Optimality and duality in nonsmooth composite vector optimization and applications (Q828870)

Let \(X,W,Y,V,Z\) be finite-dimensional spaces, let \(K\subset Y\) be a closed, convex, and pointed cone with nonempty interior, and \(S\subset Z\) be a closed convex cone. Let let \(F:X\to W\), \(f:W\to Y\), \(G:X\to V\), \(g:V\to Z\) be Lipschitz functions and consider the (composite vector/multiobjective optimization) problem \[ \min_K\{(f\circ F)(x) : x\in C\}, \] where \[ C=\{ x\in X: (g\circ G)(x)\in -S\}. \] A point \(\bar x\) is said to be a weakly efficient solution if \[ \forall x \in C, \quad (f \circ F)(x) - ( f \circ F)(\bar x) \not\in -\mathrm{int } K. \] The author presents first, in full generality, necessary conditions for \(\bar x\) to be a weakly efficient solution of the form \(0\) belongs to the sum of suitable limiting/Mordukhovich subdifferentials involving the above maps, together with a complementary slackness property. Second, a point satisfying necessary conditions is proved to be weakly efficient if a constraint qualification and a suitable generalized convexity (i.e., a separation condition) are assumed. Then weak and strong formulation of duality properties are proposed and proved, together with some corollaries that express the above results for the particular case of linear maps. The relevance of most assumptions is illustrated through counterexamples. The authors point out that they do not use scalarization arguments.