非光滑规划全局优化的填充函数法
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摘要
全局优化方法广泛应用于工程设计、金融管理、生物工程和社会科学等领域,已成为优化领域中非常有意义的研究方向。全局优化研究的是多变量非线性函数在某个约束区域上的全局最优解的特性和构造寻求全局最优解的计算方法,以及求解方法的理论性质和计算性质。由于问题本身存在多个不同的局部极小点,在求解过程中面临两个困难:如何跳出当前局部极小点得到函数值更小的局部极小点和如何判断当前的极小点是全局极小点,因此无法直接用传统的非线性规划方法来解决。全局优化方法可分为两类:随机性方法和确定性方法。填充函数法是一类较为有效的确定性全局优化方法,是解决第一个困难的实用方法之一。它在当前局部极小点处构造填充函数,通过极小化填充函数迅速跳出当前局部极小点达到函数值更小的局部极小点,循环运算直至找到全局极小点。填充函数法提供了一个利用局部优化工具解决全局优化问题的途径。
非光滑优化也是当前计算数学、应用数学以及工程优化中较为活跃的研究领域之一。但大多数非光滑优化的有效算法都是寻求它的局部解,而对于全局非光滑优化问题,其理论和算法相对讨论得比较少。因此,鉴于填充函数法在光滑全局优化问题上的有效操作性,有学者对其在非光滑全局优化问题中进行了推广。但推广的填充函数有一些缺陷,如要求目标函数只有有限个局部极小点、填充函数的参数严格依赖于局部极小点盆谷的最小半径、且要求填充函数满足在线上存在极小点的条件等,从而对算法的实施带来困难。构造形式简单以及具有较少参数的填充函数并使其具有良好的性质,以便节约许多冗长的计算步骤及调整参数的时间,提高算法的效率,是理论和实际工作者继续研究填充函数的目的。
本文基于当前学者的研究现状,针对一些突出问题进行分析,寻求解决方案,力图在理论方面有所深化,在算法效果方面有所提高。本文结构安排如下。本文包含五章内容:第一章主要介绍了目前国内外主要的几种全局优化和非光滑优化的算法。这些方法包括填充函数法、打洞函数法、罚函数法等,重点介绍了填充函数法在光滑优化领域的已有成果。第二章讨论非光滑规划无约束全局优化的填充函数法。在对已有文献中非光滑无约束规划填充函数的定义进行改进的基础上,提出一类新的双参数填充函数。对该类填充函数的分析无需目标函数的可微性假设以及局部极小点的个数有限的假设,且参数易于调节,即参数的选取与局部极小点的谷域的半径无关。由于双参数的相互调节比较复杂,在算法具体实施时或多或少地会带来一些影响,改进的想法就是减少一个参数,从而提出了改进的单参数填充函数。在理论分析的基础上分别给出了相应的双参数填充函数算法和单参数填充函数算法,数值实验结果证明算法是有效的。区间方法和打洞函数法是全局优化算法中几类行之有效的算法,把我们所提出的填充函数法与此进行比较,指出了改进的方向。对非光滑全局优化的收敛性判别准则、算法中如何确定搜索方向,我们分别给出了建议。第三章把非光滑规划无约束全局优化的填充函数法推广到不等式约束全局优化问题。在非光滑约束规划填充函数的定义下,提出了一类双参数填充函数和一类单参数填充函数,设计了相应的算法并进行数值实验。结果表明算法也是有效的。第四章是非光滑规划等式约束全局优化的填充函数法。在本章提出了一类双参数填充函数并设计了算法。第五章是填充函数法的应用性研究。把填充函数法引入应用领域,如多目标规划、人脸识别系统、车牌识别系统、粒计算等,是进一步研究的方向。
Frequently, practitioners need to solve global optimization problems in many fields such as engineering design, financial management, bioengineering, and social science. So global optimization becomes a crucial computational task for researchers, which discusses the characters of global optimal choice of multivariate nonlinear functions on a constrained region and constructs computing approaches to find the global optimal solution, as well as discusses the theoretical properties and calculation properties of the solutions. However, due to the existence of multiple local minimiz-ers that differ from the global solution, we have to face two difficulties: how to jump from a local minimizer to a smaller one and how to judge that the current minimizer is a global one. Hence all these problems cannot be solved by classical nonlinear programming techniques directly. Generally speaking, the global optimization methods can be divided into two types: stochastic methods and deterministic methods. The filled function method, first proposed for smooth optimization by Ge and Qin (1987), is one of the effective deterministic global optimization methods for settling the first difficulty. It modifies the objective function as a filled function, and then finds a better local minimizer gradually by optimizing the filled function constructed on the minimizer previously found. The filled function method provides us with a good idea to use the local optimization techniques to solve global optimization problems.
Nonsmooth optimization is also one of the active research areas in computational mathematics, applied mathematics, and engineering design optimization. Most of the existing nonsmooth optimization methods are actually to seek a local minimizer. Several researchers have extended the filled function method from smooth global optimization to nonsmooth global optimization. The existing filled functions have some drawbacks such as requiring that the objective function has only a finite number of local minimizers, or the parameters of filled functions heavily restricted by the minimal basin radius of local minimizers, or requiring that the filled function has a minimizer on the line. All of these characteristics are strongly undesirable in numerical applications as they are liable to the illness of computation. Therefore, further research is worthy of continuing on how we can construct filled functions with simple forms, better properties and more efficient algorithms.
Based on the current status of research scholars, for a number of outstanding issues, the aim of this paper is to develop the filled function with certain satisfactory properties in nonsmooth global optimization. This paper mainly consists of five chapters. In Chapter 1, we give a brief introduction to the existing research work on global optimization and nonsmooth optimization. These methods include the filled function method, the tunnelling method, the penalty function method, and so on, with an emphasis on the filled function method in smooth optimization.
In Chapter 2, we discuss the filled function to find a global minimizer for a general class of nonsmooth programming problems with a closed bounded domain. Based on the new definition for the filled function, we propose a two-parameter filled function to improve the efficiency of numerical computation. This filled function needs not the assumption that the objective function is diflFerentiable and has a finite number of local minimizers, moreover, its parameters are easy to set. Owing to the complicated co-adjustment of the two-parameters, attempts have been made to improve the properties of the filled function. The basic idea for the modification is to cancel a parameter. Then we propose a new one-parameter filled function to eliminate these drawbacks. Based on these analyses, two corresponding filled function algorithms are presented. Numerical results obtained indicate the efficiency and reliability of the proposed filled function methods. The performance of the proposed filled function methods is compared to the performance of some well-known global optimization methods, the interval method and the tunneling method. For the questions on how to evaluate the convergence and how to decide a search direction in finding another better local minimizer, we give the corresponding suggestions.
In Chapter 3, we extend the idea for nonsmooth unconstrained global optimization to nonsmooth inequalities constrained global optimization. Under the def- inition for the filled function in nonsmooth constrained optimization, we give a two-parameter filled function and a one-parameter filled function, and present two corresponding filled function algorithms. The implementation of algorithms on several test problems is reported with satisfactory numerical results.
In Chapter 4, we extend the idea for nonsmooth unconstrained global optimization to nonsmooth equalities constrained global optimization. We proposes a two-parameter filled function and the corresponding filled function algorithm.
In Chapter 5, extension conceivable applications are given in order to evaluate the merits of the filled function method. The idea of finding a global minimizer by using filled function can be explored in a number of fields such as multiobjective programming, license plate recognition system, face recognition system, granular computing. Prom this point of view, a lot of operations can be defined and relations among them can be studied. This opens an extensive area for research and, hopingly, puts forward an interesting way for utilization of global optimization to modeling of phenomena.
非光滑优化也是当前计算数学、应用数学以及工程优化中较为活跃的研究领域之一。但大多数非光滑优化的有效算法都是寻求它的局部解,而对于全局非光滑优化问题,其理论和算法相对讨论得比较少。因此,鉴于填充函数法在光滑全局优化问题上的有效操作性,有学者对其在非光滑全局优化问题中进行了推广。但推广的填充函数有一些缺陷,如要求目标函数只有有限个局部极小点、填充函数的参数严格依赖于局部极小点盆谷的最小半径、且要求填充函数满足在线上存在极小点的条件等,从而对算法的实施带来困难。构造形式简单以及具有较少参数的填充函数并使其具有良好的性质,以便节约许多冗长的计算步骤及调整参数的时间,提高算法的效率,是理论和实际工作者继续研究填充函数的目的。
本文基于当前学者的研究现状,针对一些突出问题进行分析,寻求解决方案,力图在理论方面有所深化,在算法效果方面有所提高。本文结构安排如下。本文包含五章内容:第一章主要介绍了目前国内外主要的几种全局优化和非光滑优化的算法。这些方法包括填充函数法、打洞函数法、罚函数法等,重点介绍了填充函数法在光滑优化领域的已有成果。第二章讨论非光滑规划无约束全局优化的填充函数法。在对已有文献中非光滑无约束规划填充函数的定义进行改进的基础上,提出一类新的双参数填充函数。对该类填充函数的分析无需目标函数的可微性假设以及局部极小点的个数有限的假设,且参数易于调节,即参数的选取与局部极小点的谷域的半径无关。由于双参数的相互调节比较复杂,在算法具体实施时或多或少地会带来一些影响,改进的想法就是减少一个参数,从而提出了改进的单参数填充函数。在理论分析的基础上分别给出了相应的双参数填充函数算法和单参数填充函数算法,数值实验结果证明算法是有效的。区间方法和打洞函数法是全局优化算法中几类行之有效的算法,把我们所提出的填充函数法与此进行比较,指出了改进的方向。对非光滑全局优化的收敛性判别准则、算法中如何确定搜索方向,我们分别给出了建议。第三章把非光滑规划无约束全局优化的填充函数法推广到不等式约束全局优化问题。在非光滑约束规划填充函数的定义下,提出了一类双参数填充函数和一类单参数填充函数,设计了相应的算法并进行数值实验。结果表明算法也是有效的。第四章是非光滑规划等式约束全局优化的填充函数法。在本章提出了一类双参数填充函数并设计了算法。第五章是填充函数法的应用性研究。把填充函数法引入应用领域,如多目标规划、人脸识别系统、车牌识别系统、粒计算等,是进一步研究的方向。
Frequently, practitioners need to solve global optimization problems in many fields such as engineering design, financial management, bioengineering, and social science. So global optimization becomes a crucial computational task for researchers, which discusses the characters of global optimal choice of multivariate nonlinear functions on a constrained region and constructs computing approaches to find the global optimal solution, as well as discusses the theoretical properties and calculation properties of the solutions. However, due to the existence of multiple local minimiz-ers that differ from the global solution, we have to face two difficulties: how to jump from a local minimizer to a smaller one and how to judge that the current minimizer is a global one. Hence all these problems cannot be solved by classical nonlinear programming techniques directly. Generally speaking, the global optimization methods can be divided into two types: stochastic methods and deterministic methods. The filled function method, first proposed for smooth optimization by Ge and Qin (1987), is one of the effective deterministic global optimization methods for settling the first difficulty. It modifies the objective function as a filled function, and then finds a better local minimizer gradually by optimizing the filled function constructed on the minimizer previously found. The filled function method provides us with a good idea to use the local optimization techniques to solve global optimization problems.
Nonsmooth optimization is also one of the active research areas in computational mathematics, applied mathematics, and engineering design optimization. Most of the existing nonsmooth optimization methods are actually to seek a local minimizer. Several researchers have extended the filled function method from smooth global optimization to nonsmooth global optimization. The existing filled functions have some drawbacks such as requiring that the objective function has only a finite number of local minimizers, or the parameters of filled functions heavily restricted by the minimal basin radius of local minimizers, or requiring that the filled function has a minimizer on the line. All of these characteristics are strongly undesirable in numerical applications as they are liable to the illness of computation. Therefore, further research is worthy of continuing on how we can construct filled functions with simple forms, better properties and more efficient algorithms.
Based on the current status of research scholars, for a number of outstanding issues, the aim of this paper is to develop the filled function with certain satisfactory properties in nonsmooth global optimization. This paper mainly consists of five chapters. In Chapter 1, we give a brief introduction to the existing research work on global optimization and nonsmooth optimization. These methods include the filled function method, the tunnelling method, the penalty function method, and so on, with an emphasis on the filled function method in smooth optimization.
In Chapter 2, we discuss the filled function to find a global minimizer for a general class of nonsmooth programming problems with a closed bounded domain. Based on the new definition for the filled function, we propose a two-parameter filled function to improve the efficiency of numerical computation. This filled function needs not the assumption that the objective function is diflFerentiable and has a finite number of local minimizers, moreover, its parameters are easy to set. Owing to the complicated co-adjustment of the two-parameters, attempts have been made to improve the properties of the filled function. The basic idea for the modification is to cancel a parameter. Then we propose a new one-parameter filled function to eliminate these drawbacks. Based on these analyses, two corresponding filled function algorithms are presented. Numerical results obtained indicate the efficiency and reliability of the proposed filled function methods. The performance of the proposed filled function methods is compared to the performance of some well-known global optimization methods, the interval method and the tunneling method. For the questions on how to evaluate the convergence and how to decide a search direction in finding another better local minimizer, we give the corresponding suggestions.
In Chapter 3, we extend the idea for nonsmooth unconstrained global optimization to nonsmooth inequalities constrained global optimization. Under the def- inition for the filled function in nonsmooth constrained optimization, we give a two-parameter filled function and a one-parameter filled function, and present two corresponding filled function algorithms. The implementation of algorithms on several test problems is reported with satisfactory numerical results.
In Chapter 4, we extend the idea for nonsmooth unconstrained global optimization to nonsmooth equalities constrained global optimization. We proposes a two-parameter filled function and the corresponding filled function algorithm.
In Chapter 5, extension conceivable applications are given in order to evaluate the merits of the filled function method. The idea of finding a global minimizer by using filled function can be explored in a number of fields such as multiobjective programming, license plate recognition system, face recognition system, granular computing. Prom this point of view, a lot of operations can be defined and relations among them can be studied. This opens an extensive area for research and, hopingly, puts forward an interesting way for utilization of global optimization to modeling of phenomena.
引文
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