First and Second Order Optimality Conditions for Nonsmooth Multiobjective Problems with Equilibrium Constraints

In this paper, we first extend the constant positive linear dependence (CPLD) condition in terms of convexificators given by Rimpi and Lalitha [Constraint qualifications in terms of convexificators for nonsmooth programming problems with mixed constraints. Optimization. 2023;72(8):2019-2038] for nonsmooth scalar optimization problems to nonsmooth multiobjective optimization problems with mixed constraints (MOP) which we denote by MOP-CPLD. It also extends the CPLD condition given by Andreani et al. [On the relation between constant positive linear dependence condition and quasinormality constraint qualification. J Optim Theory Appl. 2005;125(2):473-483] involving continuously differentiable functions. We establish a strong Karush-Kuhn-Tucker (KKT) optimality condition to identify local Pareto efficient solutions under the MOP-CPLD framework. We also introduce a suitable CPLD condition for a nonsmooth multiobjective optimization problem with equilibrium constraints in terms of convexificators which is denoted by MOPEC-CPLD. We introduce several nonsmooth strong Pareto stationary points for the MOPEC which extend the notions of strong Pareto stationary points given by Zhang et al. [Constraint qualifications and proper Pareto optimality conditions for multiobjective problems with equilibrium constraints. J Optim Theory Appl. 2018;176:763-782] for continuously differentiable functions. We provide necessary and sufficient optimality conditions to identify a stationary point as a Pareto efficient solution of the MOPEC under the MOPEC-CPLD condition. Further, we introduce Abadie constraint qualifications for MOPEC which is denoted by MOPEC-SOACQ in terms of Clarke generalized derivative and second-order upper directional derivative given by Páles and Zeidan. This notion utilizes second-order ACQ given by Anchal and Lalita [Second-order optimality conditions for locally Lipschitz vector optimization problems. Optimization. 2023;1-20] for multiobjective optimization problems. We derive second-order necessary optimality conditions in both the primal and the dual forms to identify weak Pareto efficient solutions and strict Pareto efficient solutions of order two for MOPEC by utilizing MOPEC-SOACQ. We give some applications of the results in interval-valued multiobjective optimization problems with equilibrium constraints and in portfolio optimization.