# Subdifferentials and Stability Analysis of Feasible Set and Pareto Front Mappings in Linear Multiobjective Optimization

## TL;DR

This paper analyzes how the feasible set and Pareto front in linear multiobjective optimization change with perturbations, using subdifferential calculus to assess stability and compute Lipschitz moduli.

## Contribution

It introduces a novel epigraphical approach to measure stability of feasible and Pareto front mappings under perturbations, including a method for linear program value computation without solutions.

## Key findings

- Subdifferentials of feasible and Pareto front mappings are characterized.
- Lipschitz stability of the mappings is verified and moduli are computed.
- A method for linear program value estimation without optimal solutions is proposed.

## Abstract

The paper concerns multiobjective linear optimization problems in R^n that are parameterized with respect to the right-hand side perturbations of inequality constraints. Our focus is on measuring the variation of the feasible set and the Pareto front mappings around a nominal element while paying attention to some specific directions. This idea is formalized by means of the so-called epigraphical multifunction, which is defined by adding a fixed cone to the images of the original mapping. Through the epigraphical feasible and Pareto front mappings we describe the corresponding vector subdifferentials, and employ them to verifying Lipschitzian stability of the perturbed mappings with computing the associated Lipschitz moduli. The particular case of ordinary linear programs is analyzed, where we show that the subdifferentials of both multifunctions are proportional subsets. We also provide a method for computing the optimal value of linear programs without knowing any optimal solution. Some illustrative examples are also given in the paper.