同伦方法求解无界域上非凸规划问题的收敛性定理

利用同伦方法求解非凸规划时,一般只能得到问题的K-K-T点.本文得到无界域上同伦方法求解非凸规划的几个收敛性定理,证明在一定条件下,通过构造合适的同伦方程,同伦算法收敛到问题的局部最优解.     

求解一类非线性规划问题的K-K-T点的全局收敛性算法

提出了一组无界性条件,在此基础上给出了求解一类无界非凸非线性规划问题的K-K-T点的一种高效的全局收敛性算法,在适当的条件下,给出了算法的收敛性证明.结果把已有的研究结果推广到无界区域上,进一步扩大了算法的求解区域.     

一类复杂非凸区域的拟法锥构造方法及其在非凸规划求解中的应用

本文针对基于一般的凸集与"契型"的余集相交形成的一类满足拟法锥条件的复杂非凸区域,给出一种拟法锥的构造方法,在给定的拟法锥条件下,建立求解在该类非凸区域上规划问题的K-K-T点的组合同伦方程,并证明了该同伦内点法的整体收敛性,并通过数值例子证明算法是可行的和有效的.     

非凸区域上的一些Hardy型不等式

研究非凸区域上的Hardy型不等式.通过选取特殊的向量值函数以及仔细的分析与计算,得到了各向异性Heisenberg群上一类非凸区域上的Hardy不等式,更进一步得到了非凸区域上与Greiner型向量场相关的几类Hardy型不等式.     

无界区域上基于自然边界归化的一种重叠型区域分解法及其离散化

1．引言由于科学技术的迅猛发展，人们遇到许多大规模科学和工程计算问题．随着并行计算机的出现和应用，并行技术越来越得到人们的重视和研究．区域分解法成为并行计算和处理这类问题的主要方法之一．但是，对于无界区域上的椭圆边值问题，因进行区域分解后至少有一个区域仍为无界区域，故仅应用通常的区域分解算法求解是不够的．由于边界归化是处理无界区域问题的有效手段，通常采用边界元和有限元耦合的方法求解此类问题IZ，6。8。121．或片什适当的人工边界并在此边界上加近似边界条件，再在有限区域应用有限元方法求解【人习．近年来…     

同伦方法求解非凸区域Brouwer不动点问题

本文构造了一个新的求解非凸区域上不动点问题的内点同伦算法,并在弱法锥(见定义2．1(2))和适当的条件下,证明了算法的全局收敛性．本文所给的条件比外法锥条件更加一般．     

连续化方法求解约束凸规划问题

1．引言大型规划问题数值求解一直是计算数学工作者感兴趣的课题之一．针对大型约束规划问题,1991年李兴斯山提出凝聚函数法,该方法用光滑的凝聚函数逼近非光滑的极大值函数,从而把多个约束函数转化为带参数的单个光滑函数约束,从而降低了问题的规模．近年来,K3］研究了凸规划问题的凝聚函数法的收敛性,在目标函数强凸性及对一般凸规划研究了收敛性质．向讨论了可行解集有界的线性规划问题的凝聚函数求解算法并证明了收效性定理．上述文章均预先把凝聚参数取得充分小,然后对固定参数的单约束近似问题进行求解．一般地,凝聚参数取得…     

无界域上不定二次规划的一个算法

本给出了无界域上不定二次规划一个算法，该算法将不定二次规划转化为一系列凸二次规划，并证明了算法的收敛性。     

凸区域上椭圆方程弱解的边界唯一延拓性和B_p权特性

本文研究非光滑凸区域上的散度型二阶椭圆方程 ｉ（ａｉｊ（Ｘ）　ｊｕ（Ｘ））＝０的非零弱解的近边无穷次消失性和双倍性质，刻划弱解梯度在区域边界上的Ｂｐ权特性，建立弱解和弱解的梯度在凸区域边界的任意开子集上不可能同时消失的边界唯一延拓性定理．     

不可微规划的各种约束规格

利用Ｗａｒｄ等人给出的广义锥方向导数和广义次梯度等概念，建立了一类非凸非光滑数学规划的各种约束规格．     

Exact penalty functions method for mathematical programming problems involving invex functions

In this paper, some new results on the exact penalty function method are presented. Simple optimality characterizations are given for the differentiable nonconvex optimization problems with both inequality and equality constraints via exact penalty function method. The equivalence between sets of optimal solutions in the original mathematical programming problem and its associated exact penalized optimization problem is established under suitable invexity assumption. Furthermore, the equivalence between a saddle point in the invex mathematical programming problem and an optimal point in its exact penalized optimization problem is also proved.     

一类非凸区域的拟法锥构造方法及其在非凸规划求解中的应用

本文给出基于球形的一类满足拟法锥条件区域的拟法锥构造方法,基于该可行域的拟法锥,建立求解在该类非凸区域上的规划问题的K-K-T点的部分凝聚同伦组合方程,并证明了该同伦内点法的整体收敛性,给出实现同伦内点法的具体数值跟踪算法步骤,并通过数值例子证明算法是可行的和有效的．     

A Branch-and-Bound Approach for Solving a Class of Generalized Semi-infinite Programming Problems

A nonconvex generalized semi-infinite programming problem is considered, involving parametric max-functions in both the objective and the constraints. For a fixed vector of parameters, the values of these parametric max-functions are given as optimal values of convex quadratic programming problems. Assuming that for each parameter the parametric quadratic problems satisfy the strong duality relation, conditions are described ensuring the uniform boundedness of the optimal sets of the dual problems w.r.t. the parameter. Finally a branch-and-bound approach is suggested transforming the problem of finding an approximate global minimum of the original nonconvex optimization problem into the solution of a finite number of convex problems.     

Stability achievement scalarization function for multiobjective nonlinear programming problems

In this paper, we present a method to determine the stability of nondominated criterion vectors using a modified weighted achievement scalarization metric. This method is based on the application of a particular objective function which scalarizes and parameterizes the original multiobjective nonlinear programming problem. Also, we show that this modified weighted achievement metric coincides with the metric introduced by Choo and Atkins [E.-U. Choo, D.R. Atkins, Proper efficiency in nonconvex multicriteria programming, Math. Oper. Res. 8 (1983) 467–470] and Kaliszewski [I. Kaliszewski, A modified weighted Tchebycheff metric for multiple objective programming, Comput. Oper. Res. 14 (1987) 315–323] in cases when sets of all criterion vectors are finite or polyhedral.     

Extended duality for nonlinear programming

Duality is an important notion for nonlinear programming (NLP). It provides a theoretical foundation for many optimization algorithms. Duality can be used to directly solve NLPs as well as to derive lower bounds of the solution quality which have wide use in other high-level search techniques such as branch and bound. However, the conventional duality theory has the fundamental limit that it leads to duality gaps for nonconvex problems, including discrete and mixed-integer problems where the feasible sets are generally nonconvex.     

Saddle Points and Pareto Points in Multiple Objective Programming

In this paper relationships between Pareto points and saddle points are studied in convex and nonconvex multiple objective programming. The analysis is based on partitioning the index sets of objectives and constraints and splitting the original problem into subproblems having a special structure. The results are based on scalarizations of multiple objective programs and related linear and augmented Lagrangian functions. In the nonconvex case, a saddle point characterization of Pareto points is possible under assumptions that guarantee existence of Pareto points and stability conditions of single objective problems. Essentially, these conditions are not stronger than those in analogous results for single objective programming.This research was partially supported by ONR Grant N00014-97-1-784AMS Subject Classification: 90C29, 90C26     

路径跟踪方法求解无界非凸区域上的不动点问题

给出了求解一类无界非凸区域上不动点问题的路径跟踪方法.在适当的条件下,给出了不动点存在性的构造性证明,从而得到了路径跟踪方法的全局收敛性结果.研究结果为计算无界非凸区域上不动点问题提供了一种全局收敛性方法.     

Conical Algorithm in Global Optimization for Optimizing over Efficient Sets

The problem of optimizing some contiuous function over the efficient set of a multiple objective programming problem can be formulated as a nonconvex global optimization problem with special structure. Based on the conical branch and bound algorithm in global optimization, we establish an algorithm for optimizing over efficient sets and discuss about the implementation of this algorithm for some interesting special cases including the case of biobjective programming problems.     

Implementations of special ordered sets in MP software

Special ordered sets (SOS) have been introduced as a practical device for efficiently handling special classes of nonconvex optimization problems. They are now implemented in most commercial codes for mathematical programming (MP software). The paper gives a survey of possible applications as multiple choice restrictions, conditional multiple choice restrictions, discrete variables, discontinuous variables and piecewise linear functions, global optimization of separable programming problems, alternative right-hand sides, overlapping special ordered sets and the solution of quadratic programming problems. Alternative problem formulations are discussed. Since special ordered sets are not defined uniquely modelling facilities depend on the definition of a special orderedset in a code. The paper demonstrates the superiority of SOS to the application of binary variables if they are treated judiciously.     

Stochastic dual dynamic programming applied to nonconvex hydrothermal models

In this paper we apply stochastic dual dynamic programming decomposition to a nonconvex multistage stochastic hydrothermal model where the nonlinear water head effects on production and the nonlinear dependence between the reservoir head and the reservoir volume are modeled. The nonconvex constraints that represent the production function of a hydro plant are approximated by McCormick envelopes. These constraints are split into smaller regions and the McCormick envelopes are used for each region. We use binary variables for this disjunctive programming approach and solve the problem with a decomposition method. We resort to a variant of the L-shaped method for solving the MIP subproblem with binary variables at any stage inside the stochastic dual dynamic programming algorithm. A realistic large-scale case study is presented.