Optimality and solutions for conic robust multiobjective programs

Abstract

The authors would like to thank the referees for valuable comments and suggestions. Research was supported by a research grant from Australian Research Council under Discovery Program Grant DP200101197. The main results of this paper were presented at the 5th IMA and OR Society Conference on Mathematics of Operational Research (Birmingham, United Kingdom, 2025), and the first author would like to acknowledge the support of the Mid and Early Career Academic Research Support Scheme (Brunel University of London, United Kingdom, 2024-2025), which made this possible.Mathematics Subject Classification: 65K10; 49K99; 90C46; 90C29.This paper presents a robust framework for handling a conic multiobjective linear optimization problem, where the objective and constraint functions are involving affinely parameterized data uncertainties. More precisely, we examine optimality conditions and calculate efficient solutions of the conic robust multiobjective linear problem. We provide necessary and sufficient linear conic criteria for efficiency of the underlying conic robust multiobjective linear program. It is shown that such optimality conditions can be expressed in terms of linear matrix inequalities and second-order conic conditions for a multiobjective semidefinite program and a multiobjective second order conic program, respectively. We show how efficient solutions of the conic robust multiobjective linear problem can be found via its conic programming reformulation problems including semidefinite programming and second-order cone programming problems. Numerical examples are also provided to illustrate that the proposed conic programming reformulation schemes can be employed to find efficient solutions for concrete problems including those arisen from practical applications.The first author would like to acknowledge the support of the Mid and Early Career Academic Research Support Scheme (Brunel University of London, United Kingdom, 2024-2025), which made this possible