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Optimality conditions for a class of relaxed quasiconvex minimax problems

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

Control and Cybernetics

2014

Vol. 43, no. 2

249--260

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.

Calculus and applications of Studniarski’s derivatives to sensitivity and implicit function theorems

Anh N. L. H., Khanh P. Q.

Control and Cybernetics

2014

Vol. 43, no. 1

33--57

We first discuss basic calculus rules for Studniarski’s derivatives. Then, we apply these derivatives to sensitivity analysis of solutions to inclusions and to computing the derivative of implicit multifunctions.

Existence conditions in symmetric multivalued vector quasiequilibrium problems

Anh L. Q., Khanh P. Q.

Control and Cybernetics

2007

Vol. 36, no 3

519-530

We consider symmetric multivalued vector quasiequilibrium problems in topological vector spaces. Sufficient conditions for the solution existence are established under relaxed assumptions, which are shown by examples to be essential, easy to check and more advantageous than recent known results. Applications to lower and upper bounded quasiequilibrium problems and to coincidence point problems are given.