A differential evolution approach for solving constrained min–max optimization problems

Many problems occurring in engineering can be formulated as min–max optimization problems, for instance, in game theory, robust optimal control and many others. Min–max problems are considered difficult to solve, specially constrained min–max optimization problems. Approaches using co-evolutionary algorithms have successfully been used to solve min–max optimization problems without constraints. We propose a novel differential evolution approach consisting of three populations with a scheme of copying individuals for solving constrained min–max problems. Promising results have been obtained showing the suitability of the approach.     

Self-adaptive differential evolution algorithm with α-constrained-domination principle for constrained multi-objective optimization

Real-world problems are inherently constrained optimization problems often with multiple conflicting objectives. To solve such constrained multi-objective problems effectively, in this paper, we put forward a new approach which integrates self-adaptive differential evolution algorithm with α-constrained-domination principle, named SADE-αCD. In SADE-αCD, the trial vector generation strategies and the DE parameters are gradually self-adjusted adaptively based on the knowledge learnt from the previous searches in generating improved solutions. Furthermore, by incorporating domination principle into α-constrained method, α-constrained-domination principle is proposed to handle constraints in multi-objective problems. The advantageous performance of SADE-αCD is validated by comparisons with non-dominated sorting genetic algorithm-II, a representative of state-of-the-art in multi-objective evolutionary algorithms, and constrained multi-objective differential evolution, over fourteen test problems and four well-known constrained multi-objective engineering design problems. The performance indicators show that SADE-αCD is an effective approach to solving constrained multi-objective problems, which is basically enabled by the integration of self-adaptive strategies and α-constrained-domination principle.     

A co-evolutionary differential evolution algorithm for solving min–max optimization problems implemented on GPU using C-CUDA

Several areas of knowledge are being benefited with the reduction of the computing time by using the technology of graphics processing units (GPU) and the compute unified device architecture (CUDA) platform. In case of evolutionary algorithms, which are inherently parallel, this technology may be advantageous for running experiments demanding high computing time. In this paper, we provide an implementation of a co-evolutionary differential evolution (DE) algorithm in C-CUDA for solving min–max problems. The algorithm was tested on a suite of well-known benchmark optimization problems and the computing time has been compared with the same algorithm implemented in C. Results demonstrate that the computing time can significantly be reduced and scalability is improved using C-CUDA. As far as we know, this is the first implementation of a co-evolutionary DE algorithm in C-CUDA.     

Hybrid whale optimization algorithm enhanced with Lévy flight and differential evolution for job shop scheduling problems

The job shop scheduling problem (JSSP) has been a hot issue in manufacturing. For the past few decades, scholars have been attracted to research JSSP and proposed many novel meta-heuristic algorithms to solve it. Whale optimization algorithm (WOA) is such a novel meta-heuristic algorithm and has been proven to be efficient in solving real-world optimization problems in the literature. This paper proposes a hybrid WOA enhanced with Lévy flight and differential evolution (WOA-LFDE) to solve JSSP. By changing the expression of Lévy flight and DE search strategy, Lévy flight enhances the abilities of global search and convergence of WOA in iteration, while DE algorithm improves the exploitation and local search capabilities of WOA and keeps the diversity of solutions to escape local optima. It is then applied to solve 88 JSSP benchmark instances and compared with other state-of-art algorithms. The experimental results and statistical analysis show that the proposed algorithm has superior performance over contesting algorithms.     

An improved firefly algorithm for numerical optimization problems and it’s application in constrained optimization

Engineering with Computers - Metaheuristic algorithms are successful methods of optimization. The firefly algorithm is one of the known metaheuristic algorithms used in a variety of applications....     

Constrained min–max optimization via the improved constraint-activated differential evolution with escape vectors

In system design, the best system designed under a simple experimental environment may not be suitable for application in real world if dramatic changes caused by uncertainties contained in the real world are considered. To deal with the problem caused by uncertainties, designers should try their best to get the most robust solution. The most robust solution can be obtained by constrained min–max optimization algorithms. In this paper, the scheme of generating escape vectors has been proposed to solve the problem of premature convergence of differential evolution. After applying the proposed scheme to the constrained min–max optimization algorithm, the performance of the algorithm could be greatly improved. To evaluate the performance of constrained min–max optimization algorithms, more complex test problems have also been proposed in this paper. Experimental results show that the improved constrained min–max optimization algorithm is able to achieve a quite satisfied success rate on all considered test problems under limited accuracy.     

Necessary and sufficient conditions for the inequality constrained optimization problem using directional derivatives †

The directional derivatives approach is used in the solution, of the classical optimization problem with inequality constraints. Necessary and second-order sufficient conditions are obtained. A procodure to generate all the stationary points is developed, and the further evaluation of those points up to sufficient conditions is presented. The general non-linear programming problem is discussed and an example illustrating the technique is presented.     

A hybridization of teaching–learning-based optimization and differential evolution for chaotic time series prediction

Chaotic time series prediction problems have some very interesting properties and their prediction has received increasing interest in the recent years. Prediction of chaotic time series based on the phase space reconstruction theory has been applied in many research fields. It is well known that prediction of a chaotic system is a nonlinear, multivariable and multimodal optimization problem for which global optimization techniques are required in order to avoid local optima. In this paper, a new hybrid algorithm named teaching–learning-based optimization (TLBO)–differential evolution (DE), which integrates TLBO and DE, is proposed to solve chaotic time series prediction. DE is incorporated into update the previous best positions of individuals to force TLBO jump out of stagnation, because of its strong searching ability. The proposed hybrid algorithm speeds up the convergence and improves the algorithm’s performance. To demonstrate the effectiveness of our approaches, ten benchmark functions and three typical chaotic nonlinear time series prediction problems are used for simulating. Conducted experiments indicate that the TLBO–DE performs significantly better than, or at least comparable to, TLBO and some other algorithms.     

Parameterized eigensolution technique for solving constrained least squares problems ∗

This paper aims to introduce an algorithm for solving large scale least squares problems subject to quadratic inequality constraints. The algorithm recasts the least squares problem in terms of a parameterized eigenproblem. A variant of k-step Arnoldi method is determined to be well suited for computing the parameterized eigenpair. A two-point interpolating scheme is developed for updating the parameter. A local convergence theory for this algorithm is presented. It is shown that this algorithm is superlinearly convergent.     

Subgradient-based feedback neural networks for non-differentiable convex optimization problems

1 Introduction Optimization problems arise in a broad variety of scientific and engineering applica- tions. For many practice engineering applications problems, the real-time solutions of optimization problems are mostly required. One possible and very pr…     

Differential evolution and quantum-inquired differential evolution for evolving Takagi–Sugeno fuzzy models

The differential evolution (DE) is a global optimization algorithm to solve numerical optimization problems. Recently the quantum-inquired differential evolution (QDE) has been proposed for binary optimization. This paper proposes DE/QDE to learn the Takagi–Sugeno (T–S) fuzzy model. DE/QDE can simultaneously optimize the structure and the parameters of the model. Moreover a new encoding scheme is given to allow DE/QDE to be easily performed. The two benchmark problems are used to validate the performance of DE/QDE. Compared to some existing methods, DE/QDE shows the competitive performance in terms of accuracy.     

Hybrid differential evolution and Nelder�CMead algorithm with re-optimization

Nonlinear optimization algorithms could be divided into local exploitation methods such as Nelder–Mead (NM) algorithm and global exploration ones, such as differential evolution (DE). The former searches fast yet could be easily trapped by local optimum, whereas the latter possesses better convergence quality. This paper proposes hybrid differential evolution and NM algorithm with re-optimization, called as DE-NMR. At first a modified NM, called NMR is presented. It re-optimizes from the optimum point at the first time and thus being able to jump out of local optimum, exhibits better properties than NM. Then, NMR is combined with DE. To deal with equal constraints, adaptive penalty function method is adopted in DE-NMR, which relaxes equal constraints into unequal constrained functions with an adaptive relaxation parameter that varies with iteration. Benchmark optimization problems as well as engineering design problems are used to experiment the performance of DE-NMR, with the number of function evaluation times being employed as the main index of measuring convergence speed, and objective function values as the main index of optimum’s quality. Non-parametric tests are employed in comparing results with other global optimization algorithms. Results illustrate the fast convergence speed of DE-NMR.     

A novel hybrid PSO–GWO algorithm for optimization problems

Engineering with Computers - In this study, we propose a new hybrid algorithm fusing the exploitation ability of the particle swarm optimization (PSO) with the exploration ability of the grey wolf...     

The minimal time control of multi-system problems with energy constrained inputs†

This paper presents an. application, of functional analysis to the minimal?time control problem where two or more energy constrained systems of the same order must be taken to an initially unknown common point in state space. Two different types of problems with different forms of the energy constraint are described in detail. The development depends on the fact that the reachable region of a system with an energy constraint on the input vector is an ellipsoid.     

Non-linear programming using least pth optimization with extrapolation†

We present a general approach for solving minimax and non-linear programming problems through a sequence of least pth approximations with extrapolation. Several numerical examples illustrate the effectiveness of the present approach. A comparison with the woll-known SUMT method of Fiaeco and McCormick is made under the same conditions and employing Fletcher's quasi.Newton programme.     

Chebyshev series approximation for solving differential–algebraic equations (DAEs)

The numerical solution of differential–algebraic equations (DAEs) using the Chebyshev series approximation is considered in this article. Two different problems are solved using the Chebyshev series approximation and the solutions are compared with the exact solutions. First, we calculate the power series of a given equation system and then transform it into Chebyshev series form, which gives an arbitrary order for solving the DAE numerically.     

Solving continuous optimization problems using the ımproved Jaya algorithm (IJaya)

Jaya algorithm is one of the heuristic algorithms developed in recent years. The most important difference from other heuristic algorithms is that it updates its position according to its best and worst position. In addition to its simplicity, there is no algorithm-specific parameter. Because of these advantages, it has been preferred by researchers for problem-solving in the literature. In this study, the random walk phase of the original Jaya algorithm is developed and the Improved Jaya Algorithm (IJaya) is proposed. IJaya has been tested for success in eighteen classic benchmark test functions. Although the performance of the original Jaya algorithm has been tested at low dimensions in the literature, its success in large sizes has not been tested. In this study, IJaya's success in 10, 20, 30, 100, 500, and 1000 dimensions was examined. Also, the success of IJaya was tested in different population sizes. It has been proven that IJaya's performance has increased with the tests performed. Test results show that IJaya displays good performance and can be used as an alternative method for constrained optimization. In addition, three different engineering design problems were tested in different population sizes to demonstrate the achievements of Jaya and IJaya. According to the results, IJaya can be used as an optimization algorithm in the literature for continuous optimization and large-scale optimization problems.