Gap functions for quasi-equilibria
The authors present an approach for solving quasi-equilibrium problems (QEPs) relying on gap functions. The reformulation of QEPs as an optimization problem is shown and the smoothness properties of gap functions are analysed. An upper estimate of its Clarke directional derivative is given which provides a key tool in devising the descent method. Monotonicity assumptions on both the equilibrium and constraining bifunctions are key tools to guarantee that all the stationary points of a gap function actually solve the QEP. A few classes of constraints satisfying such assumptions are identified covering a wide range of situations. Relying on these results, a descent method for solving the QEP is devised and its convergence is proved. Error bounds are also given that guarantee the boundedness of the sequence generated by the algorithm. Finally, possible improvements of the results and directions of future improvements are discussed. The authors conclude the paper with a remark that a new class of gap functions based on linearization of the constraining functions \(g_i\)'s could be developed in order to simplify the computation of the gap function at a given point and penalization techniques properly applied to this kind of gap function could drive to further solution algorithms.
scientific article; zbMATH DE number 6708403
scientific article