Abstract

The theory of optimality conditions serves as a fundamental basis for finding solutions to optimization problems, both analytically and numerically. Beyond the optimality conditions for exact solutions, researchers are often interested in optimality conditions for approximate solutions. In this paper, we focus on obtaining these conditions for multi-objective optimization problems with infinitely many constraints via a modified scalarizing method. 
Researchers often study approximate solutions within the feasible sets of optimization problems, such as "ε" -solutions or "ε" -quasi-solutions. Attention has also been directed toward approximate solutions that lie outside the feasible set, such as almost "ε" -solutions or almost "ε" -quasi-solutions. These concepts were initially proposed by P. Loridan for single-objective optimization problems and were later extended to the multi-objective case. Such types of solutions are well suited for numerical methods based on iterative sequences. 
Recently, new concepts of "ε" -quasi-subdifferentials for locally Lipschitz functions and "ε" -quasi-normal sets have been proposed. By employing a modification of the "ε" -constrained scalarization method, and combining it with the new concepts above, conditions for approximate solutions of nonconvex multi-objective optimization problems with infinitely many constraints are established. These results are expressed via inclusions involving a new form of approximate subdifferentials for locally Lipschitz functions together with the new notion of approximate quasi-normal sets. Throughout this paper, we contribute alternative representations of optimality conditions to obtain approximate solutions of nonconvex multi-objective optimization problems.