On a Smooth Dual Gap Function for a Class of Quasi-Variational Inequalities

Nadja Harms (), Tim Hoheisel () and Christian Kanzow ()
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Nadja Harms: University of Würzburg
Tim Hoheisel: University of Würzburg
Christian Kanzow: University of Würzburg

Journal of Optimization Theory and Applications, 2014, vol. 163, issue 2, No 4, 413-438

Abstract: Abstract A well-known technique for the solution of quasi-variational inequalities (QVIs) consists in the reformulation of this problem as a constrained or unconstrained optimization problem by means of so-called gap functions. In contrast to standard variational inequalities, however, these gap functions turn out to be nonsmooth in general. Here, it is shown that one can obtain an unconstrained optimization reformulation of a class of QVIs with affine operator by using a continuously differentiable dual gap function. This extends an idea from Dietrich (J. Math. Anal. Appl. 235:380–393 [24]). Some numerical results illustrate the practical behavior of this dual gap function approach.

Keywords: Quasi-variational inequality; Set-valued mapping; Regularized gap function; DC optimization; Conjugate function; Dual gap function; $$\mathrm{PC}^1$$ PC 1 function; Nonconvex duality; 49M29; 65K10; 90C33 (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/s10957-014-0536-4

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