Optimality conditions for robust nonsmooth multiobjective optimization problems in asplund spaces (Q2139119)

The most powerful results are already established for robust optimization in the finite dimension case, both for under risk and uncertain multi-objective optimization. The result in this paper is to state and analyze problems that deal with infinite-dimensional frameworks. The robust optimization is known as the problem that the uncertain objective and constraint are satisfied for all possible scenarios within a prescribed uncertainty set. The notation is basically standard in variational analysis. The main purpose in this paper is to investigate a non-smooth/nonconvex multi-objective optimization problem with uncertain constraints in arbitrary Asplund spaces under the pseudo convexity assumptions. The authors first establish a necessary optimality theorem for weakly robust efficient solutions of the problem (UP) by employing a fuzzy optimality condition of a non-smooth/nonconvex multi-objective optimization problem without any constrained qualification in the sense of the Fréchet subdifferential, and then derive a necessary optimality condition for properly robust efficient solutions of the problem. Problem (UP) is defined by the authors as followed: Suppose that \(f:Z\to Y\) is a locally Lipschitzian vector-valued function between Asplund spaces, and that \(K\subset Y\) is a pointed closed convex cone. (P) is the following multiobjective optimization problem: \(min_Kf(z)\) s.t. \(g_i(z)\leq 0\), \(i=1,2,\dots,n\), where the functions \(g_i:Z\to \mathbb{R}\), \(i=1,2,\dots,n\), define the constraints. Problem (P) in the face of data uncertainty in the constraints can be captured by the following uncertain multiobjective optimization problem: (UP) \(min_K f(z)\) s.t. \(g_i(z,u)\leq 0\), \(i=1,2,\dots,n\), where \(z\in Z\) is the vector of decision variables, \(u\) is the vector of uncertain parameters and \(u\in U\) for some sequentially compact topological space \(U\), and \(g_i:Z\times U\to \mathbb{R}\), \(i=1,2,\dots,n\), are given functions. Sufficient conditions for weakly and properly efficient solutions as well as for robust efficient solutions to such a problem are also provided by means of applying the new concepts of generalized pseudo convex functions. Along with optimality conditions, the authors address a Mond-Weir-type robust dual problem to the problem (UP) and explore weak, strong, and converse duality properties under assumptions of pseudo convexity. Motivated by the concept of pseudo-quasi generalized convexity due to \textit{M. Fakhar} et al. [Eur. J. Oper. Res. 265, No. 1, 39--48 (2018; Zbl 1374.90335)], they introduce a similar concept of pseudo-quasi convexity type for objective functions. The outline of the paper is organized as follows: \begin{itemize} \item[--] Introduction \item[--] some preliminary definitions (type I pseudo convex and type II pseudo convex for \((f,g)\)) and several auxiliary results are recalled into Section 2; \item[--] Section 3 is devoted to study necessary optimality conditions for weakly and properly robust efficient solutions of problem (UP) by exploiting the non-smooth version of Fermat's rule, the sum rule for the limiting subdifferential and the scalarization formulae of the coderivatives, and to discuss sufficient optimality conditions for such solutions as well as for robust efficient solutions by imposing the pseudo convexity assumptions. \item[--]Section 4 is devoted to presenting duality (weak, strong, converse) relations between the corresponding problems. The paper also contains examples for the usage of the new results. \end{itemize}