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Optimality conditions for a class of relaxed quasiconvex minimax problems

Diem H. T. H., Khanh P. Q.

Control and Cybernetics

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2014

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Vol. 43, no. 2

249--260

EN

A class of minimax problems is considered. We approach it with the techniques of quasiconvex optimization, which includes most important nonsmooth and relaxed convex problems and has been intensively developed. Observing that there have been many contributions to various themes of minimax problems, but surprisingly very few on optimality conditions, the most traditional and developed topic in optimization, we establish both necessary and sufficient conditions for solutions and unique solutions. A main feature of this work is that the involved functions are relaxed quasi- convex in the sense that the sublevel sets need to be convex only at the considered point. We use star subdifferentials, which are slightly bigger than other subdifferentials applied in many existing results for minimization problems, but may be empty or too small in various situations. Hence, when applied to the special case of minimization problems, our results may be more suitable. Many examples are provided to illustrate the applications of the results and also to discuss the imposed assumptions.

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Analytic improper intersections. [Pt] 2, Deformation to an algebraic bicone and applications

Nowak K.

Bulletin of the Polish Academy of Sciences. Mathematics

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2000

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Vol. 48, no 2

131-140

EN

This article, being the second part of our paper [7], continues to study the analytic intersection algorithm. We present a certain method of deformation of an analytic set to an algebraic bicone, and next we express the result of the analytic intersection algorithm and multiplicity for improper intersections as a degree sequence and the Samuel multiplicity of that bicone, respectively (cf. [6]). Also produced are many important consequences (cf. [6]), among others the coincidence between the intersection indices for analytic improper intersections defined by Tworzewski [12] and those defined by Achilles and Manaresi [1] (Corollary 3), the linear testing theorem (Corollary 6) or a generalization of the classical reduction theorem to the case of analytic improper intersections (Corollary 7) which ensures the canonical character of the diagonal procedure.

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Analytic improper intersections. [Pt] 1, Deformation to the normal cone

Nowak K.

Bulletin of the Polish Academy of Sciences. Mathematics

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2000

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Vol. 48, no 2

121-130

EN

In this paper we present in the algebraic setting an intersection algorithm considered by Tworzewski [22], which is a local analytic counterpart of the Stueckrad-Vogel intersection algorithm from global algebraic geometry (cf. [20, 5)]. Some other local algebraic counterparts have been investigated by Achilles and Manaresi [1, 2]. The main purpose is to provide in this analytic context the significant method of deformation to the normal cone, which is one of the most powerful tools of intersection theory (cf. [4, 5, 2, 10]). For some important refinements and applications of this method we refer the reader to our next article [11].