Directional compactness, approximations and efficiency conditions for nonsmooth vector equilibrium problems with constraints (Q6542428)

The constrained nonsmooth vector equilibrium problem (CVEP) \(F_{\bar x}(x)\notin -\mathrm{int} Q\) \((\forall x\in K)\) with feasible set \(K=\{x\in C\mid g(x)\in -S\}\) is considered where \(X,Y,Z\) are Banach spaces, \(\emptyset \neq C\subset X\), \(S\subset Z\) is a convex cone with \(\mathrm{int} S \neq \emptyset\), \(Q\subset Y\) is a closed convex cone defining a partial order in \(Y\), the vector objective \( F_{\bar x}:X\rightarrow Y\) satisfies \(F_{\bar x}(\bar x)=0\) for all \(\bar x \in X\), \(g:X\rightarrow Z\) is the constraint function and \(U\) is some neighborhood of \(\bar x\). Optimality conditions for local (global \(U=X)\) \textit{weakly efficient solution} (\(F_{\bar x}(x)\notin -\mathrm{int} Q\,\forall x\in K\cap U\)), \textit{ideal efficient solution} (\(F_{\bar x}(x)\in Q\,\forall x\in K\cap U\)), \textit{first-order strict Pareto solution} are proven. The main assumption in each theorem is some point-wise sequential relative compactness of a set of linear continuous operators which arise after some first-order approximation of \(g\) and \(F_{\bar x}\) at \(\bar x\) using e.g. generalized Hadamard directional derivatives at \(\bar x\) in direction \(u\in X\).\N\NTheorem 1: If \(\bar x\) is a locally weakly efficient / ideal efficient / first-order strict Pareto solution then \(0\in Y\) is weakly efficient/ ideal efficient/first order strict Pareto solution w.r.t. the set \(d_GF_{\bar x}\vert_C(\bar x,u)\subset Y\), the generalized Hadamard directional derivative of \(F_{\bar x}\) at \(\bar x\) relatively to \(C\) in direction \(u\), for all \(u\) belonging to the tangent cone \( T(C;\bar x)\) and satisfying \(g(\bar x)+a(u)\in -\mathrm{int} S\) where \(a\in \mathcal{L}(X,Y)\) is an element of some first-order approximation of \(g\) at \(\bar x\) in the above sense.\N\NFurther KKT theorems for weakly efficient /ideal efficient solutions are proven under some local Slater CQ. Finally, local first-order sufficient optimality conditions are shown for strict local Pareto solution and ideal efficient solution using the above first-order approximation and the KKT form.\N\NSeveral examples in the Hilbert space \(l^2\) illustrate essential steps.\N\NThe presented optimality conditions generalize known results in nonsmooth vector equilibrium problems in infinite dimensional Banach spaces. The breakthrough for such results is caused by the point-wise relative sequential compactness (p-compactness) and asymptotic and bounded point-wise compactness (abp-compactness) of sets of linear continuous operators and the generalized Hadamard directional derivative.