Improved Smoothed Analysis of Multiobjective Optimization

TL;DR

This paper advances the understanding of smoothed analysis in multiobjective optimization by providing tighter bounds on the number of Pareto-optimal solutions, including for zero-preserving perturbations and polynomial objectives.

Contribution

It introduces improved bounds for the smoothed number of Pareto-optimal solutions and extends analysis to zero-preserving perturbations and polynomial objectives.

Findings

Bound of O(n^{2d} phi^d) for Pareto-optimal solutions

Polynomial bound on the moments of the smoothed number

Polynomial bound for zero-preserving perturbations

Abstract

We present several new results about smoothed analysis of multiobjective optimization problems. Motivated by the discrepancy between worst-case analysis and practical experience, this line of research has gained a lot of attention in the last decade. We consider problems in which d linear and one arbitrary objective function are to be optimized over a subset S of {0,1}^n of feasible solutions. We improve the previously best known bound for the smoothed number of Pareto-optimal solutions to O(n^{2d} phi^d), where phi denotes the perturbation parameter. Additionally, we show that for any constant c the c-th moment of the smoothed number of Pareto-optimal solutions is bounded by O((n^{2d} phi^d)^c). This improves the previously best known bounds significantly. Furthermore, we address the criticism that the perturbations in smoothed analysis destroy the zero-structure of problems by showing that the smoothed number of Pareto-optimal solutions remains polynomially bounded even for zero-preserving perturbations. This broadens the class of problems captured by smoothed analysis and it has consequences for non-linear objective functions. One corollary of our result is that the smoothed number of Pareto-optimal solutions is polynomially bounded for polynomial objective functions.