Constraint Qualifications and Optimality Conditions for Nonsmooth Semidefinite Multiobjective Programming Problems with Mixed Constraints Using Convexificators

In this article, we investigate a class of non-smooth semidefinite multiobjective programming problems with inequality and equality constraints (in short, NSMPP). We establish the convex separation theorem for the space of symmetric matrices. Employing the properties of the convexificators, we establish Fritz John (in short, FJ)-type necessary optimality conditions for NSMPP. Subsequently, we introduce a generalized version of Abadie constraint qualification (in short, NSMPP-ACQ) for the considered problem, NSMPP. Employing NSMPP-ACQ, we establish strong Karush-Kuhn-Tucker (in short, KKT)-type necessary optimality conditions for NSMPP. Moreover, we establish sufficient optimality conditions for NSMPP under generalized convexity assumptions. In addition to this, we introduce the generalized versions of various other constraint qualifications, namely Kuhn-Tucker constraint qualification (in short, NSMPP-KTCQ), Zangwill constraint qualification (in short, NSMPP-ZCQ), basic constraint qualification (in short, NSMPP-BCQ), and Mangasarian-Fromovitz constraint qualification (in short, NSMPP-MFCQ), for the considered problem NSMPP and derive the interrelationships among them. Several illustrative examples are furnished to demonstrate the significance of the established results.