Negative Effects of Modeling Techniques on Search Performance

Symmetries occur in many combinatorial problems, and a great deal of research has been done on symmetry breaking techniques for backtrack search. However, few results have been reported on the use of symmetry breaking with local search. On four classes of problem we find that adding symmetry breaking constraints to a model impairs local search performance, in terms of both execution time and search steps. We also find that implied constraints can impair backtrack search performance. These results show that modeling techniques and search heuristics should be combined with caution. They also motivate a novel modeling technique for local search: removing constraints to add new symmetries.     

Optimizing the periodic pick-up of raw materials for a manufacturer of auto parts

We describe a solution procedure for a special case of the periodic vehicle routing problem (PVRP). Operation managers at an auto parts manufacturer in the north of Spain described the optimization problem to the authors. The manufacturer must pick up parts (raw material) from geographically dispersed locations. The parts are picked up periodically at scheduled times. The problem consists of assigning a pickup schedule to each of its supplier’s locations and also establishing daily routes in order to minimize total transportation costs. The time horizon under consideration may be as long as 90 days. The resulting PVRP is such that the critical decision is the assignment of locations to schedules, because once this is done, the daily routing of vehicles is relatively straightforward. Through extensive computational experiments, we show that the metaheuristic procedure described in this paper is capable of finding high-quality solutions within a reasonable amount of computer time. Our main contribution is the development of a procedure that is more effective at handling PVRP instances with long planning horizons when compared to those proposed in the literature.     

三项记忆梯度法及其投影算法的收敛性分析

对于无约束优化问题,提出了一类新的三项记忆梯度算法．这类算法是在参数满足某些假设的条件下,确定它的取值范围,从而保证三项记忆梯度方向是使目标函数充分下降的方向．在非单调步长搜索下讨论了算法的全局收敛性．为了得到具有更好收敛性质的算法,结合Solodov and Svaiter(2000)中的部分技巧,提出了一种新的记忆梯度投影算法,并证明了该算法在函数伪凸的情况下具有整体收敛性．     

Sets in Abelian groups with distinct sums of pairs

A subset S={s₁,…,sk} of an Abelian group G is called an St-set of size k if all sums of t different elements in S are distinct. Let s(G) denote the cardinality of the largest S₂-set in G. Let v(k) denote the order of the smallest Abelian group for which s(G)?k. In this article, bounds for s(G) are developed and v(k) is determined for k?15 by computing s(G) for Abelian groups of order up to 183 using exhaustive backtrack search with isomorph rejection.     

An unconstrained optimization method using nonmonotone second order Goldstein's line search

In this paper,an unconstrained optimization method using the nonmonotone second order Goldstein's line search is proposed.By using the negative curvature information from the Hessian, the sequence generated is shown to converge to a stationary point with the second order optimality conditions.Numerical tests on a set of standard test problems confirm the efficiency of our new method.     

Frame-Based Ray Search Algorithms in Unconstrained Optimization

This paper describes a class of frame-based direct search methods for unconstrained optimization without derivatives. A template for convergent direct search methods is developed, some requiring only the relative ordering of function values. At each iteration, the template considers a number of search steps which form a positive basis and conducts a ray search along a step giving adequate decrease. Various ray search strategies are possible, including discrete equivalents of the Goldstein–Armijo and one-sided Wolfe–Powell ray searches. Convergence is shown under mild conditions which allow successive frames to be rotated, translated, and scaled relative to one another.     

遗传算法的物理解释

提出了一种理想化的模拟仿生搜索算法——扰动算法 ,以此方法为基础 ,分析了遗传算法的搜索过程和效率问题 ,阐明了遗传算法作为一种次优算法的有效性 .相对于遗传算法的生物解释 ,本文给出了相应的物理解释 .同时 ,本文为遗传算法、进化策略和模拟退火算法找到了一种统一的物理解释 ,揭示了这些重要的仿生类算法实质上的相似性 .     

On the Convergence of a Population-Based Global Optimization Algorithm

In global optimization, a typical population-based stochastic search method works on a set of sample points from the feasible region. In this paper, we study a recently proposed method of this sort. The method utilizes an attraction-repulsion mechanism to move sample points toward optimality and is thus referred to as electromagnetism-like method (EM). The computational results showed that EM is robust in practice, so we further investigate the theoretical structure. After reviewing the original method, we present some necessary modifications for the convergence proof. We show that in the limit, the modified method converges to the vicinity of global optimum with probability one.     

Global Convergence Properties of Nonlinear Conjugate Gradient Methods with Modified Secant Condition

Conjugate gradient methods are appealing for large scale nonlinear optimization problems. Recently, expecting the fast convergence of the methods, Dai and Liao (2001) used secant condition of quasi-Newton methods. In this paper, we make use of modified secant condition given by Zhang et al. (1999) and Zhang and Xu (2001) and propose a new conjugate gradient method following to Dai and Liao (2001). It is new features that this method takes both available gradient and function value information and achieves a high-order accuracy in approximating the second-order curvature of the objective function. The method is shown to be globally convergent under some assumptions. Numerical results are reported.     

A gradient-related algorithm with inexact line searches

In this paper, a new gradient-related algorithm for solving large-scale unconstrained optimization problems is proposed. The new algorithm is a kind of line search method. The basic idea is to choose a combination of the current gradient and some previous search directions as a new search direction and to find a step-size by using various inexact line searches. Using more information at the current iterative step may improve the performance of the algorithm. This motivates us to find some new gradient algorithms which may be more effective than standard conjugate gradient methods. Uniformly gradient-related conception is useful and it can be used to analyze global convergence of the new algorithm. The global convergence and linear convergence rate of the new algorithm are investigated under diverse weak conditions. Numerical experiments show that the new algorithm seems to converge more stably and is superior to other similar methods in many situations.