Dual and bidual problems for a Lipschitz optimization problem based on quasi-conjugation
This paper deals with a Lipschitz optimization problem (LOP) constrained by linear functions in \(\mathbb R^{n}\) which is one of the most important problems in global optimization. The Lipschitz optimization problem is transformed into a certain problem (MP) constrained by a ball in \(\mathbb R^{n+1}\). Besides, for (MP), the authors define the dual problem (DP) and the bidual problem (BDP) by applying the quasi-conjugate and quasi-biconjugate functions defined by \textit{P. T. Thach} [J. Math. Anal. Appl. 159, No. 2, 299--322 (1991; Zbl 0734.90073)]. By the properties of quasi-conjugation, the objective functions of (DP) and (BDP) are quasi-convex. Despite there is no guarantee that the objective function of (MP) is quasi-convex, the authors prove that each optimal value of (MP) and (DP) coincides with the multiplication of the other by \(-1\), and that each optimal solution of them is computed by dividing the other by the square of its norm. Moreover, it is also shown that both sets of all globally optimal solutions of (MP) and (BDP) coincide. Furthermore, by using the quasi-convexity of the objective function of (DP), the authors propose the overestimation procedure and an outer approximation algorithm for (DP).
scientific article; zbMATH DE number 5865650
scientific article