一类连续函数模拟退火算法及其收敛性分析

高维连续函数的全局优化问题普遍存在于计算生物学、计算化学等领域．针对这类问题和现有连续函数模拟退火算法的某些不足,本文给出了一类改进的模拟退火算法．采用一种简单的方法证明了算法的全局收敛性．数值结果表明,对于高维连续函数,该算法能够快速有效地收敛到全局最优点,比较了两种新解产生方法的试验结果。     

混合互补问题的光滑逼近法

提出了求解混合互补问题的一个光滑逼近算法，并在一定条件下证明了该算法的全局收敛性．     

在退化情况下的一个可行方向法

本文给出了一个新的非线性约束优化的可行方向法．该算法适用于退化问题(积极约束梯度线性相关)，算法结构简单，在适当条件下，证明此算法具有全局收敛性．数值实验表明算法是有效的．     

线性互补问题的并行多分裂松弛迭代算法

运用矩阵多重分裂理论,同时考虑并行计算与松弛迭代法,得到一类求解线性互补问题的高效数值算法．当问题的系数矩阵为对角元为正的H-矩阵或对称半正定矩阵时,证明了算法的全局收敛性；该算法与已有算法相比,具有计算量小、计算速度快等特点,因而特别适于求解大规模问题．数值试验的结果说明了算法的有效性．     

一类新的非单调信赖域算法

通过改变预计下降量，使其与实际下降量对应起来，对无约束最优化问题提出一类新的非单调信赖域算法．可以证明，在一定的条件下，该算法具有全局收敛性．     

半无限极大极小问题的全局收敛方法

利用广义伪方向导数，在较弱的条件下，给出了半无限极大极小问题（P）的全局收敛性理论算法模型；利用离散策略给出了问题（P）全局收敛的可实现算法．数值结果表明本文给出的可实现算法是有效的．     

求解线性互补问题的一种新算法

本文针对线性互补问题,提出了与其等价的非光滑方程的逐次逼近阻尼牛顿法,并在一定条件下证明了该算法具有的全局收敛性．同时给出了一些数值例子,得到很好的数值结果．     

求解一类二阶段有补偿问题的对偶梯度法

利用对偶理论，本文给出了求解一类具有简单补偿的非线性二阶段问题的新对偶梯度法．在假设目标函数为可分连续可微凸函数的条件下，在每一选代步可将原二阶段有补偿问题转化为几个一维凸规划问题，大大简化了问题的求解．所给算法简单易行，文中还证明了该算法的全局收敛性．     

求解一般约束优化问题的修正BFGS信赖域算法

借鉴无约束优化问题的BFGS信赖域算法,建立了非线性一般约束优化问题的BFGS信赖域算法,并证明了算法的全局收敛性．数值实验表明,算法是有效的．     

不等式约束优化一个新的SQP算法

本文提出了一个处理不等式约束优化问题的新的SQP算法．和传统的SQP算法相比，该算法每步只需求解一个仅含等式约束的子二次规划，从而减少了算法的计算工作量．在适当的条件下，证明算法是全局收敛的且具有超线性收敛速度．数值实验表明算法是有效的．     

Global and Local Search Algorithms for Concave Cost Transshipment Problems

Traditionally, the minimum cost transshipment problems have been simplified as linear cost problems, which are not practical in real applications. Recently, some advanced local search algorithms have been developed that can directly solve concave cost bipartite network problems. However, they are not applicable to general transshipment problems. Moreover, the effectiveness of these modified local search algorithms for solving general concave cost transshipment problems is doubtful. In this research, we propose a global search algorithm for solving concave cost transshipment problems. Effecient methods for encoding, generating initial populations, selection, crossover and mutation are proposed, according to the problem characteristics. To evaluate the effectiveness of the proposed global search algorithm, four advanced local search algorithms based on the threshold accepting algorithm, the great deluge algorithm, and the tabu search algorithm, are also developed and are used for comparison purpose. To assist with the comparison of the proposed algorithms, a randomized network generator is designed to produce test problems. All the tests are performed on a personal computer. The results indicate that the proposed global search algorithm is more effective than the four advanced local algorithms, for solving concave cost transshipment problems.     

Global Optimization by Monotonic Transformation

This paper addresses the problem of global optimization by means of a monotonic transformation. With an observation on global optimality of functions under such a transformation, we show that a simple and effective algorithm can be derived to search within possible regions containing the global optima. Numerical experiments are performed to compare this algorithm with one that does not incorporate transformed information using several benchmark problems. These results are also compared to best known global search algorithms in the literature. In addition, the algorithm is shown to be useful for several neural network learning problems, which possess much larger parameter spaces.     

Proof of Convergence for a Global Optimization Algorithm for Problems with Ordinary Differential Equations

A deterministic spatial branch and bound global optimization algorithm for problems with ordinary differential equations in the constraints has been developed by Papamichail and Adjiman [A rigorous global optimization algorithm for problems with ordinary differential equations. J. Glob. Optim. 24, 1–33]. In this work, it is shown that the algorithm is guaranteed to converge to the global solution. The proof is based on showing that the selection operation is bound improving and that the bounding operation is consistent. In particular, it is shown that the convex relaxation techniques used in the algorithm for the treatment of the dynamic information ensure bound improvement and consistency are achieved.     

A discrete filled function algorithm embedded with continuous approximation for solving max-cut problems

In this paper, a discrete filled function algorithm embedded with continuous approximation is proposed to solve max-cut problems. A new discrete filled function is defined for max-cut problems, and properties of the function are studied. In the process of finding an approximation to the global solution of a max-cut problem, a continuation optimization algorithm is employed to find local solutions of a continuous relaxation of the max-cut problem, and then global searches are performed by minimizing the proposed filled function. Unlike general filled function methods, characteristics of max-cut problems are used. The parameters in the proposed filled function need not to be adjusted and are exactly the same for all max-cut problems that greatly increases the efficiency of the filled function method. Numerical results and comparisons on some well known max-cut test problems show that the proposed algorithm is efficient to get approximate global solutions of max-cut problems.     

Aspiration Based Simulated Annealing Algorithm

An aspiration based simulated annealing algorithm for continuousvariables has been proposed. The new algorithm is similar to the one givenby Dekkers and Aarts (1991) except that a kind of memory is introduced intothe procedure with a self-regulatory mechanism. The algorithm has beenapplied to a set of standard global optimization problems and a number ofmore difficult, complex, practical problems and its performance comparedwith that of the algorithm of Dekkers and Aarts (1991). The new algorithmappears to offer a useful alternative to some of the currently availablestochastic algorithms for global optimization.     

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.     

A new genetic algorithm for global optimization of multimodal continuous functions

In this paper a new genetic algorithm is developed to find the near global optimal solution of multimodal nonlinear optimization problems. The algorithm defined makes use of a real encoded crossover and mutation operator. The performance of GA is tested on a set of twenty-seven nonlinear global optimization test problems of variable difficulty level. Results are compared with some well established popular GAs existing in the literature. It is observed that the algorithm defined performs significantly better than the existing ones.     

Simplicial Branch-and-Reduce Algorithm for Convex Programs with a Multiplicative Constraint

Optimization problems that involve products of convex functions in the objective function or in the constraints arise in a variety of applications. These problems are difficult global optimization problems. During the past 15 years, however, a number of practical algorithms have been proposed for globally solving these types of problems. In this article, we present and validate a branch-and-reduce algorithm for finding a global optimal solution to a convex program that contains an additional constraint on the product of several convex functions. To globally solve this problem, the algorithm instead globally solves an equivalent master problem. At any stage of the algorithm, a disconnected set consisting of a union of simplices is constructed. This set is guaranteed to contain a portion of the boundary of the feasible region of the master problem where a global optimal solution lies. The algorithm uses a new branch-and-reduce scheme to iteratively reduce the sizes of these sets until a global optimal solution is found. Several potential computational advantages of the algorithm are explained, and a numerical example is solved.     

A Locally-Biased form of the DIRECT Algorithm

In this paper we propose a form of the DIRECT algorithm that is strongly biased toward local search. This form should do well for small problems with a single global minimizer and only a few local minimizers. We motivate our formulation with some results on how the original formulation of the DIRECT algorithm clusters its search near a global minimizer. We report on the performance of our algorithm on a suite of test problems and observe that the algorithm performs particularly well when termination is based on a budget of function evaluations.     

A hybrid global optimization algorithm for non-linear least squares regression

A hybrid global optimization algorithm is proposed aimed at the class of objective functions with properties typical of the problems of non-linear least squares regression. Three components of hybridization are considered: simplicial partition of the feasible region, indicating and excluding vicinities of the main local minimizers from global search, and computing the indicated local minima by means of an efficient local descent algorithm. The performance of the algorithm is tested using a collection of non-linear least squares problems evaluated by other authors as difficult global optimization problems.