Efficient parameter-free adaptive penalty method with balancing the objective function value and the constraint violation

A fast constrained state transition algorithm

Any evolutionary algorithm tends to end up in a local optimum. A new approach based on an evolutionary algorithm named as Diversity Driven Multi-Parent Evolutionary Algorithm with Adaptive non-uniform mutation is presented. In the proposed algorithm, Non-uniform mutation is used to maintain diversity in the explored solutions. Fitness variance, which signifies solution space aggregation, is used to detect the premature convergence of the population to a local optimum. The term multi-parent is used in the context of more than two parents participating in crossover operation. After multi-parent selection for cross-over to generate new solutions, the non-uniform adaptive mutation is performed, which in turn is triggered by the diminishing value of fitness variance of candidate solutions and pushes solutions out of local optimum. Hence, it can be said that the algorithm is driven by the diversity of the population and overcomes the tendency of evolutionary algorithms to stuck in local optimum. The performance of this algorithm is tested on 23 basic benchmarks, CEC05 functions, and CEC17 functions. As CEC17 benchmark functions include constraint problems, a constraint-handling technique is proposed based on the fuzzy set theory. In the proposed constrained handling strategy, constraint violation is also taken as another objective along with the main objective. The decision to accept or discard the solution is based on the fuzzy set theory. The values of constraint violation and objective function are calculated and fuzzified by calculating membership values by considering the main objective and constraint violation as triangular fuzzy functions. The best solutions are selected based on cardinal priority ranking. The obtained results from the proposed algorithm are compared with the results available in the literature. The result indicates that this algorithm is competitive, even with a smaller number of function evaluations.

How to balance diversity and convergence as well as constraint violation and objective function in Constrained Optimization Problems (COPs) is an important and key issue when designing a new Constrained Optimization Evolutionary Algorithms (COEAs). C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> oDE set a good example on this point, and obtains promising results. In C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> oDE, the two Constraint Handling Techniques (CHTs) are adopted fixed in two phases. To verify whether there are some other more efficient combinations, Stochastic Ranking (SR) is adopted to balance the combination. Experimental results show a good performance, which also give some inspiration for other research.

The relationship between Constraint Handling Techniques (CHTs) and Constrained Optimization Problems (COPs) is an important issue as it gives some direction on deciding when and how to use these CHTs effectively. In the previous studies, the relationship between problems and algorithms as well as the CHTs are studied from the qualitative and quantitative aspects. This paper tries to study the relationship between COPs and two simple CHTs, i.e., Deb’s feasibility-based rule and multiobjective optimization technique, with a new set of benchmark functions adopted. The experimental results show the importance of the classification of problems, which is a basis for related research.

To improve distribution and convergence of the obtained solution set in constrained multi-objective optimization problems, this paper presents an adaptive ε-constraint multi-objective evolutionary algorithm based on decomposition and differential evolution (ε-MOEA/D-DE). First, an adaptive ε-constraint strategy based on both evolution generation and constraint violation is designed to make better use of excellent evolution individuals and improve population diversity. Then, an adaptive differential evolution (DE) mutation strategy with full utilization of infeasible individuals is proposed to increase search efficiency and avoid falling into the local optimum. Finally, a replacement mechanism is suggested to take advantage of the infeasible individuals in the population with better objective function values and constraint violation degree, and thus both diversity and convergence are well coordinated. A comparative experiment with four other excellent constrained multi-objective algorithms was implemented on standard constrained multi-objective optimization problems (CF series), and the results showed that the diversity and convergence of our algorithm were both improved.

Generally, constraint-handling techniques are designed for evolutionary algorithms to solve Constrained Multiobjective Optimization Problems (CMOPs). Most Multiojective Particle Swarm Optimization (MOPSO) designs adopt these existing constraint-handling techniques to deal with CMOPs. In this chapter, the authors present a constrained MOPSO in which the information related to particles' infeasibility and feasibility status is utilized effectively to guide the particles to search for feasible solutions and to improve the quality of the optimal solution found. The updating of personal best archive is based on the particles' Pareto ranks and their constraint violations. The infeasible global best archive is adopted to store infeasible nondominated solutions. The acceleration constants are adjusted depending on the personal bests' and selected global bests' infeasibility and feasibility statuses. The personal bests' feasibility statuses are integrated to estimate the mutation rate in the mutation procedure. The simulation results indicate that the proposed constrained MOPSO is highly competitive in solving selected benchmark problems.

Currently, rehabilitation systems with closed kinematic chain mechanisms are low-cost alternatives for treatment and health care. In designing these systems, the dimensional synthesis is commonly stated as a constrained optimization problem to achieve repetitive rehabilitation movements, and metaheuristic algorithms for constrained problems are promising methods for searching solutions in the complex search space. The Constraint Handling Techniques (CHTs) in metaheuristic algorithms have different capacities to explore and exploit the search space. However, the study of the relationship in the CHT performance of the mechanism dimensional synthesis for rehabilitation systems has not been addressed, resulting in an important gap in the literature of such problems. In this paper, we present a comparative empirical study to investigate the influence of four CHTs (penalty function, feasibility rules, stochastic-ranking, and ϵ-constraint) on the performance of ten representative algorithms that have been reported in the literature for solving mechanism synthesis for rehabilitation (four-bar linkage, eight-bar linkage, and cam-linkage mechanisms). The study involves analysis of the overall performance, six performance metrics, and evaluation of the obtained mechanism. This identified that feasibility rules usually led to efficient optimization for most analyzed algorithms and presented more consistency of the obtained results in these kinds of problems.

A trust region filter method for general non-linear programming

A priori penalty factor determination for (trimmed) NURBS-based shells with Dirichlet and coupling constraints in isogeometric analysis

In this paper, we consider a multi-fidelity optimization under two types of constraints: prior constraints and posterior constraints. The prior constraints are prerequisite to execution of the simulation that computes the objective function value and the posterior constraint violation values, and are evaluated independently from the simulation with significantly lower computational time than the simulation. We have several simulators that approximately simulate the objective and constraint violation values with different trade-offs between accuracy and computational time. We propose an approach to solve the described constrained optimization problem with as little computational time as possible by utilizing multiple simulators. Based on a covariance matrix adaptation evolution strategy, we combines three algorithmic components: prior constraint handling technique, posterior constraint handling technique, and adaptive simulator selection technique for multi-fidelity optimization. We apply the proposed approach to a compliance minimization problem and show a promising convergence behavior.

We propose and analyze a class of penalty-function-free nonmonotone trust-region methods for nonlinear equality constrained optimization problems. The algorithmic framework yields global convergence without using a merit function and allows nonmonotonicity independently for both, the constraint violation and the value of the Lagrangian function. Similar to the Byrd–Omojokun class of algorithms, each step is composed of a quasi-normal and a tangential step. Both steps are required to satisfy a decrease condition for their respective trust-region subproblems. The proposed mechanism for accepting steps combines nonmonotone decrease conditions on the constraint violation and/or the Lagrangian function, which leads to a flexibility and acceptance behavior comparable to filter-based methods. We establish the global convergence of the method. Furthermore, transition to quadratic local convergence is proved. Numerical tests are presented that confirm the robustness and efficiency of the approach.

Due to significant improvements in the performance of computers, Genetic Algorithms are actively applied to actual engineering problems. Most engineering problems are constrained optimization problems that are used to optimize objective functions under many constraints. The penalty method is well-known solution to such constrained optimization problems. However, the benchmarks of constrained optimization problems have only a small number of constraints. Thus, the effectiveness of the penalty method has not been investigated in numerous constrained problems. In many constrained optimization problems, penalty method has a risk whereby all constraints cannot be satisfied. This paper proposes the stepwise satisfaction method of constraints to satisfy conditions that involve many constraints. In the proposed method, the priority of constraints to be satisfied is defined based on the initial population and the objective functions are optimized after the satisfaction of all constraints. Furthermore, this paper studies the effects of classifying constraints into difficult and easy ones as well as combining the proposed method with the penalty method. In the experiment, the performance of the proposed method and the penalty method was compared in two problems with more than 50 constraints.

Constrained optimization is a computationally difficult task, particularly if the constraint functions are nonlinear and non-convex. As a generic classical approach, the penalty function approach is a popular methodology which degrades the objective function value by adding a penalty proportional to the constraint violation. However, the penalty function approach has been criticized for its sensitivity to the associated penalty parameters. Since its inception, evolutionary algorithms have been modified in various ways to solve constrained optimization problems. Of them, the recent use of a bi-objective evolutionary algorithm in which the minimization of the constraint violation is included as an additional objective has received significant attention. In this article, a combination of a bi-objective evolutionary approach with the classical penalty function methodology is proposed, in a manner complementary to each other. The evolutionary approach provides an appropriate estimate of the penalty parameter, while the solution of an unconstrained penalized function by a classical method induces a convergence property to the overall hybrid algorithm. The working of the procedure on a number of standard numerical test problems and an engineering design problem is demonstrated. In most cases, the proposed hybrid methodology is observed to take one or more orders of magnitude fewer function evaluations to find the constrained minimum solution accurately than some of the best reported existing methodologies.

Most of real-world optimization problems correspond to constrained multi-objective optimization problems (CMOPs). Multi-objective evolutionary algorithms (MOEAs) are useful to find out diverse Pareto-optimal solutions in CMOPs, and has been applied in various engineering fields. In order to expand the range of application of MOEAs further, we propose an improved evolutionary algorithm with parallel evaluation strategy (EAPES). In EAPES, feasible solutions and infeasible solution are separately stored in different populations, and infeasible solutions are evaluated in an unusual manner where not only feasible solutions but also useful infeasible solutions are used as parents to reproduce the populations for the next generation. Infeasible solutions are ranked based on a scalarizing function determined by objective function values and a total constraint violation value. This paper investigates the performance of the proposed EAPES to search for Pareto-optimal solutions compared to NSGA-II and the previous EAPES. The proposed EAPES with a well-tuned parameter is most capable to explore Pareto-optimal solutions with good diversity, spread, and convergence to the true Pareto front. The proposed EAPES assigns bad rank to the infeasible solutions that exist away from the true Pareto front, and does not store such solutions. Thus the proposed EAPES exhibits a higher searching capability than the previous EAPES by evaluating infeasible solutions in an appropriate balance between objective functions and total constraint violation. In addition, the results suggest that the proposed EAPES may exhibit high solution search performance even in more difficult CMOPs, which have more objective functions and/or more constraints.

An evolutionary algorithm with directed weights for constrained multi-objective optimization