利用非可导性条件判别函数的凸性

本文先给出凸函数两个常用定义,并对连续函数证明其等价性.然后给出函数凸性的一些几何判别法和积分判别法.最后给出一个有用的微分判别法.     

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。     

关于对偶问题及对偶性质

本文考虑非可微凸规划的一个对偶问题，它使用目标函数的扰动函数的次微分及外法向量锥，它不同于已知结果．我们给出相应的对偶性质．     

非凸规划的对偶问题极值

本文对非凸规划的对偶问题的目标函数极值给出一个表达式 ,从而得出对偶间隙 ,使用的方法是扰动函数的凸色 ,而不使用任何有关凸性的假定     

凸整数规划最优点的判定条件

本文得到判别已知可行整值点为凸整数规划最优解的一个充分条件,此条件只涉及目标函数在该整值点为中心的边长为2的超立方体上的性态.     

多目标半定规划的最优性条件及对偶理论

在不变凸的假设下来讨论多目标半定规划的最优性条件、对偶理论以及非凸半定规划的最优性条件．首先给出了非凸半定规划的一个KKT条件成立的充分必要条件, 并利用此定理证明了其最优性必要条件．其次讨论了多目标半定规划的最优性必要条件、充分条件, 并对其建立Wolfe对偶模型, 证明了弱对偶定理和强对偶定理．     

E-凸函数的若干特征

讨论了一类广义的凸集和凸函数:E-凸集和E-凸函数的若干性质,并给出E-凸函数的一个判别准则．     

外形设计中检验凸曲面的判别条件

关于判别三维欧氏空间中一张曲面是否为凸曲面的问题,我们曾经给出过一些判别法。而在工程设计上却有这样的一种判别法:沿着直角坐标系的三个坐标轴作一系列的平行平面与坐标轴相垂直,于是得到三组平行平面,在此之外再另取一平面,如果所有这些平面与曲面相截得出的都是凸曲线,那么就认为曲面是凸曲面。     

拟凸函数判别准则的一个注记

我们在上半连续的条件下,给出了拟凸函数的一个新的判别准则,即：凸集上的一个上半连续函数是拟凸的充分必要条件是这个函数是中间拟凸的。     

求解群体多目标凸规划的一个交互规划算法

本文给出一个由决策群体求解带有非线性约束的多目标凸规划问题的交互规划算法。此法通过决策群体与分析者之间以及各决策者之间的交互，最终得到群体满意的有效解，本文还证明了这一算法的收敛性定理。     

A new approximation approach to optimality and duality for a class of nonconvex differentiable vector optimization problems

Computational Management Science - In this paper, a new approximation method for a characterization of (weak) Pareto solutions in some class of nonconvex differentiable multiobjective programming...     

Analysis of a smoothing method for symmetric conic linear programming

This paper proposes a smoothing method for symmetric conic linear programming (SCLP). We first characterize the central path conditions for SCLP problems with the help of Chen-Harker-Kanzow-Smale smoothing function. A smoothing-type algorithm is constructed based on this characterization and the global convergence and locally quadratic convergence for the proposed algorithm are demonstrated.     

pth Power Lagrangian Method for Integer Programming

When does there exist an optimal generating Lagrangian multiplier vector (that generates an optimal solution of an integer programming problem in a Lagrangian relaxation formulation), and in cases of nonexistence, can we produce the existence in some other equivalent representation space? Under what conditions does there exist an optimal primal-dual pair in integer programming? This paper considers both questions. A theoretical characterization of the perturbation function in integer programming yields a new insight on the existence of an optimal generating Lagrangian multiplier vector, the existence of an optimal primal-dual pair, and the duality gap. The proposed pth power Lagrangian method convexifies the perturbation function and guarantees the existence of an optimal generating Lagrangian multiplier vector. A condition for the existence of an optimal primal-dual pair is given for the Lagrangian relaxation method to be successful in identifying an optimal solution of the primal problem via the maximization of the Lagrangian dual. The existence of an optimal primal-dual pair is assured for cases with a single Lagrangian constraint, while adopting the pth power Lagrangian method. This paper then shows that an integer programming problem with multiple constraints can be always converted into an equivalent form with a single surrogate constraint. Therefore, success of a dual search is guaranteed for a general class of finite integer programming problems with a prominent feature of a one-dimensional dual search.     

A compact formulation of an elastoplastic analysis problem

Two important problems in the area of engineering plasticity are limit load analysis and elastoplastic analysis. It is well known that these two problems can be formulated as linear and quadratic programming problems, respectively (Refs. 1–2). In applications, the number of variables in each of these mathematical programming problems tends to be large. Consequently, it is important to have efficient numerical methods for their solution. The purpose of this paper is to present a method which allows the quadratic programming formulation of the elastoplastic analysis to be reformulated as an equivalent quadratic programming problem which has significantly fewer variables than the original formulation. Indeed, in Section 4, we will present details of an example for which the original quadratic programming formulation required 297 variables and for which the equivalent formulation presented here required only two variables. The method is based on a characterization of the entire family of optimal solutions for a linear programming problem.This research was supported by the Natural Science and Engineering Council of Canada under Grant No. A8189 and by a Leave Fellowship from the Social Sciences and Humanities Research Council of Canada. The author takes pleasure in acknowledging many stimulating discussions with Professor D. E. Grierson.     

The nonnegative MINQUE estimate

The purpose of this paper is to give a characterization of the nonnegative MINQUE estimate for variance components. A similar characterization has been given by Pukelsheim but only in some special cases. The proof presented here uses results from convex programming and emphasizes certain geometrical aspects of the nonnegative MINQUE estimate problem.     

Convergence property of the Iri-Imai algorithm for some smooth convex programming problems

In this paper, the Iri-Imai algorithm for solving linear and convex quadratic programming is extended to solve some other smooth convex programming problems. The globally linear convergence rate of this extended algorithm is proved, under the condition that the objective and constraint functions satisfy a certain type of convexity, called the harmonic convexity in this paper. A characterization of this convexity condition is given. The same convexity condition was used by Mehrotra and Sun to prove the convergence of a path-following algorithm.The Iri-Imai algorithm is a natural generalization of the original Newton algorithm to constrained convex programming. Other known convergent interior-point algorithms for smooth convex programming are mainly based on the path-following approach.     

齐次多目标规划问题的KKT点

通过构造原问题的辅助问题,得到多目标规划问题的一些性质.并且给出目标函数是齐次函数的多目标优化问题KKT点的一个等价性质.     

Lower subdifferentiable functions and their minimization by cutting planes

This paper introduces lower subgradients as a generalization of subgradients. The properties and characterization of boundedly lower subdifferentiable functions are explored. A cutting plane algorithm is introduced for the minimization of a boundedly lower subdifferentiable function subject to linear constraints. Its convergence is proven and the relation is discussed with the well-known Kelley method for convex programming problems. As an example of application, the minimization of the maximum of a finite number of concave-convex composite functions is outlined.The author thanks the referees for several constructive remarks.     

Duality and optimality for convex extremal problems described by discrete inclusions

This paper is devoted to the Hamiltonian approach for extremal problems concerning convex (multi-valued) mapping. The approach exploits the concept of a Hamiltonian function permitting simplified proofs and useful mathematical insights. Moreover it provides in a duality framework a common point ox view upon the methods used. by Rockafellar, (the theory of convex processes), Pshenichnyi (the conjugate transformation method) and CASS (the symmetric duality scheme) to construct optimality conditions. The theory is used to develop a complete characterization of optimal solutions for multi-period convex programming problems.     

The problem of optimal control with reflection studied through a linear optimization problem stated on occupational measures

We obtain a linear programming characterization for the minimum cost associated with finite dimensional reflected optimal control problems. In order to describe the value functions, we employ an infinite dimensional dual formulation instead of using the characterization via Hamilton-Jacobi partial differential equations. In this paper we consider control problems with both infinite and finite horizons. The reflection is given by the normal cone to a proximal retract set.