Non-monotone steepest descent algorithms for robust counterparts of uncertain multiobjective problems

In this paper, a steepest descent method (SDM) with a non-monotone line search strategy is developed to solve the robust counterpart of uncertain multiobjective optimization problems (UMOPs) under finite uncertainty. The uncertainty in the objective functions is addressed using an objective-wise maximum-type robust reformulation, which converts the original uncertain problem into a deterministic form. To solve this robust deterministic problem, a descent direction is computed by solving an auxiliary subproblem at each iteration. Subsequently, a non-monotone line search technique is applied to determine an appropriate step length, allowing flexibility in the acceptance of trial points and improving convergence behaviour. Two variants of non-monotonicity, maximum-type and average-type, are considered, and corresponding algorithms are formulated. The convergence of the proposed methods is established under standard assumptions. Finally, the effectiveness of both non-monotone algorithms is validated through a set of numerical experiments and compared with the existing SDM with monotone line search (an Armijo-type inexact line search technique) developed for UMOPs, as well as with an SDM equipped with non-monotone line search designed for multiobjective optimization problems. The comparison is carried out using performance profiles.