First- and second-order optimality conditions of nonsmooth sparsity multiobjective optimization via variational analysis (Q6541380)

In this paper, the authors investigate optimality conditions of a nonsmooth sparsity multi-objective optimization problem (shortly, SMOP) by advanced variational analysis. The problem under investigation is defined as follows: \[ \min f (x),\quad \text{s.t.}\; \|x\|_0\le s,\,x\in \mathbb{R}^n, \] where \(f :\mathbb{R}^n\to \mathbb{R}^m\) with \(f:=( f_1, f_2,\dots, f_m) \), \(f_i : \mathbb{R}^n\to \mathbb{R}\) are locally Lipschitz continuous for all \(i=1,2,\dots, m\), \(s < n\) is a positive integer, and \(\|x\|_0\) counts non-zero entries of the vector \(x = (x_1, x_2,\dots , x_n),\) i.e., \(\|x\|_0 = |\{i \in \{1, 2,\dots, n\} : x_i = 0\}|.\) The sparse feasible set of SMOP is denoted by \(S :=\{x\in \mathbb{R}^n:\, \|x\|_0\le s\}\). The authors present the variational analysis characterizations, such as tangent cones, normal cones, dual cones and second-order tangent set, of the sparse set, and give the relationships among the sparse set and its tangent cones and second-order tangent set. A point \(\overline{x}\in S\) is called an \(N^\divideontimes\)-stationary point of SMOP if there exists \(\overline{\lambda}\in \Lambda^+\) such that \(0 \in \partial_c f(\overline{x})^\top\overline{\lambda}+N^\divideontimes_S(\overline{x}),\) where \(\divideontimes \in \{B,C,M\}\). A first-order necessary condition for local weakly Pareto efficient solution of SMOP is established under some suitable conditions in Theorem 1: Let \(\overline{x}\in S\) be a local weakly Pareto efficient solution of SMOP. Then there exists \(\overline{\lambda}\in \Lambda^+\) such that \(\overline{x}\) is an \(N^C\)-stationary point of SMOP. Furthermore, if \(\partial_c f(\overline{x})^\top \overline{\lambda} \subseteq T^C_S(\overline{x})\) holds, then \(\overline{x}\in S\) is also an \(N^\divideontimes\)-stationary point of SMOP, where \(\divideontimes\) denotes either \(B\) or \(M\). The sufficient optimality conditions of SMOP are derived under the pseudoconvexity assumptions. Moreover, the second-order necessary and sufficient optimality conditions of SMOP are established by the Dini directional derivatives of the objective function and the Bouligand tangent cone and second-order tangent set of the sparse set.