Some relations between variational-like inequalities and efficient solutions of certain nonsmooth optimization problems

The connections between variational inequalities and optimization problems is well known, and many investigators have discussed them along many years; see, for instance, [1, 8, 10, 13]. This last article, which was authored by Giannessi, in particular, is one of the main works that study these connections in the finite-dimensional context. In recent years, the interest in the investigation on the relationships between these two classes of problems has increased, resulting in several different conditions for the existence of solutions for many variational-type inequalities (e.g., [5, 16, 19, 20]). Connections among variational inequalities and vectorial optimization problems have also been studied in [11, 18, 25], for instance. By using a variational-like inequality, Lee et al. [19] obtained some results of existence of solutions for nonsmooth invex problems, which are generalizations of those obtained by Chen and Craven [4] for differentiable convex problems. Recently, Giannessi [11] showed the equivalence between efficient solutions of a differentiable and convex optimization problem and the solutions of a variational inequality of Minty type. He also proved the equivalence between weak efficient solutions of a differentiable convex optimization problem and solutions of a variational inequality of weak Minty type. Following this last line of investigation, Lee [17] was able to establish the equivalence between the solutions of the inequalities of Minty and Stampacchia types for subdifferential (in the convex analysis sense) and efficient solutions and weakly efficient solutions, respectively, in the case of vectorial nonsmooth convex optimization problems. Moreover, using these characterizations, he proved a theorem on existence of weakly efficient solutions for the vectorial nonsmooth convex optimization problem, under hypothesis of compactness.