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

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

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 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

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

Constrained optimal control theory for differential linear repetitive processes

Differential repetitive processes are a distinct class of continuous-discrete two-dimensional linear systems of both systems theoretic and applications interest. These processes complete a series of sweeps termed passes through a set of dynamics defined over a finite duration known as the pass length, and once the end is reached the process is reset to its starting position before the next pass begins. Moreover the output or pass profile produced on each pass explicitly contributes to the dynamics of the next one. Applications areas include iterative learning control and iterative solution algorithms, for classes of dynamic nonlinear optimal control problems based on the maximum principle, and the modeling of numerous industrial processes such as metal rolling, long-wall cutting, etc. In this paper we develop substantial new results on optimal control of these processes in the presence of constraints where the cost function and constraints are motivated by practical application of iterative learning control to robotic manipulators and other electromechanical systems. The analysis is based on generalizing the well-known maximum and $\epsilon$-maximum principles to the

Mirror instability in a plasma with cold gyrating dust particles

In this work linear stability analysis of a magnetized dusty plasma with an anisotropic dust component having transversal motions much stronger than motions parallel to the external magnetic field, and isotropic light plasma components is described. Such a situation presumably establishes in a shock compressed space dusty plasma downstream the shock front. Oblique low-frequency magneto-hydrodynamic waves ($\omega\ll \omega_{cd}$, $\omega_{cd}$ being the dust cyclotron frequency) are shown to be undergone to the mirror instability. Consequences for nonthermal dust destruction behind shock fronts in the interstellar medium are discussed.Comment: 13 pages, 5 figs, accepted to Phys. Pasma

The Motives for Corporate Identification of the Contemporary Russian University

AbstractThe paper addresses the problem of reexamining the identification of the contemporary Russian university in connection with its new corporate tendencies. We propose to see in corporatism and corporate culture of the university its value-oriented, spiritually-moral meaning that ensures ethical norms of human behavior in the network society. Seen from this perspective, corporate culture does not contradict the professional university culture, and appears to be necessary for training graduates and professionals who are capable of answering the challenges of the contemporary professional world

On the algorithm of determination of immobile indices for convex SIP problems

We consider convex Semi-Infinite Programming (SIP) problems with a continuum of constraints. For these problems we introduce new concepts of immobility orders and immobile indices. These concepts are objective and important characteristics of the feasible sets of the convex SIP problems since they make it possible to formulate optimality conditions for these problems in terms of optimality conditions for some NLP problems (with a finite number of constraints). In the paper we describe a finite algorithm (DIO algorithm) of determination of immobile indices together with their immobility orders, study some important properties of this algorithm, and formulate the Implicit Optimality Criterion for convex SIP without any constraint qualification conditions (CQC). An example illustrating the application of the DIO algorithm is provided