On equivalent stability properties in semi-infinite optimization

This paper studies noncompact feasible sets of a semi-infinite optimization problem which are defined by finitely many equality constraints and infinitely many inequality constraints. The main result is the equivalence of the overall validity of the Extended Mangasarian Fromovitz Constraint Qualification with certain (topological) stability conditions. Furthermore, two perturbation theorems being of independent interest are presented.This work was supported by the Deutsche Forschungsgemeinschaft under grant Gu 304/1-2.     

Theorems of the alternative and optimality conditions

A theorem of the alternative is stated for generalized systems. It is shown how to deduce, from such a theorem, known optimality conditions like saddle-point conditions, regularity conditions, known theorems of the alternative, and new ones. Exterior and interior penalty approaches, weak and strong duality are viewed as weak and strong alternative, respectively.     

Universal duality in conic convex optimization

Given a primal-dual pair of linear programs, it is well known that if their optimal values are viewed as lying on the extended real line, then the duality gap is zero, unless both problems are infeasible, in which case the optimal values are +∞ and −∞. In contrast, for optimization problems over nonpolyhedral convex cones, a nonzero duality gap can exist when either the primal or the dual is feasible. For a pair of dual conic convex programs, we provide simple conditions on the ``constraint matrices' and cone under which the duality gap is zero for every choice of linear objective function and constraint right-hand side. We refer to this property as ``universal duality'. Our conditions possess the following properties: (i) they are necessary and sufficient, in the sense that if (and only if) they do not hold, the duality gap is nonzero for some linear objective function and constraint right-hand side; (ii) they are metrically and topologically generic; and (iii) they can be verified by solving a single conic convex program. We relate to universal duality the fact that the feasible sets of a primal convex program and its dual cannot both be bounded, unless they are both empty. Finally we illustrate our theory on a class of semidefinite programs that appear in control theory applications. This work was supported by a fellowship at the University of Maryland, in addition to NSF grants DEMO-9813057, DMI0422931, CUR0204084, and DoE grant DEFG0204ER25655. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation or those of the US Department of Energy.     

一类非光滑广义分式规划的Kuhn-Tucker型最优性必要条件

考虑一类非线性不等式约束的非光滑minimax分式规划问题;目标函数中的分子是可微函数与凸函数之和形式而分母是可微函数与凸函数之差形式,且约束函数是可微的.在Arrow- Hurwicz-Uzawa约束品性下,给出了这类规划的最优解的Kuhn-Tucker型必要条件.所得结果改进和推广了已有文献中的相应结果.     

一类不可微广义分式规划的K-T型必要条件

本文对一类在Rn的开子集X上的非线性不等式约束的广义分式规划问题: 目标函数中的分子是可微函数与凸函数之和而分母是可微函数与凸函数之差,且约束函数是可微的,在Abadie约束品性或Calmness约束品性下,给出了最优解的Kuhn-Tucker 型必要条件,所得结果改进和推广了已有文献中的相应结果．     

Computational Experience with Ill-Posed Problems in Semidefinite Programming

Many theoretical and algorithmic results in semidefinite programming are based on the assumption that Slater's constraint qualification is satisfied for the primal and the associated dual problem. We consider semidefinite problems with zero duality gap for which Slater's condition fails for at least one of the primal and dual problem. We propose a numerically reasonable way of dealing with such semidefinite programs. The new method is based on a standard search direction with damped Newton steps towards primal and dual feasibility.     

Subdifferential Evolution Inclusion in Nonconvex Analysis

Under quite general assumptions, we prove existence, uniqueness and regularity of a solution U to the evolution equation U'(t) + (g F)(U(t)) 0, U(0)=u₀, where g : X {} is a closed convex proper function, F : Y X is a continuously differentiable mapping whose Jacobian is locally Lipschitz continuous, X and Y being two Hilbert spaces. We also study the stability and the asymptotic behavior of U, and give various examples.     

Optimality conditions in multiobjective differentiable programming

Necessary conditions not requiring convexity are based on the convergence of a vector at a point and on Motzkin's theorem of the alternative. A constraint qualification is also involved in the establishment of necessary conditions. Three theorems on sufficiency require various levels of convexity on the component of the functions involved, and the equality constraints are not necessarily linear. Scalarization of the objective function is used only in the last sufficiency theorem.The author is thankful to the unknown referce whose comments improved the quality of the paper.     

Nonlinear programming in normed linear spaces

Necessary conditions in the form of multiplier rules are given for a function to have a constrained minimum. First-order differentiability conditions are imposed, and various combinations of set, equality, and inequality constraints are considered in arbitrary normed linear spaces.This paper is based upon part of the author's doctoral dissertation at Ohio University, Athens, Ohio.     

Some relationships between lagrangian and surrogate duality in integer programming

Lagrangian dual approaches have been employed successfully in a number of integer programming situations to provide bounds for branch-and-bound procedures. This paper investigates some relationship between bounds obtained from lagrangian duals and those derived from the lesser known, but theoretically more powerful surrogate duals. A generalization of Geoffrion's integrality property, some complementary slackness relationships between optimal solutions, and some empirical results are presented and used to argue for the relative value of surrogate duals in integer programming. These and other results are then shown to lead naturally to a two-phase algorithm which optimizes first the computationally easier lagrangian dual and then the surrogate dual.