Exponential type vector variational-like inequalities and nonsmooth vector optimization problems

Abstract

This paper is devoted to study a new class of exponential form of vector variational-like inequality problems comprised of locally Lipschitz functions having exponential type invexities. In the setting of Banach space, we investigate the relationship of nonsmooth exponential type vector variational-like inequality problems with vector optimization problems involving nonsmooth ( p, r )-invex functions. Also, we explore the conditions for solvability of the aforesaid nonsmooth exponential type vector variational-like inequality problems using Fan-KKM Theorem. Moreover, we provide examples to elucidate our results.Keywords Locally Lipschitz function · Nonsmooth vector optimization problems · Nonsmooth exponential type vector variational-like inequalities · (p, r)-invexity · Fan-KKM Theorem Mathematics Subject Classification 90C26 · 90C29 · 49J52 · 58E17 · 58E35

Generalized Nonsmooth Exponential-Type Vector Variational-Like Inequalities and Nonsmooth Vector Optimization Problems in Asplund Spaces

“…(i) If θ ≡ 0 and ∂ L f (·) = ∂ f (·), i.e., the Clarke subdifferential operator, then (P 2 ) and (P 3 ) reduces to nonsmooth exponential-type vector variational like inequality problem and nonsmooth exponential-type weak vector variational like inequality problem considered and studied by Jayswal and Choudhury [17]. (ii) For p = 0, a similar analogue of problems (P 2 ) and (P 3 ) was introduced and studied by Oveisiha and Zafarani [13].…”

“…After that, Mandal and Nahak [16] introduced (p, r)-ρ-(η, θ)-invexity mapping which is the generalization of the result of Antczak [15]. By using (p, r)-invexity, Jayaswal and Choudhury [17] introduced exponential type vector variational-like inequality problem involving locally Lipschitz mappings.…”