Journal of Advances in Mathematics and Computer Science

We study a nonsmooth vector optimization problem (VP) with locally Lipschitz equality and inequality constraints on a real Banach space X. The Michel-Penot (MP) subdifferential ∂^⋄, which is strictly smaller than the Clarke generalized gradient ∂^◦ and reduces to the classical gradient under Gâteaux differentiability, serves as the primary subdifferential tool. This choice yields sharper necessary optimality conditions and avoids the extraneous subgradients that may appear in the Clarke framework. Under the locally Lipschitz assumption, Fritz John necessary conditions at weakly efficient solutions of (VP) are recalled (Theorem 3.2). To guarantee that all objective multipliers λ_k are nonzero, a requirement that fails under Fritz John conditions alone in the vector setting appropriate constraint qualifications (CQs) are needed. An alternative theorem (Theorem 3.4) for convex functions on Banach spaces is established; it serves as the cornerstone for proving the equivalence MFCQ ⇔ WBCQ, and for deriving KKT conditions under each of the CQs in Definition 4.1. Under the mixed differentiability assumption (A1), three new regularity conditions for (VP) are introduced: the Mangasarian-Fromovitz regularity condition (MFRC), the nondifferentiable Slater regularity condition (SRC), and the weak Guignard regularity condition (WGRC). Using Theorem 3.4, we establish the complete hierarchy LIRC ⇒ MFRC ⇔ WBRC ⇒ WGRC, SRC ⇒ WBRC, with all implications shown to be strict via counterexample. Under WBRC, the proper KKT multiplier sets L_τ(x_◦) are nonempty and bounded. On reflexive Banach spaces, KKT necessary conditions at efficient solutions of (VP) are derived under WGRC (Theorem 4.5), via a characterization of the polar of the MP-linearization cone (Lemma 4.4). The results extend and unify the scalar multiplier theory of Gauvin (1977) and Ye (2001, 2004), and the smooth vector theory of Chandra et al. (2004) and Dutta and Lalitha (2006), to the fully nonsmooth vector setting with both equality and inequality constraints.