables subject to one nonconvex inequality constraint and six boundary conditions, as follows: sM.ti.nimize𝑔𝑓₁(𝐱)=5𝑥₁+50000𝑥¹⁻¹+20𝑥₂+72000𝑥²⁻¹+10𝑥₃+144000𝑥³⁻¹(𝐱)=4𝑥¹⁻¹+32𝑥²⁻¹+120𝑥³⁻¹≤1,1≤𝑥_𝑛≤1000,𝑛=1,2,3.(J.1) The global solution to TP 10 is 𝐱^∗=(107.4,84.9,204.5),𝑓𝐱^∗=6300.(J.2)

K. TP 11 [1, 34]

TP 11 involves five decision variables, six inequality constraints, and ten boundary conditions, as follows: sM.ti.nimize𝑔𝑔₀₁(𝐱)=5.3578𝑥²³+0.8357𝑥₁𝑥₅+37.2392𝑥₁𝑔(𝐱)=0.00002584𝑥₃𝑥₅−0.00006663𝑥₂𝑥₅−0.0000734𝑥₁𝑥₄≤1,₂𝑔(𝐱)=0.000853007𝑥₂𝑥₅+0.00009395𝑥₁𝑥₄−0.00033085𝑥₃𝑥₅≤1,₄𝑔(𝐱)=0.00024186𝑥₂𝑥₅+0.00010159𝑥₁𝑥₂+0.00007379𝑥²³≤1,₃𝑔(𝐱)=1330.3294𝑥²⁻¹𝑥⁵⁻¹−0.42𝑥₁𝑥⁵⁻¹−0.30586𝑥²⁻¹𝑥²³𝑥⁵⁻¹≤1,₅𝑔(𝐱)=2275.1327𝑥³⁻¹𝑥⁵⁻¹−0.2668𝑥₁𝑥⁵⁻¹−0.40584𝑥₄𝑥⁵⁻¹≤1,₆(𝐱)=07.080≤02𝑥9₁955𝑥₃𝑥₅+0.00007992𝑥₁𝑥₃+0.00012157𝑥₃𝑥₄≤1,≤102,33≤𝑥₂≤45,27≤𝑥₃≤45,27≤𝑥₄≤45,27≤𝑥₅≤45.(K.1)

The global solution to TP 11 is 𝐱^∗=(78.0,33.0,29.998,45.0,36.7673),𝑓𝐱^∗=10122.6964.(K.2)

L. TP 12 (a Tension/Compression String Design Problem) [23]

TP 12 involves three decision variables, six inequality constraints, and six boundary conditions. This problem is taken from Huang et al. [23]. This problem attempts to minimize the weight (i.e., 𝑓(𝐱)) of a tension/compression spring subject to constraints on minimum deflection, shear stress, and surge frequency. The design variables are the mean coil diameter 𝑥₂, wire diameter 𝑥₁, and number of active coils 𝑥₃: Minimize𝑓(𝐱)=𝑥₃+2𝑥₂𝑥²¹s.t.𝑔₁(𝐱)=1−𝑥³²𝑥₃71785𝑥⁴¹𝑔≤0,₂(𝐱)=4𝑥²²−𝑥₁𝑥₂12566𝑥₂𝑥³¹−𝑥⁴¹+51018𝑥²¹𝑔−1≤0,₃(𝐱)=1−140.45𝑥¹𝑥²²𝑥₃𝑔≤0,₄(𝐱)=𝑥¹+𝑥₂1.5−1≤0,0.05≤𝑥¹≤2,0.25≤𝑥₂≤1.3,2≤𝑥₃≤15.(L.1)

M. TP 13 (Pressure Vessel Design Problem) [23]

TP 13 involves four decision variables, four inequality constraints, and eight boundary conditions. This problem attempts to minimize the total cost (𝑓(𝐱)), including cost of materials forming and welding. A cylindrical vessel is capped at both ends by hemispherical heads. Four design variables exist: thickness of the shell 𝑥₁, thickness of the head 𝑥₂, inner radius 𝑥₃, and length of the cylindrical section of the vessel, excluding the head 𝑥₄: Minimize𝑓(𝐱)=0.6224𝑥₁𝑥₃𝑥₄+1.7781𝑥₂𝑥²³+3.1661𝑥²¹𝑥₄+19.84𝑥²¹𝑥₃s.t.𝑔𝑔₁(𝐱)=−𝑥₁+0.0193𝑥₃≤0,₂𝑔(𝐱)=−𝑥₂+0.00954𝑥₃≤0,₃(𝐱)=−𝜋𝑥³²𝑥₄−43𝜋𝑥³³𝑔+1296000≤0,₄(𝐱)=𝑥₄−240≤0,0≤𝑥₁≤100,0≤𝑥₂≤100,10≤𝑥₃≤200,10≤𝑥₃≤200.(M.1)

Acknowledgment

The author would like to thank the National Science Council of the Republic of China, Taiwan, for financially supporting this research under Contract no. NSC 100-2622-E-262-006-CC3.