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1
Content available Second-order optimality conditions in nonsmooth vector optimization
Yadav Priyanka
Control and Cybernetics
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2023
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Vol. 52, No. 1
35--51
EN
In this paper, we introduce new classes of nonsmooth second-order cone-convex functions and respective generalizations in terms of first and second-order directional derivative. These classes encapsulate several already existing classes of cone-convex functions and their weaker variants. Second-order KKT type sufficient optimality conditions and duality results for a nonsmooth vector optimization problem are proved using these functions. The results have been supported by examples.
2
Content available Recommendations as a result of decision evaluations based on reference examples
Kwiatkowski Włodzimierz
Przegląd Teleinformatyczny
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2019
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T. 7, Nr 1-2 (48)
1--23
EN
The problem of evaluation of decisions is considered, which evaluation consists in selecting from the set of possible decisions those that meet the decision-maker's preferences. The added value of solving this problem lies in the reduction of the number of decisions one can choose. Evaluation of decisions is based on their complete characteristics, rather than on a pre-defined quality indicator. The basis for the quality assessment are given pattern examples of decisions made. These are decisions that the decision maker has found to be exemplary or acceptable. They are used as defining his preferences. The methods proposed in this article concern the ordering and clustering of decisions based on their characteristics. The set of decisions selected by an algorithm is interpreted as recommended for the decision maker. Presented solutions can find a variety of applications, for example in investment planning, routing, diagnostics or searching through multimedia databases.
PL
Rozpatrywany jest problem ewaluacji decyzji polegający na wytypowaniu spośród możliwych decyzji tych, które spełniają preferencje decydenta. Użyteczność rozwiązania problemu polega na zredukowaniu liczby możliwych do wyboru decyzji. Ewaluacja decyzji bazuje na ich kompletnych charakterystykach, a nie na wcześniej zdefiniowanym wskaźniku jakości. Podstawą oceny jakości są wzorcowe przykłady decyzji. Są to decyzje, które decydent uznał za doskonałe lub akceptowalne. Wskazane przez decydenta przykłady są wykorzystywane jako określające jego preferencje. Proponowane w artykule metody dotyczą porządkowania i grupowania decyzji na podstawie ich charakterystyk. Wytypowany zbiór decyzji jest interpretowany jako rekomendowany dla decydenta. Przedstawione rozwiązania mogą znaleźć różnorakie zastosowania, np. w planowaniu inwestycji, trasowaniu, diagnostyce czy przeszukiwaniu multimedialnych baz danych.
3
Higher-order conditions for strict local Pareto minima for problems with partial order introduced by a polyhedral cone
Stasiak A.
Journal of Applied Analysis
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2018
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Vol. 24, nr 1
45--54
EN
Using the definitions of μ-th order lower and upper directional derivatives of vector-valued functions, introduced in Rahmo and Studniarski (J. Math. Anal. Appl. 393 (2012), 212-221), we provide some necessary and sufficient conditions for strict local Pareto minimizers of order μ for optimization problems where the partial order is introduced by a pointed polyhedral cone with non-empty interior.
4
Content available remote On weak sharp minima in vector optimization with applications to parametric problems
Bednarczuk E.
Control and Cybernetics
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2007
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Vol. 36, no 3
563-570
EN
In the paper we discuss the concepts of weak sharp solutions to vector optimization problems. As an application we provide sufficient conditions for stability of solutions in perturbed vector optimization problems.
5
Content available remote On stability of some lexicographic multicriteria Boolean problem
Emelichev V. A., Gurevsky E. E.
Control and Cybernetics
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2007
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Vol. 36, no 2
333-346
EN
We consider a multicriteria lexicographic Boolean problem of minimizing absolute deviations of linear functions from zero. We investigate the stability radius which can be understood as a limit level of independent perturbations of the parameters, for which new lexicographic optima do not appear. Lower and upper accessible bounds of the stability radius are obtained.
6
Content available remote C^1'1 vector optimization problems and Riemann derivatives
Ginchev I., Guerraggio A., Rocca M.
Control and Cybernetics
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2004
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Vol. 33, no 2
259-273
EN
In this paper we introduce a generalized second-order Riemann-type derivative for C^1'1 vector functions and use it to establish necessary and sufficient optimality conditions for vector optimization problems. We show that, these conditions are stronger than those obtained by means of the second-order subdinerential in Clarke sense considered in Guerraggio, Luc (2001) and also to some extent than those obtained in Guerraggio, Luc, Minh (2001).
7
Content available remote Order-Lipschitzian properties of multifunctions with applications to stability of efficient points
Bednarczuk E. M.
Control and Cybernetics
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2003
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Vol. 32, no 3
491-502
EN
We define order-Lipschitzian properties of multifunctions and we investigate local upper order-lipschitzness and order-calmness of efficient points of a set depending upon a parameter.
8
Content available Hoelder-like properties of minimal points in vector optimization
Bednarczuk E.
Control and Cybernetics
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2002
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Vol. 31, no 3
423-438
EN
We derive conditions for Hoelder calmness of minimal points of a given set, as a function of a parameter appearing in the description of the set. Different criteria are proved depending on whwther the ordering cone has a nonempty interior or not.
9
Content available Optimality conditions and a classification of duality schemes in vector optimization
El Ghali B.
Control and Cybernetics
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2000
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Vol. 29, no 4
839-860
EN
Vector minimization of a relation F valued in an ordered vector space under a constraint A consists in finding x[0] belongs to A, w[0] belongs to Fx[0] such that w[0] is minimal in FA. To a family of vector minimization problems minimize[x belongs to X] F(x, y), y [belongs to] Y, one associates a Lagrange relation [L(x, [xi], y[0]) = union of sets y belongs to Y(F(x, y)-xi(y)+(y[0]))] where [xi] belongs to an arbitrary class [Xi] of mappings. For this type of problem, there exist several notions of solutions. Some useful characterizations of existential solutions are established and, consequently, some necessary conditions of optimality are derived. One result of intermediate duality is proved with the aid of the scalarization theory. Existence theorems for existential solutions are given and a comparison of several exact duality schemes is established, more precisely in the convex case it is shown that the majority of exact duality schemes can be obtained from one result of S. Dolecki and C. Malivert.
10
Content available Contingent epiderivative and its applications to set-valued optimization
Bednarczuk E., Song W.
Control and Cybernetics
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1998
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Vol. 27, no 3
375-386
EN
In the present paper we give an alternative definition of contingent epiderivative for a set-valued map. We use our concept of contingent epiderivative to formulate necessary and/ or sufficient optimality conditions for a set-valued optimization problem and to study sensivity of a family of parametrized vector optimization problems.
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