凸规划的新算法

对一般凸目标函数和一般凸集约束的凸规划问题新解法进行探讨,它是线性规划一种新算法的扩展和改进,此算法的基本思想是在规划问题的可行域中由所建－的一个切割面到另一个切割面的不断推进来求取最优的。文章对目标函数是二次的且约束是一般凸集和二次目标函数且约束是线性的情形,给出了更简单的算法。     

不等式约束优化基于新型积极识别集的SQCQP算法

本文提出一个新的求解非线性不等式约束优化问题的罚函数型序列二次约束二次规划(SQCQP)算法.算法每次迭代只需求解一个凸二次约束二次规划(QCQP)子问题,且通过引入新型积极识别集技术,QCQP子问题的规模显著减小,从而降低计算成本.在不需要函数凸性等较弱假设下,算法具有全局收敛性.初步的数值试验表明算法是稳定有效的.     

一个关于二次规划问题的分段线性同伦算法

本文发展了一个关于二次规划问题的分段线性同伦算法。该算法可看作是外点罚函数法的一个变体。凡是符合外点罚函数法收敛条件的二次规划问题用该算法均可经有限次轮回运算得到稳定解。大量的关于随机的凸二次规划问题的数值实验结果表明它的计算效率是高的，在某些条件下可能是多项式时间算法。     

凸二次交叉规划的等价形式

利用参数规划逆问题考虑凸二次交叉规划与多目标规划的关系 ,把交叉规划转变为同变量规划组 ,再把同变量规划组变为多目标规划 ,证明了凸二次交叉规划的均衡解与多目标规划的最优解的关系。     

一类值型双层凸规划的Johri一般对偶

本文首先给出一类特殊的值型凸二次双层规划一其下层子规划只含有线性约束(简记为VBCP)；然后证明了一般形式的VBCP可以等价变换为非增值型凸二次双层规划的形式；最后给出该类双层规划VBCP的Johri对偶规划及其对偶性质．     

边界约束非凸二次规划问题的分枝定界方法

本文是研究带有边界约束非凸二次规划问题,我们把球约束二次规划问题和线性约束凸二次规划问题作为子问题,分明引用了它们的一个求整体最优解的有效算法,我们提出几种定界的紧、松驰策略,给出了求解原问题整体最优解的分枝定界算法,并证明了该算法的收敛性,不同的定界组合就可以产生不同的分枝定界算法,最后我们简单讨论了一般有界凸域上非凸二次规划问题求整体最优解的分枝与定界思想。     

基于半定规划松弛的凸二层规划问题算法研究

提出使用凸松弛的方法求解二层规划问题,通过对一般带有二次约束的二次规划问题的半定规划松弛的探讨,研究了使用半定规划(SDP)松弛结合传统的分枝定界法求解带有凸二次下层问题的二层二次规划问题,相比常用的线性松弛方法,半定规划松弛方法可快速缩小分枝节点的上下界间隙,从而比以往的分枝定界法能够更快地获得问题的全局最优解.     

一类二层多目标规划的若干性质

本文对于下层为线性多目标规划的二层规划问题,在约束域非空有界的条件下证明了可行集的弱拟凸性、连通性,为算法设计提供了理论依据．     

目标控制型线性三级规划的基本性质

本文讨论了一类以下级目标函数最优值为反馈的线性三级递阶优化问题，按照参数规划的方法给出了可行集、最优解等概念，得到了可靠集的弱拟凸性，连通性等性质，为算法设计了基础。     

凸线性合成对策解的结构

两个对策的凸合成首先是由Neumann和Morganstem在[1]中引进的。Owen证明合成对策的稳定集可由其子对策的稳定集表示出来。本文一般化两个对策的合成到多个对策的凸线性合成,并且证明了凸线性合成对策的优化解(稳定集,核Shapley值,Banzkaf势指标)也可以通过其子对策的优化解表达出来。     

Maximization of the Ratio of Two Convex Quadratic Functions over a Polytope

In this paper, we will develop an algorithm for solving a quadratic fractional programming problem which was recently introduced by Lo and MacKinlay to construct a maximal predictability portfolio, a new approach in portfolio analysis. The objective function of this problem is defined by the ratio of two convex quadratic functions, which is a typical global optimization problem with multiple local optima. We will show that a well-designed branch-and-bound algorithm using (i) Dinkelbach's parametric strategy, (ii) linear overestimating function and (iii) -subdivision strategy can solve problems of practical size in an efficient way. This algorithm is particularly efficient for Lo-MacKinlay's problem where the associated nonconvex quadratic programming problem has low rank nonconcave property.     

New semidefinite programming relaxations for box constrained quadratic program

We establish in this paper optimal parametric Lagrangian dual models for box constrained quadratic program based on the generalized D.C. (difference between convex) optimization approach, which can be reformulated as semidefinite programming problems. As an application, we propose new valid linear constraints for rank-one relaxation.     

Inequalities involving partial derivatives

This paper presents a method for obtaining closed form solutions to serial and nonserial dynamic programming problems with quadratic stage returns and linear transitions. Global parametric optimum solutions can be obtained regardless of the convexity of the stage returns. The closed form solutions are developed for linear, convex, and nonconvex quadratic returns, as well as the procedure for recursively solving each stage of the problem. Dynamic programming is a mathematical optimization technique which is especially powerful for certain types of problems. This paper presents a procedure for obtaining analytical solutions to a class of dynamic programming problems. In addition, the procedure has been programmed on the computer to facilitate the solution of large problems.     

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.     

One modification of the logarithmic barrier function method in linear and convex programming

A novel modification of the logarithmic barrier function method is introduced for solving problems of linear and convex programming. The modification is based on a parametric shifting of the constraints of the original problem, similarly to what was done in the method of Wierzbicki-Hestenes-Powell multipliers for the usual quadratic penalty function (this method is also known as the method of modified Lagrange functions). The new method is described, its convergence is proved, and results of numerical experiments are given.     

Über lokale stabilitätsmengen in der parametrischen quadratischen optimierung

In the present paper the concept of so-called local stability sets introduced by F. No?i?ka in parametric linear programming is extend to the case of convex quadratic programs parametrized in the objective function and in the right-hand side of the linear constraints. The continuity of the optimal-set-map and of the extreme value function holds on these sets. It is shown that the local stability sets are connected. Examples which illustrate the necessity of certain assumptions are given.     

单输入多时滞离散系统的线性二次调节问题

研究了单输入多时滞的离散时间系统的线性二次调节问题(LQR问题),给出了求解最优控制输入序列的一种简单有效而又新颖的方法.将该动态的离散时滞系统的LQR最优控制问题最终转化成了一个静态的、不带时滞的数学规划模型——带等式线性约束的严格凸二次规划问题,并利用两种方法解这个二次规划问题,均成功地导出了系统的最优控制输入序列.仿真结果验证了我们的方法的正确有效性.     

Constructive Solution of Inverse Parametric Linear/Quadratic Programming Problems

Parametric convex programming has received a lot of attention, since it has many applications in chemical engineering, control engineering, signal processing, etc. Further, inverse optimality plays an important role in many contexts, e.g., image processing, motion planning. This paper introduces a constructive solution of the inverse optimality problem for the class of continuous piecewise affine functions. The main idea is based on the convex lifting concept. Accordingly, an algorithm to construct convex liftings of a given convexly liftable partition will be put forward. Following this idea, an important result will be presented in this article: Any continuous piecewise affine function defined over a polytopic partition is the solution of a parametric linear/quadratic programming problem. Regarding linear optimal control, it will be shown that any continuous piecewise affine control law can be obtained via a linear optimal control problem with the control horizon at most equal to 2 prediction steps.