Proper approximate solutions and \(\varepsilon\)-subdifferentials in vector optimization: Moreau-Rockafellar type theorems (Q2922480)

This paper is mainly dedicated to the derivation of two theorems of the Moreau-Rockafellar type, i.e., exact rules for calculating a particular type of a proper \(\epsilon\)-subdifferential of vector-valued mappings are given. The results are obtained by a scalarization using known properties of the Brøndsted-Rocefellar \(\epsilon\)-subdifferential on an extended real-valued convex mapping. Some results extend earlier results by \textit{M. El Maghri} [Optim. Lett. 6, No. 4, 763--781 (2012; Zbl 1280.90104)] and \textit{M. El Maghri} and \textit{M. Laghdir} [SIAM J. Optim. 19, No. 4, 1970--1994 (2009; Zbl 1176.49023)] since the new results are based on a more general proper \(\epsilon\)-efficiency and since they were obtainbed under weaker convexity assumptions. After giving some preliminaries in Section 2, a new approximate strong solution concept of vector optimization problems is introduced which generalizes an earlier concept by \textit{S. S. Kutateladze} [Sov. Math., Dokl. 20, 391--393 (1979); translation from Dokl. Akad. Nauk SSSR 245, 1048--1050 (1979; Zbl 0425.49027)]. In Section 4, a new notion of a strong \(\epsilon\)-subdifferential for extended vector-valued mappings is given and investigated. In particular, it is characterized by means of \(\epsilon\)-subgradients of associated scalar mappings. Then a new regularity condition for extended vector-valued defined in terms of the strong \(\epsilon\)-subdifferential from Section 4 is introduced and studied in Section 5. Finally, Section 6 proves the two theorems of Moreau-Rockefellar type for the introduced strong \(\epsilon\)-subdifferential. As a consequence, two formulas revealing the gap between the strong and the proper \(\epsilon\)-subdifferential are presented.