On a Class of Second-Order PDE &PDI Constrained Robust Optimization Problems

Abstract

AbstractOver the last decade, the modified objective function method and saddle-point efficiency criteria have been intensively investigated by many researchers (see, for instance, Jayswal et al. [1, 2], Antczak et al. [3], Jayswal and Preeti [4], Sun et al. [5], Treanţă [6,7,8,9], Preeti et al. [10]). More concretely, new auxiliary optimization problems have been constructed for solving the original optimization problems. These auxiliary problems are easier to study than the original ones. For various aspects on interval and robust optimization problems, including efficiency conditions, Lagrange multiplier characterizations, robust algorithms, the reader is directed, for instance, to Liu and Yuan [11], Jeyakumar et al. [12], Wei et al. [13], and Treanţă [14]. In this chapter, motivated and inspired by the above mentioned works, by using vector multiple integral cost functionals and the notion of convexity associated with a multi-time uncertain controlled second-order Lagrangian, we develop a new mathematical framework on multi-dimensional multi-objective variational control problems with mixed constraints implying second-order partial differential equations and inequations. More specifically, we start with a multi-dimensional multi-objective optimization problem with uncertainty in the objective and constraint functionals, denoted by (MOP), and introduce a modified optimization problem for which we formulate and prove conditions of efficiency. As can be easily noticed, the modified variational control problem is simpler and easier to solve than the original optimization problem.