Convex Semi-Infinite programming: explicit optimality conditions

We consider the convex Semi-In¯nite Programming (SIP) problem where objec- tive function and constraint function are convex w.r.t. a ¯nite-dimensional variable x and all of these functions are su±ciently smooth in their domains. The constraint function depends also on so called time variable t that is de¯ned on the compact set T ½ R. In our recent paper [15] the new concept of immobility order of the points of the set T was introduced and the Implicit Optimality Criterion was proved for the convex SIP problem under consideration. In this paper the Implicit Optimality Criterion is used to obtain new ¯rst and second order explicit optimality conditions. We consider separately problems that satisfy and that do not satisfy the the Slater condition. In the case of SIP problems with linear w.r.t. x constraints the optimal- ity conditions have a form of the criterion. Comparison of the results obtained with some other known optimality conditions for SIP problems is provided as well

Rigidity of abnormal extrema in the problem of non-linear programming with mixed constraints

We study abnormal extremum in the problem of non-linear pro- gramming with mixed constraints. Abnormal extremum occurs when in necessary optimality conditions the Lagrange multiplier, which cor- responds to the objective function, vanishes. We demonstrate that in this case abnormal second-order su±cient optimality conditions guar- antee rigidity of the corresponding extremal point, which means iso- latedness of this point in the set determined by the constraints

A constructive algorithm for determination of immobile indices in convex SIP problems with polyhedral index sets

We consider convex Semi-Infinite Programming (SIP) problems with polyhedral index sets. For these problems, we generalize the concepts of immobile indices and their immobility orders (that are objective and important characteristics of the feasible sets permitting to formulate new efficient optimality conditions. We describe and justify a finite constructive algorithm (DIIPS algorithm) that determines immobile indices and their immobility orders along the feasible directions. This algorithm is based on a representation of the cones of feasible directions of polyhedral index sets in the form of linear combinations of the extremal rays {and on the approach described in our previous papers for the cases of multidimensional immobile sets of more simple structure. A constructive procedure of determination of the extremal rays is described and an example illustrating the application of the DIIPS algorithm is provided

On equivalent representations and properties of faces of the cone of copositive matrices

The paper is devoted to a study of the cone COPp of copositive matrices. Based on the known from semi-infinite optimization concept of immobile indices, we define zero and minimal zero vectors of a subset of the cone COPp and use them to obtain different representations of faces of COPp and the corresponding dual cones. We describe the minimal face of COPp containing a given convex subset of this cone and prove some propositions that allow to obtain equivalent descriptions of the feasible sets of a copositive problems and may be useful for creating new numerical methods based on their regularization.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

On strong duality in linear copositive programming

The paper is dedicated to the study of strong duality for a problem of linear copositive programming. Based on the recently introduced concept of the set of normalized immobile indices, an extended dual problem is deduced. The dual problem satisfies the strong duality relations and does not require any additional regularity assumptions such as constraint qualifications. The main difference with the previously obtained results consists in the fact that now the extended dual problem uses neither the immobile indices themselves nor the explicit information about the convex hull of these indices. The strong duality formulations presented in the paper have similar structure and properties as that proposed in the works of M. Ramana, L. Tuncel, and H. Wolkovicz, for semidefinite programming, but are obtained using different techniques.publishe

ЗАДАЧИ ЛИНЕЙНОГО ПОЛУОПРЕДЕЛЕННОГО ПРОГРАММИРОВАНИЯ: РЕГУЛЯРИЗАЦИЯ И ДВОЙСТВЕННЫЕ ФОРМУЛИРОВКИ В СТРОГОЙ ФОРМЕ

Regularisation consists in reducing a given optimisation problem to an equivalent form where certain regularity conditions, which guarantee the strong duality, are fulfilled. In this paper, for linear problems of semidefinite programming (SDP), we propose a regularisation procedure which is based on the concept of an immobile index set and its properties. This procedure is described in the form of a finite algorithm which converts any linear semidefinite problem to a form that satisfies the Slater condition. Using the properties of the immobile indices and the described regularisation procedure, we obtained new dual SDP problems in implicit and explicit forms. It is proven that for the constructed dual problems and the original problem the strong duality property holds true.Регуляризация задачи оптимизации состоит в ее сведении к эквивалентной задаче, удовлетворяющей условиям регулярности, которые гарантируют выполнение соотношений двойственности в строгой форме. В настоящей статье для линейных задач полуопределенного программирования предлагается процедура регуляризации, основанная на понятии неподвижных индексов и их свойствах. Эта процедура описана в виде алгоритма, который за конечное число шагов преобразует любую задачу линейного полубесконечного программирования в эквивалентную задачу, удовлетворяющую условию Слейтера. В результате использования свойств неподвижных индексов и предложенной процедуры регуляризации получены новые двойственные задачи полубесконечного программирования в явной и неявной формах. Доказано, что для этих двойственных задач и исходной задачи соотношения двойственности выполняются в строгой форме.publishe

Optimality conditions for linear copositive programming problems with isolated immobile indices

In the present paper, we apply our recent results on optimality for convex semi-infinite programming to a problem of linear copositive programming (LCP). We prove explicit optimality conditions that use concepts of immobile indices and their immobility orders and do not require the Slater constraint qualification to be satisfied. The only assumption that we impose here is that the set of immobile indices consists of isolated points and hence is finite. This assumption is weaker than the Slater condition; therefore, the optimality conditions obtained in the paper are more general when compared with those usually used in LCP. We present an example of a problem in which the new optimality conditions allow one to test the optimality of a given feasible solution while the known optimality conditions fail to do this. Further, we use the immobile indices to construct a pair of regularized dual copositive problems and show that regardless of whether the Slater condition is satisfied or not, the duality gap between the optimal values of these problems is zero. An example of a problem is presented for which the standard strict duality fails, but the duality gap obtained by using the regularized dual problem vanishes

Convex semi-infinite programming: implicit optimality criterion based on the concept of immobile points

The paper deals with convex Semi-In¯nite Programming (SIP) problems. A new concept of immobility order is introduced and an algorithm of determination of the immobility orders (DIO algorithm) and so called immobile points is suggested. It is shown that in the presence of the immobile points SIP problems do not satisfy the Slater condition. Given convex SIP problem, we determine all its immobile points and use them to formulate a Nonlinear Programming (NLP) problem in a special form. It is proved that optimality conditions for the (in¯nite) SIP problem can be formulated in terms of the analogous conditions for the corresponding (¯nite) NLP problem. The main result of the paper is the Implicit Optimality Criterion that permits to obtain new e±cient optimality conditions for the convex SIP problems (even not satisfying the Slater condition) using the known results of the optimality theory of NLP

Generalized problem of linear copositive programming

Статья посвящена изучению оптимизационных задач, в которых целевая функция линейна по конечномерной переменной х, в то время как ограничения линейны по х и квадратичны по индексу t, принадлежащему заданному конусу. Задачи такого вида могут интерпретироваться как обобщение задач полуопределенного и коположительного программирования. Для рассматриваемой задачи формулируется эквивалентная задача полубесконечного программирования и вводится множество неподвижных индексов, которое либо пусто, либо является объединением конечного числа выпуклых ограниченных многогранников. Изучение свойств множества допустимых планов позволило сформулировать и доказать новые эффективные условия оптимальности, которые не требуют дополнительных условий на ограничения и имеют форму критериев.We consider a special class of optimization problems where the objective function is linear w.r.t. decision variable х and the constraints are linear w.r.t. х and quadratic w.r.t. index t defined in a given cone. The problems of this class can be considered as a generalization of semi-definite and copositive programming problems. For these problems, we formulate an equivalent semi-infinite problem and define a set of immobile indices that is either empty or a union of a finite number of convex bounded polyhedra. We have studied properties of the feasible sets of the problems under consideration and use them to obtain new efficient optimality conditions for generalized copositive problems. These conditions are CQ-free and have the form of criteria.publishe