- 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
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 L2-Control Problem in Coefficients for a Linear Elliptic Equation. Submitted to Mathematical Control and Related Fields, 2013, 1-60. DOI: arXiv:1306.2513
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 L2(Ω; RN×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/
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; RN×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/
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; RN×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  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
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.
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