On a constructive approach to optimality conditions for convex SIP problems with polyhedral index sets

In the paper, we consider a problem of convex Semi-Infinite Programming with an infinite index set in the form of a convex polyhedron. In study of this problem, we apply the approach suggested in our recent paper [Kostyukova OI, Tchemisova TV. Sufficient optimality conditions for convex Semi Infinite Programming. Optim. Methods Softw. 2010;25:279–297], and based on the notions of immobile indices and their immobility orders. The main result of the paper consists in explicit optimality conditions that do not use constraint qualifications and have the form of criterion. The comparison of the new optimality conditions with other known results is provided

CQ-free optimality conditions and strong dual formulations for a special conic optimization problem

In this paper, we consider a special class of conic optimization problems, consisting of set-semidefinite (orK-semidefinite) programming problems, where the setKis a polyhedral convex cone. For these problems, we introduce theconcept of immobile indices and study the properties of the set of normalized immobile indices and the feasible set. Thisstudy provides the main result of the paper, which is to formulate and prove the new first-order optimality conditions inthe form of a criterion. The optimality conditions are explicit and do not use any constraint qualifications. For the case of alinear cost function, we reformulate theK-semidefinite problem in a regularized form and construct its dual. We show thatthe pair of the primal and dual regularized problems satisfies the strong duality relation which means that the duality gap is vanishing.publishe

Implicit optimality criterion for convex SIP problem with box constrained index set

We consider a convex problem of Semi-Infinite Programming (SIP) with multidimensional index set. In study of this problem we apply the approach suggested in [20] for convex SIP problems with one-dimensional index sets and based on the notions of immobile indices and their immobility orders. For the problem under consideration we formulate optimality conditions that are explicit and have the form of criterion. We compare this criterion with other known optimality conditions for SIP and show its efficiency in the convex case

γ-Active constraints in convex semi-infinite programming

In this article, we extend the definition of γ-active constraints for linear semi-infinite programming to a definition applicable to convex semi-infinite programming, by two approaches. The first approach entails the use of the subdifferentials of the convex constraints at a point, while the second approach is based on the linearization of the convex inequality system by means of the convex conjugates of the defining functions. By both these methods, we manage to extend the results on γ-active constraints from the linear case to the convex case

Sufficient optimality conditions for convex semi-infinite programming

We consider a convex semi-infinite programming (SIP) problem whose objective and constraint functions are convex w.r.t. a finite-dimensional variable x and whose constraint function also depends on a so-called index variable that ranges over a compact set inR. In our previous paper [O.I.Kostyukova,T.V. Tchemisova, and S.A.Yermalinskaya, On the algorithm of determination of immobile indices for convex SIP problems, IJAMAS Int. J. Math. Stat. 13(J08) (2008), pp. 13–33], we have proved an implicit optimality criterion that is based on concepts of immobile index and immobility order. This criterion permitted us to replace the optimality conditions for a feasible solution x0 in the convex SIP problem by similar conditions for x0 in certain finite nonlinear programming problems under the assumption that the active index set is finite in the original semi-infinite problem. In the present paper, we generalize the implicit optimality criterion for the case of an infinite active index set and obtain newfirst- and second-order sufficient optimality conditions for convex semi-infinite problems. The comparison with some other known optimality conditions is provided

Estudo prático de regularidade de problemas de programação semidefinida

Mestrado em Matemática e Aplicações - Matemática Empresarial e TecnológicaUm problema linear de Programa c~ao Semide nida (SDP) consiste na minimiza c~ao de uma fun c~ao linear sujeita a condi c~ao de que a fun c~ao matricial linear seja semide nida. Um problema de SDP considera-se regular se certas condi c~oes est~ao satisfeitas. H a diferentes caracteriza c~oes de regularidade de um problema, sendo uma delas a veri ca c~ao da condi c~ao de Slater. Os problemas regulares de SDP t^em sido estudados e as condi c~oes de optimalidade para estes problemas t^em a forma de teoremas cl assicos do tipo Karush-Kuhn-Tucker, e s~ao facilmente veri cadas. Na pr atica, e frequente encontrar problemas n~ao regulares. O estudo destes problemas e bem mais complicado. Por isso, tem surgido o interesse em estudar e testar a regularidade dos problemas de SDP e deduzir condi c~oes de optimalidade e m etodos de resolu c~ao dos problemas n~ao regulares. Em Kostyukova e Tchemisova [32] e proposto um algoritmo, chamado Algoritmo DIIS (Algorithm of Determination of the Immobile Index Subspace), que permite veri car se as restri c~oes de um dado problema de SDP satisfazem a condi c~ao de Slater. A teoria que serve de base a constru c~ao deste algoritmo assenta nas no c~oes de ndices e subespa co de ndices im oveis, originalmente usadas em Programa c~ao Semi-In nita (SIP), e transpostas em [32] para SDP. Este algoritmo constr oi uma matriz b asica do subespa co de ndices im oveis, caso a condi c~ao de Slater n~ao seja veri cada. A dimens~ao desta matriz caracteriza o grau de n~ao regularidade do problema. O objectivo deste trabalho e estudar o Algoritmo DIIS, implement a- -lo e test a-lo usando v arios problemas de teste de diferentes bases de dados de problemas de SDP. O Algoritmo DIIS foi implementado e executado a partir do MatLab e os testes num ericos efectuados permitiram concluir que o programa constru do veri ca com sucesso a maioria dos problemas teste. Al em disso, o algoritmo permite caracterizar o grau de n~ao regularidade dos problemas de SDP e pode ser usado para constru c~ao de algoritmos de resolu c~ao dos problemas de SDP n~ao regulares.A linear problem of Semide nite Programming (SDP) consists of minimizing a linear function subject to the constraint that the linear matrix function is semide nite. An SDP problem is considered regular if certain conditions are satis- ed. There are several characterizations of a problem regularity, one of which is checking the Slater condition. The regular SDP problems have been studied and the optimality conditions for these problems have the form of Karush-Kuhn-Tucker type theorems, and are easily veri ed. In practice it is common to nd problems that are not regular. The study of these problems is far more complicated. Therefore, there has been interest in studying and testing the regularity of the problems and deduct the SDP optimality conditions and methods for solving non-regular problems. In Kostyukova e Tchemisova [32] is proposed an algorithm, called Algorithm DIIS (Algorithm of Determination of the Immobile Index Subspace), which allows to check if the constraints of a given SDP problem satisfy the Slater condition. The theory that underlies the construction of this algorithm is based on the notions of subspace of immobile indices and immobile indices properties, originally used in Semi-In nite Programming (SIP), and implemented in [32] for SDP. This algorithm constructs a basic matrix of the subspace of immobile indices if the Slater condition is not veri ed. The size of this matrix characterizes the degree of non-regularity of the problem. The purpose of this work is to study the DIIS algorithm, implement and test it using several test problems of di erent databases of SDP problems. The DIIS algorithm was implemented and executed from MatLab and numerical tests carried out showed that the program checks successfully the majority of test problems. Moreover, the algorithm allows to characterize the degree of non-regular problems of SDP and can be used to construct algorithms for solving non-regular SDP problems

On a constructive approach to optimality conditions for convex SIP problems with polyhedral index sets

In the paper,we consider a problem of convex Semi-Infinite Programming with an infinite index set in the form of a convex polyhedron. In study of this problem, we apply the approach suggested in our recent paper [Kostyukova OI, Tchemisova TV. Sufficient optimality conditions for convex Semi Infinite Programming. Optim. Methods Softw. 2010;25:279–297], and based on the notions of immobile indices and their immobility orders. The main result of the paper consists in explicit optimality conditions that do not use constraint qualifications and have the form of criterion. The comparison of the new optimality conditions with other known results is provided