# Main publications

**Buttazzo, G. and Kogut, P.I.,**On quadratic scalarization of vector optimization problems in Banach spaces.*Applicable Analysis, Vol. 93, No. 5, 2014, 994-1009.**DOI: 10.1080/00036811.2013.809068*

**Abstract**

We study vector optimization problems in partially ordered Banach spaces and suppose that the objective mapping possesses a weakened property of lower semicontinuity and make no assumptions on the interior of the ordering cone. We discuss the so-called adaptive scalarization of such problems and show that the corresponding scalar non-linear optimization problems can be by-turn approximated by quadratic minimization problems.

**Horsin, T. and Kogut, P.I.,**Optimal*L*^{2}-Control Problem in Coefficients for a Linear Elliptic Equation.*Submitted to Mathematical Control and Related Fields, 2013, 1-60.**DOI: arXiv:1306.2513*

**Abstract**

In this paper we study an optimal control problem (OCP) associated
to a linear elliptic equation on a bounded domain Ω. The matrixvalued
coefficients A of such systems is our control in Ω and will be taken in
*L*^{2}(Ω; R^{N×N}) which in particular may comprise some cases of unboundedness.
Concerning the boundary value problems associated to the equations of this
type, one may face non-uniqueness of weak solutions— namely, approximable
solutions as well as another type of weak solutions that can not be obtained
through the L^{∞}-approximation of matrix A. Following the direct method in
the calculus of variations, we show that the given OCP is well-posed in the
sense that it admits at least one solution. At the same time, optimal solutions
to such problem may have a singular character in the above sense. In view
of this, we indicate two types of optimal solutions to the above problem: the
so-called variational and non-variational solutions, and show that some of that
optimal solutions can be attainable by solutions of special optimal boundary
control problems.

**Kogut, P.I., Kogut, O.P. and Leugering, G.,**Optimal Control in Matrix-Valued Coefficients for Nonlinear Monotone Problems: Optimality Conditions. Part I.*Journal of Analysis and its Applications (ZAA), Volume 34, Issue 1, 2015, 85-108.**DOI: 10.4171/ZAA/*

**Abstract**

In this article we study an optimal control problem for a nonlinear monotone
Dirichlet problem where the controls are taken as matrix-valued coefficients in *L*^{∞}(Ω; R^{N×N}).
For the exemplary case of a tracking cost functional, we derive first order optimality conditions.
This is the first part out of two articles. This first part is concerned with the general
case of matrix-valued coefficients under some hypothesis, while the second part focuses on
the special class of diagonal matrices.

**Kogut, P.I., Kogut, O.P. and Leugering, G.,**Optimal Control in Matrix-Valued Coefficients for Nonlinear Monotone Problems: Optimality Conditions. Part II.*Journal of Analysis and its Applications (ZAA), Volume 34, Issue 2, 2015, 199-219.**DOI: DOI: 10.4171/ZAA/*

**Abstract**

In this paper we study an optimal control problem for a nonlinear monotone Dirichlet problem where the controls are taken as the matrix-valued coefficients in
*L*^{∞}(Ω; R^{N×N}). Given a suitable cost function, the objective is to provide a substantiation
of the first order optimality conditions using the concept of convergence in variable
spaces. While in the first part [2] optimality conditions have been derived an analysed in
the general case under some assumptions on the quasi-adjoint states, in this second part,
we consider diagonal matrices and analyse the corresponding optimality system without
such assumptions.

**Dovzhenko, A.V., Kogut, P.I., and Manzo, R.,**Epi and Coepi-Analysis of One Class of Vector-Valued Mappings.*Optimization. A Journal of Mathematical Programming and Operations Research, 2014, Vol 63, Issue 4, P. 535-557.**DOI: DOI: 10.1080/0233193YY*

**Abstract**

This paper deals with a new characterization of lower semicontinuity of vector-valued mappings in normed spaces. We study the link between the lower semicontinuity property of vector-valued mappings and the topological properties of their epigraphs and coepigraphs, respectively. We show that if the objective space is partially ordered by a pointed cone with nonempty interior, then coepigraphs are stable with respect to the procedure of their closure and, moreover, the locally semicompact vector-valued mappings with closed coepigraphs are lower semicontinuous. Using these results we propose some regularization schemes for vector-valued functions. In the case when there are no assumptions on the topological interior of the ordering cone, we introduce a new concept of lower semicontinuity for vector-valued mappings, the so-called epi-lower semicontinuity, which is closely related with the closedness of epigraphs of such mappings, and study their main properties. All principal notions and assertions are illustrated by numerous examples.

Страницы: 1 2 3 4 5 6 7 8 9 10