ABS methods for continuous and integer linear equations and optimization

ABS methods are a large class of algorithms for solving continuous and integer linear algebraic equations, and nonlinear continuous algebraic equations, with applications to optimization. Recent work by Chinese researchers led by Zunquan Xia has extended these methods also to stochastic, fuzzy and infinite systems, extensions not considered here. The work on ABS methods began almost thirty years. It involved an international collaboration of mathematicians especially from Hungary, England, China and Iran, coordinated by the university of Bergamo. The ABS method are based on the rank reducing matrix update due to Egerváry and can be considered as the most fruitful extension of such technique. They have led to unification of classes of methods for several problems. Moreover they have produced some special algorithms with better complexity than the standard methods. For the linear integer case they have provided the most general polynomial time class of algorithms so far known; such algorithms have been extended to other integer problems, as linear inequalities and LP problems, in over a dozen papers written by Iranian mathematicians led by Nezam Mahdavi-Amiri. ABS methods can be implemented generally in a stable way, techniques existing to enhance their accuracy. Extensive numerical experiments have shown that they can outperform standard methods in several problems. Here we provide a review of their main properties, for linear systems and optimization. We also give the results of numerical experiments on some linear systems. This paper is dedicated to Professor Egerváry, developer of the rank reducing matrix update, that led to ABS methods.     

A particle swarm optimization-based hybrid algorithm for minimum concave cost network flow problems

Traditionally, minimum cost transshipment problems have been simplified as linear cost problems, which are not practical in real applications. Some advanced local search algorithms have been developed to solve concave cost bipartite network problems. These have been found to be more effective than the traditional linear approximation methods and local search methods. Recently, a genetic algorithm and an ant colony system algorithm were employed to develop two global search algorithms for solving concave cost transshipment problems. These two global search algorithms were found to be more effective than the advanced local search algorithms for solving concave cost transshipment problems. Although the particle swarm optimization algorithm has been used to obtain good results in many applications, to the best of our knowledge, it has not yet been applied in minimum concave cost network flow problems. Thus, in this study, we employ an arc-based particle swarm optimization algorithm, coupled with some genetic algorithm and threshold accepting method techniques, as well as concave cost network heuristics, to develop a hybrid global search algorithm for efficiently solving minimum cost network flow problems with concave arc costs. The proposed algorithm is evaluated by solving several randomly generated network flow problems. The results indicate that the proposed algorithm is more effective than several other recently designed methods, such as local search algorithms, genetic algorithms and ant colony system algorithms, for solving minimum cost network flow problems with concave arc costs.     

Geometric branch-and-bound methods for constrained global optimization problems

Geometric branch-and-bound methods are popular solution algorithms in deterministic global optimization to solve problems in small dimensions. The aim of this paper is to formulate a geometric branch-and-bound method for constrained global optimization problems which allows the use of arbitrary bounding operations. In particular, our main goal is to prove the convergence of the suggested method using the concept of the rate of convergence in geometric branch-and-bound methods as introduced in some recent publications. Furthermore, some efficient further discarding tests using necessary conditions for optimality are derived and illustrated numerically on an obnoxious facility location problem.     

正交约束优化问题的一阶算法

带有正交约束的矩阵优化问题在材料计算、统计及数据分析等领域中有着广泛的应用.由于正交约束的可行域是Stiefel流形,一直以来流形上的优化方法是求解这一问题的主要方法.近年来,随着实际应用问题所要求的变量规模的扩大,传统的流形优化方法在计算上的劣势显现出来,而一些迭代简单、收敛快的新算法逐渐被提出.通过收缩方法、非收缩可行方法、不可行方法三个类别分别来介绍求解带有正交约束的矩阵优化问题的最新算法.通过分析这些方法的主要特性,以及应用问题的要求,对这类问题算法设计的研究进行了展望.     

Evaluation of the multiobjective ant colony algorithm performances on biobjective quadratic assignment problems

The difficulty of resolving the multiobjective combinatorial optimization problems with traditional methods has directed researchers to investigate new approaches which perform better. In recent years some algorithms based on ant colony optimization (ACO) metaheuristic have been suggested to solve these multiobjective problems. In this study these algorithms have been reported and programmed both to solve the biobjective quadratic assignment problem (BiQAP) instances and to evaluate the performances of these algorithms. The robust parameter sets for each 12 multiobjective ant colony optimization (MOACO) algorithms have been calculated and BiQAP instances in the literature have been solved within these parameter sets. The performances of the algorithms have been evaluated by comparing the Pareto fronts obtained from these algorithms. In the evaluation step, a multi significance test is used in a non hierarchical structure, and a performance metric (P metric) essential for this test is introduced. Through this study, decision makers will be able to put in the biobjective algorithms in an order according to the priority values calculated from the algorithms’ Pareto fronts. Moreover, this is the first time that MOACO algorithms have been compared by solving BiQAPs.     

Global solutions to a class of CEC benchmark constrained optimization problems

This paper aims to solve a class of CEC benchmark constrained optimization problems that have been widely studied by nature-inspired optimization algorithms. Based on canonical duality theory, these challenging problems can be reformulated as a unified canonical dual problem over a convex set, which can be solved deterministically to obtain global optimal solutions in polynomial time. Applications are illustrated by some well-known CEC benchmark problems, and comparisons with other methods have demonstrated the effectiveness of the proposed approach.     

Using multi-objective evolutionary algorithms for single-objective optimization

In recent decades, several multi-objective evolutionary algorithms have been successfully applied to a wide variety of multi-objective optimization problems. Along the way, several new concepts, paradigms and methods have emerged. Additionally, some authors have claimed that the application of multi-objective approaches might be useful even in single-objective optimization. Thus, several guidelines for solving single-objective optimization problems using multi-objective methods have been proposed. This paper offers a survey of the main methods that allow the use of multi-objective schemes for single-objective optimization. In addition, several open topics and some possible paths of future work in this area are identified.     

The Million-Variable “March” for Stochastic Combinatorial Optimization

Combinatorial optimization problems have applications in a variety of sciences and engineering. In the presence of data uncertainty, these problems lead to stochastic combinatorial optimization problems which result in very large scale combinatorial optimization problems. In this paper, we report on the solution of some of the largest stochastic combinatorial optimization problems consisting of over a million binary variables. While the methodology is quite general, the specific application with which we conduct our experiments arises in stochastic server location problems. The main observation is that stochastic combinatorial optimization problems are comprised of loosely coupled subsystems. By taking advantage of the loosely coupled structure, we show that decomposition-coordination methods provide highly effective algorithms, and surpass the scalability of even the most efficiently implemented backtracking search algorithms.     

Generation of efficient frontiers in multi-objective optimization problems by generalized data envelopment analysis

In many practical problems such as engineering design problems, criteria functions cannot be given explicitly in terms of design variables. Under this circumstance, values of criteria functions for given values of design variables are usually obtained by some analyses such as structural analysis, thermodynamical analysis or fluid mechanical analysis. These analyses require considerably much computation time. Therefore, it is not unrealistic to apply existing interactive optimization methods to those problems. On the other hand, there have been many trials using genetic algorithms (GA) for generating efficient frontiers in multi-objective optimization problems. This approach is effective in problems with two or three objective functions. However, these methods cannot usually provide a good approximation to the exact efficient frontiers within a small number of generations in spite of our time limitation. The present paper proposes a method combining generalized data envelopment analysis (GDEA) and GA for generating efficient frontiers in multi-objective optimization problems. GDEA removes dominated design alternatives faster than methods based on only GA. The proposed method can yield desirable efficient frontiers even in non-convex problems as well as convex problems. The effectiveness of the proposed method will be shown through several numerical examples.     

Differential evolution with multi-constraint consensus methods for constrained optimization

Constrained optimization is an important research topic that assists in quality planning and decision making. To solve such problems, one of the important aspects is to improve upon any constraint violation, and thus bring infeasible individuals to the feasible region. To achieve this goal, different constraint consensus methods have been introduced, but no single method performs well for all types of problems. Hence, in this research, for solving constrained optimization problems, we introduce different variants of the Differential Evolution algorithm, with multiple constraint consensus methods. The proposed algorithms are tested and analyzed by solving a set of well-known bench mark problems. For further improvements, a local search is applied to the best variant. We have compared our algorithms among themselves, as well as with other state of the art algorithms. Those comparisons show similar, if not better performance, while also using significantly lower computational time.     

On finding multiple Pareto-optimal solutions using classical and evolutionary generating methods

In solving multi-objective optimization problems, evolutionary algorithms have been adequately applied to demonstrate that multiple and well-spread Pareto-optimal solutions can be found in a single simulation run. In this paper, we discuss and put together various different classical generating methods which are either quite well-known or are in oblivion due to publication in less accessible journals and some of which were even suggested before the inception of evolutionary methodologies. These generating methods specialize either in finding multiple Pareto-optimal solutions in a single simulation run or specialize in maintaining a good diversity by systematically solving a number of scalarizing problems. Most classical generating methodologies are classified into four groups mainly based on their working principles and one representative method from each group is chosen in the present study for a detailed discussion and for its performance comparison with a state-of-the-art evolutionary method. On visual comparisons of the efficient frontiers obtained for a number of two and three-objective test problems, the results bring out interesting insights about the strengths and weaknesses of these approaches. The results should motivate researchers to design hybrid multi-objective optimization algorithms which may be better than each of the individual methods.     

网络路径问题及其多目标扩展形式的混合进化算法

本文就几类困难的网络路径问题及其多目标扩展形式给出相应的混合型进化算法,并在微机上予以实现,为复杂的组合优化问题提供了新的求解手段．     

On the global optimization properties of finite-difference local descent algorithms

The paper is devoted to the convergence properties of finite-difference local descent algorithms in global optimization problems with a special -convex structure. It is assumed that the objective function can be closely approximated by some smooth convex function. Stability properties of the perturbed gradient descent and coordinate descent methods are investigated. Basing on this results some global optimization properties of finite-difference local descent algorithms, in particular, coordinate descent method, are discovered. These properties are not inherent in methods using exact gradients.The paper was presented at the II. IIASA-Workshop on Global Optimization, Sopron (Hungary), December 9–14, 1990.     

A new hybrid classical-quantum algorithm for continuous global optimization problems

Grover’s algorithm can be employed in global optimization methods providing, in some cases, a quadratic speedup over classical algorithms. This paper describes a new method for continuous global optimization problems that uses a classical algorithm for finding a local minimum and Grover’s algorithm to escape from this local minimum. Such algorithms will be useful when quantum computers of reasonable size are available. Simulations with testbed functions and comparisons with algorithms from the literature are presented.     

Parallel radial basis function methods for the global optimization of expensive functions

We introduce a master–worker framework for parallel global optimization of computationally expensive functions using response surface models. In particular, we parallelize two radial basis function (RBF) methods for global optimization, namely, the RBF method by Gutmann [Gutmann, H.M., 2001a. A radial basis function method for global optimization. Journal of Global Optimization 19(3), 201–227] (Gutmann-RBF) and the RBF method by Regis and Shoemaker [Regis, R.G., Shoemaker, C.A., 2005. Constrained global optimization of expensive black box functions using radial basis functions, Journal of Global Optimization 31, 153–171] (CORS-RBF). We modify these algorithms so that they can generate multiple points for simultaneous evaluation in parallel. We compare the performance of the two parallel RBF methods with a parallel multistart derivative-based algorithm, a parallel multistart derivative-free trust-region algorithm, and a parallel evolutionary algorithm on eleven test problems and on a 6-dimensional groundwater bioremediation application. The results indicate that the two parallel RBF algorithms are generally better than the other three alternatives on most of the test problems. Moreover, the two parallel RBF algorithms have comparable performances on the test problems considered. Finally, we report good speedups for both parallel RBF algorithms when using a small number of processors.     

Approximation algorithms for nonnegative polynomial optimization problems over unit spheres

We consider approximation algorithms for nonnegative polynomial optimization problems over unit spheres. These optimization problems have wide applications e.g., in signal and image processing, high order statistics, and computer vision. Since these problems are NP-hard, we are interested in studying on approximation algorithms. In particular, we propose some polynomial-time approximation algorithms with new approximation bounds. In addition, based on these approximation algorithms, some efficient algorithms are presented and numerical results are reported to show the efficiency of our proposed algorithms.     

A dynamic genetic algorithm based on continuous neural networks for a kind of non-convex optimization problems

This paper presents a kind of dynamic genetic algorithm based on a continuous neural network, which is intrinsically the steepest decent method for constrained optimization problems. The proposed algorithm combines the local searching ability of the steepest decent methods with the global searching ability of genetic algorithms. Genetic algorithms are used to decide each initial point of the steepest decent methods so that all the initial points can be searched intelligently. The steepest decent methods are employed to decide the fitness of genetic algorithms so that some good initial points can be selected. The proposed algorithm is motivated theoretically and biologically. It can be used to solve a non-convex optimization problem which is quadratic and even more non-linear. Compared with standard genetic algorithms, it can improve the precision of the solution while decreasing the searching scale. In contrast to the ordinary steepest decent method, it can obtain global sub-optimal solution while lessening the complexity of calculation.     

Particle swarm optimization: Hybridization perspectives and experimental illustrations

Metaheuristic optimization algorithms have become popular choice for solving complex and intricate problems which are otherwise difficult to solve by traditional methods. In the present study an attempt is made to review the hybrid optimization techniques in which one main algorithm is a well known metaheuristic; particle swarm optimization or PSO. Hybridization is a method of combining two (or more) techniques in a judicious manner such that the resulting algorithm contains the positive features of both (or all) the algorithms. Depending on the algorithm/s used we made three classifications as (i) Hybridization of PSO and genetic algorithms (ii) Hybridization of PSO with differential evolution and (iii) Hybridization of PSO with other techniques. Where, other techniques include various local and global search methods. Besides giving the review we also show a comparison of three hybrid PSO algorithms; hybrid differential evolution particle swarm optimization (DE-PSO), adaptive mutation particle swarm optimization (AMPSO) and hybrid genetic algorithm particle swarm optimization (GA-PSO) on a test suite of nine conventional benchmark problems.     

基于乌鸦喝水的元启发式优化算法

启发式优化算法已成为求解复杂优化问题的一种有效方法,可用于解决传统的优化方法难以求解的问题.受乌鸦喝水寓言故事启发,提出一种新型元启发式优化算法—乌鸦喝水算法,首先建立了乌鸦喝水算法数学模型;其次,给出实现该算法的详细步骤;最后,将该算法用于基准函数优化,并将该算法与乌鸦搜索算法、粒子群优化算法、多元宇宙优化算法、花授...     

Necessary Optimality Conditions and New Optimization Methods for Cubic Polynomial Optimization Problems with Mixed Variables

Multivariate cubic polynomial optimization problems, as a special case of the general polynomial optimization, have a lot of practical applications in real world. In this paper, some necessary local optimality conditions and some necessary global optimality conditions for cubic polynomial optimization problems with mixed variables are established. Then some local optimization methods, including weakly local optimization methods for general problems with mixed variables and strongly local optimization methods for cubic polynomial optimization problems with mixed variables, are proposed by exploiting these necessary local optimality conditions and necessary global optimality conditions. A global optimization method is proposed for cubic polynomial optimization problems by combining these local optimization methods together with some auxiliary functions. Some numerical examples are also given to illustrate that these approaches are very efficient.