Conjugacy in quasi-convex programming

This paper develops a symmetric conjugate relation for quasi-convex functions. The concept of an evenly quasi-convex function is introduced and it is shown that this is the required property for a duality framework in quasi-convex programming.     

Epi-Convergence and Lower and Epi-Upper Semicontinuous Approximations in Distribution

The paper introduces an extension of the epi-convergence, the lower semicontinuous approximation and the epi-upper semicontinuous approximation of random real functions in distribution. The new notions could be helpful tools for sensitivity analyzes of stochastic optimization problems. The research is evoked by S. Vogel and continues the research started by Vogel and the author.     

Penalization in non-classical convex programming via variational convergence

A penalty method for convex functions which cannot necessarily be extended outside their effective domains by an everywhere finite convex function is proposed and combined with the proximal method. Proofs of convergence rely on variational convergence theory.     

A feasible SQP method for nonlinear programming

In this paper, a new SQP algorithm is presented to solve the general nonlinear programs with mixed equality and inequality constraints. Quoted from P. Spellucci (see [9]), this method maybe be named sequential equality constrained quadratic programming (SECQP) algorithm. Per single iteration, based on an active set strategy ( see [9]), this SECQP algorithm requires only to solve equality constrained quadratic programming subproblems or system of linear equations. The theoretical analysis shows that global and superlinear convergence can be induced under some suitable conditions.     

Penalty function theory for general convex programming problems

A class of penalty functions for solving convex programming problems with general constraint sets is considered. Convergence theorems for penalty methods are established by utilizing the concept of infimal convergence of a sequence of functions. It is shown that most existing penalty functions are included in our class of penalty functions.     

Directional derivatives of optimal value functions in mathematical programming

This note discusses the existence of the directional derivatives of the optimal value functions in a class of nonlinear programming problems and gives the expressions of the directional derivatives. In the study, it is not assumed that the optimal set at the point discussed is not empty. Many well-known results of this area can be derived as special cases of the main theorems of this note.This research was supported by the National Science Foundation of China. The authors would like to thank Professor A. V. Fiacco and the referees for their helpful suggestions.     

A recursive quadratic programming algorithm that uses differentiable exact penalty functions

In this paper, a recursive quadratic programming algorithm for solving equality constrained optimization problems is proposed and studied. The line search functions used are approximations to Fletcher's differentiable exact penalty function. Global convergence and local superlinear convergence results are proved, and some numerical results are given.     

A symmetry result for the p-Laplacian in a punctured manifold

Let M be a Riemannian manifold such that its geodesic spheres centered at a point a∈M are isoperimetric and the distance function dist(⋅,a) is isoparametric, and let Ω⊂M be a bounded domain. We prove that if there exists a lower bounded nonconstant function u which is p-harmonic (1<p?n) in the punctured domain Ω?{a} such that both u and are constant on ∂Ω, then u is radial and ∂Ω is a geodesic sphere. The proof hinges on a combination of maximum principles, isoparametricity and the isoperimetric inequality.     

Finite dimensional approximation in infinite dimensional mathematical programming

We consider the problem of approximating an optimal solution to a separable, doubly infinite mathematical program (P) with lower staircase structure by solutions to the programs (P(N)) obtained by truncating after the firstN variables andN constraints of (P). Viewing the surplus vector variable associated with theNth constraint as a state, and assuming that all feasible states are eventually reachable from any feasible state, we show that the efficient set of all solutions optimal to all possible feasible surplus states for (P(N)) converges to the set of optimal solutions to (P). A tie-breaking algorithm which selects a nearest-point efficient solution for (P(N)) is shown (for convex programs) to converge to an optimal solution to (P). A stopping rule is provided for discovering a value ofN sufficiently large to guarantee any prespecified level of accuracy. The theory is illustrated by an application to production planning.The work of Robert L. Smith was partially supported by the National Science Foundation under Grant ECS-8700836.     

A stability result concerning the obstacle problem for a plate

It is proposed here to study the free boundary of the obstacle problem in the case of an elastic plate. Under a nondegeneracy assumption, we prove a stability theorem which relates the variations of the contact zone to the variations the external forces. The statement of this result obtained and the steps in the proof are very close to those given by D.G. Schaeffer in 1975, except for the very important fact that the present study deals with the biharmonic operator.     

一种反向集值鞅型序列的收敛性

本文研究了在非空有界闭凸集的上鞅收敛性和有界闭凸集的适应可积集值序列的收敛性.引入了反向集值mil的基础上,得到了收敛性定理及定理证明.     

Lagrange multiplier theorem of multiobjective programming problems with set functions

In this paper, Lagrange multiplier theorems are developed for the cases of single-objective and multiobjective programming problems with set functions. Properly efficient solutions are also characterized by subdifferentials and zero-like functions.The authors greatly appreciate helpful and valuable comments and suggestions received from the referee.     

A correction to a result in linear programming

We show that Theorem 1 in [1, p.236] about linear programs is incorrect as stated. We provide the correct version of this theorem and an elementary proof of it.     

Forbidden sets and stability in some rational difference equations

In this paper, we solve open problem (5) submitted by Sedaghat in his paper, On third order rational difference equations with quadratic terms, J. Differ. Equ. Appl., 14(8) (2008), pp. 889–897. We also confirm conjecture (6) in the mentioned paper.     

On the stability of solutions to quadratic programming problems

We consider the parametric programming problem (Q _p ) of minimizing the quadratic function f(x,p):=x ^T Ax+b ^T x subject to the constraint Cx≤d, where x∈ℝ ⁿ , A∈ℝ ^n×n , b∈ℝ ⁿ , C∈ℝ ^m×n , d∈ℝ ^m , and p:=(A,b,C,d) is the parameter. Here, the matrix A is not assumed to be positive semidefinite. The set of the global minimizers and the set of the local minimizers to (Q _p ) are denoted by M(p) and M ^loc (p), respectively. It is proved that if the point-to-set mapping M ^loc (·) is lower semicontinuous at p then M ^loc (p) is a nonempty set which consists of at most ? _m,n points, where ? _m,n = is the maximal cardinality of the antichains of distinct subsets of {1,2,...,m} which have at most n elements. It is proved also that the lower semicontinuity of M(·) at p implies that M(p) is a singleton. Under some regularity assumption, these necessary conditions become the sufficient ones. Received: November 5, 1997 / Accepted: September 12, 2000?Published online November 17, 2000     

On convergence properties of a least-distance programming procedure for minimization problems under linear constraints

In Ref. 1, Bazaraa and Goode provided an algorithm for solving a nonlinear programming problem with linear constraints. In this paper, we show that this algorithm possesses good convergence properties.This paper was written under the guidance of Associate Professor C. Y. Wang. The author takes great pleasure in thanking him.     

A result in surrogate duality for certain integer programming problems

We consider linear programming problems with some equality constraints. For such problems, surrogate relaxation formulations relaxing equality constraints existwith zero primal-dual gap both when all variables are restricted to be integers and when no variable is required to be integer. However, for such surrogate formulations, when the variables are mixed-integer, the primal-dual gap may not be zero. We establish this latter result by a counterexample.     

A non-interior continuation method for second-order cone programming

We extend the smoothing function proposed by Huang, Han and Chen [Journal of Optimization Theory and Applications, 117 (2003), pp. 39–68] for the non-linear complementarity problems to the second-order cone programming (SOCP). Based on this smoothing function, a non-interior continuation method is presented for solving the SOCP. The proposed algorithm solves only one linear system of equations and performs only one line search at each iteration. It is shown that our algorithm is globally and locally superlinearly convergent in absence of strict complementarity at the optimal solution. Numerical results indicate the effectiveness of the algorithm.     

An optimization result for set-valued mappings and a stability property in vector problems with constraints

First, we prove an existence result relative to minimal points of set-valued mappings. Then, conditions about the upper and lower semicontinuity of constraint sets defined through set-valued mappings are given. Finally, a stability result relative to vector problems with abstract constraints is proved.The author thanks the referee for helpful comments on the first version of this paper.