非可微凸规划的对偶问题

文章建立关于非可微凸规划的一个新的对偶问题,它不同于已知的对偶问题,文中证明了弱对偶性及强对偶性。并用Ｌａｇｒａｎｇｅ正则性证明了强对偶性的充要条件。最后,讨论了等式约束的情况。     

线性规划问题罚函数方法的一种统一形式

用一种统一的方式,讨论了线性规划问题中常用的罚函数方法及其对偶性.并将这种方法应用到等式约束二次规划问题中.     

锥约束多目标规划的二阶对称对偶性质

给出一对锥约束多目标非线性规划的二阶对称对偶问题,以及二阶F凸函数类的概念.在二阶F凸假设下证明了真有效解的对偶性质———弱对偶性、强对偶性及逆对偶性.     

均衡约束数学规划问题的一类广义Mond-Weir型对偶理论

针对均衡约束数学规划模型难以满足约束规范及难于求解的问题,基于Mond和Weir提出的标准非线性规划的对偶形式,利用其S稳定性,建立了均衡约束数学规划问题的一类广义Mond-Weir型对偶,从而为求解均衡约束优化问题提供了一种新的方法.在Hanson-Mond广义凸性条件下,利用次线性函数,分别提出了弱对偶性、强对偶性和严格逆对偶性定理,并给出了相应证明.该对偶化方法的推广为研究均衡约束数学规划问题的解提供了理论依据.     

含有锥约束的多目标变分问题的广义对称对偶性

本文研究了一类含有锥约束多目标变分问题的广义对称对偶性.利用函数的(F,ρ)-不变凸性的条件,得出了多目标变分问题关于有效解的弱对偶定理、强对偶定理和逆对偶定理,将多目标变分问题的对称对偶性理论推广到含有锥约束的广义对称对偶性上来.     

半无限规划的对偶性

本文对半无限凸规划提出一个新的对偶问题,它由扰动函数及其次微分刻划.同时讨论了弱对偶性、强对偶性及逆对偶性,证明强对偶性等价于鞍点准则.     

G-(F,ρ)凸意义下多目标规划真有效解的对偶性

本文利用文［1］中的有关结果，讨论带等式约束和不等式约束的多目标规划关于Geoffrion真有效解的对偶性，建立了Wolfe及Mond－Weir型对偶问题，并在G－（F，ρ）凸型假设下，证明了弱、强及逆对偶定理．     

基于最小描述的覆盖粒计算模型

在覆盖近似空间(U,C)中,基于最小描述定义了一种新的Zoom-in算子,并讨论了其性质。在论域U上,通过复合Zoom-in和Zoom-out算子,得到了具有对偶性的两对近似算子。同样在粒化论域C上,通过复合Zoom-out和Zoom-in算子,也得到了具有对偶性的两对近似算子。     

带球（椭球）约束的不定二次规划问题

本文证明了带球（椭球）约束的不定二次规划问题具有强Ｌａｇｒａｎｇｅ对偶性，设计了一个求解这类问题的算法，本语文的结论比文「７」强，所设计的算法比文「７」简洁。     

广义凸多目标规划的Mond─Weir型对偶性

本文研究带不等式和等式约束的多目标规划的Ｍｏｎｄ－Ｗｅｉｒ型对偶性理论。在目标和约束是广义凸的假设下，证明了弱对偶定理、直接对偶定理以及逆对偶定理     

Mathematical Programming with System of Equilibrium Constraints

In this paper, we study the mathematical program with system of equilibrium constraints. This problem contains bilevel program with system of equilibrium constraints, semi-infinite program with system of equilibrium constraints, mathematical program with Nash equilibrium constraints, mathematical program with system of mixed variational like inequalities constraints. We establish the existence theorems of mathematical program with system of equilibrium constraints under various assumptions.     

Duality for multiobjective optimization problems with convex objective functions and D.C. constraints

In this paper we provide a duality theory for multiobjective optimization problems with convex objective functions and finitely many D.C. constraints. In order to do this, we study first the duality for a scalar convex optimization problem with inequality constraints defined by extended real-valued convex functions. For a family of multiobjective problems associated to the initial one we determine then, by means of the scalar duality results, their multiobjective dual problems. Finally, we consider as a special case the duality for the convex multiobjective optimization problem with convex constraints.     

Second-Order Necessary Optimality Conditions for Some State-Constrained Control Problems of Semilinear Elliptic Equations

In this paper we are concerned with some optimal control problems governed by semilinear elliptic equations. The case of a boundary control is studied. We consider pointwise constraints on the control and a finite number of equality and inequality constraints on the state. The goal is to derive first- and second-order optimality conditions satisfied by locally optimal solutions of the problem. Accepted 6 May 1997     

Dirichlet Boundary Control of Semilinear Parabolic Equations Part 1: Problems with No State Constraints

This paper is concerned with distributed and Dirichlet boundary controls of semilinear parabolic equations, in the presence of pointwise state constraints. The paper is divided into two parts. In the first part we define solutions of the state equation as the limit of a sequence of solutions for equations with Robin boundary conditions. We establish Taylor expansions for solutions of the state equation with respect to perturbations of boundary control (Theorem 5.2). For problems with no state constraints, we prove three decoupled Pontryagin's principles, one for the distributed control, one for the boundary control, and the last one for the control in the initial condition (Theorem 2.1). Tools and results of Part 1 are used in the second part to derive Pontryagin's principles for problems with pointwise state constraints. Accepted 12 July 2001. Online publication 21 December 2001.     

Dirichlet Boundary Control of Semilinear Parabolic Equations Part 2: Problems with Pointwise State Constraints

This paper is the continuation of the paper ``Dirichlet boundary control of semilinear parabolic equations. Part 1: Problems with no state constraints.' It is concerned with an optimal control problem with distributed and Dirichlet boundary controls for semilinear parabolic equations, in the presence of pointwise state constraints. We first obtain approximate optimality conditions for problems in which state constraints are penalized on subdomains. Next by using a decomposition theorem for some additive measures (based on the Stone—Cech compactification), we pass to the limit and recover Pontryagin's principles for the original problem. Accepted 21 July 2001. Online publication 21 December 2001.     

Minimax Control of Parabolic Systems with Dirichlet Boundary Conditions and State Constraints

In this paper we formulate and study a minimax control problem for a class of parabolic systems with controlled Dirichlet boundary conditions and uncertain distributed perturbations under pointwise control and state constraints. We prove an existence theorem for minimax solutions and develop effective penalized procedures to approximate state constraints. Based on a careful variational analysis, we establish convergence results and optimality conditions for approximating problems that allow us to characterize suboptimal solutions to the original minimax problem with hard constraints. Then passing to the limit in approximations, we prove necessary optimality conditions for the minimax problem considered under proper constraint qualification conditions. Accepted 7 June 1996     

Some Randomized Algorithms for Convex Quadratic Programming

We adapt some randomized algorithms of Clarkson [3] for linear programming to the framework of so-called LP-type problems, which was introduced by Sharir and Welzl [10]. This framework is quite general and allows a unified and elegant presentation and analysis. We also show that LP-type problems include minimization of a convex quadratic function subject to convex quadratic constraints as a special case, for which the algorithms can be implemented efficiently, if only linear constraints are present. We show that the expected running times depend only linearly on the number of constraints, and illustrate this by some numerical results. Even though the framework of LP-type problems may appear rather abstract at first, application of the methods considered in this paper to a given problem of that type is easy and efficient. Moreover, our proofs are in fact rather simple, since many technical details of more explicit problem representations are handled in a uniform manner by our approach. In particular, we do not assume boundedness of the feasible set as required in related methods. Accepted 7 May 1997     

大规模严格凸二次规划问题一个新算法

根据广义乘子法的思想,将具有等式约束和非负约束的凸二次规划问题转化只有非负约束的简单凸二次规划,通过简单凸二次规划来得到解等式约束一非负约束的凸二次规划新算法,新算法不用求逆矩阵,这样可充分保持矩阵的稀疏性,用来解大规模稀疏问题,数值结果表明：在微机４８６／３３上就能解较大规模的凸二次规划。     

A CLASS OF REDUCED GRADIENT METHODS FOR HANDLING OPTIMIZATION PROBLEMS WITH LINEAR INEQUALITY CONSTRAINTS

A class of reduced gradient methods for handling general optimization problems with linear equality and inequality constraints is suggested in this paper. Although a slack vector is introduced, the dimension of the problem is not increased, which is unlike the conventional way of transferring the inequality constraints into the equality constraints by introducing slack variables. When an iterate x(k) is not a K-T point of the problem under consideration, different feasible descent directions can be obtained by different choices of the slack vectors. The suggested method is globally convergent and the numerical experiment given in the paper shows that the method is efficient.     

AN EXTREMAL APPROACH TO BIRKHOFF QUADRATURE FORMULAS

1. Introduction and Main ResultsIn tfor paPer we shaJl use the ddstions and notations of [3l. Let E = (e'k)7t' kt. be anincidence matrir with entries consisting of zeros and ones and satisfying lEl:= Z.,* ei* = n + 1(here we allow a zero row ). Furthermore, in wha follOws we assume that(A) E satisfies the P6lya condition(B) all sequences of E in the interior rows, 0 < i < m + 1, are even.Let Sm denote the set of poiats X = (xo, z1 l "') xm, x.+1) fOr whichand Sm its clOusure. If some O…