Smoothed Analysis of Pareto Curves in Multiobjective Optimization

TL;DR

This paper applies smoothed analysis to multiobjective optimization, showing that the expected number of Pareto-optimal solutions is polynomially bounded, which explains the typically moderate size of Pareto sets in practice.

Contribution

It introduces a smoothed analysis framework for Pareto curves, providing polynomial bounds on their expected size and discussing related algorithmic complexity results.

Findings

Expected Pareto set size is polynomially bounded under smoothed analysis.

Provides algorithms for computing Pareto-optimal solutions.

Discusses smoothed complexity of various optimization problems.

Abstract

In a multiobjective optimization problem a solution is called Pareto-optimal if no criterion can be improved without deteriorating at least one of the other criteria. Computing the set of all Pareto-optimal solutions is a common task in multiobjective optimization to filter out unreasonable trade-offs. For most problems the number of Pareto-optimal solutions increases only moderately with the input size in applications. However, for virtually every multiobjective optimization problem there exist worst-case instances with an exponential number of Pareto-optimal solutions. In order to explain this discrepancy, we analyze a large class of multiobjective optimization problems in the model of smoothed analysis and prove a polynomial bound on the expected number of Pareto-optimal solutions. We also present algorithms for computing the set of Pareto-optimal solutions for different optimization problems and discuss related results on the smoothed complexity of optimization problems.