Non-linear Multi-objective Optimization with Probabilistic Branch and Bound

TL;DR

This paper introduces MOPBnB(so), a novel probabilistic branch and bound algorithm for stochastic multi-objective optimization that efficiently approximates the Pareto set using single observations, outperforming existing methods like NSGA-II.

Contribution

The paper presents MOPBnB(so), a new algorithm that evaluates solutions once and estimates neighboring solutions, providing finite-time bounds and asymptotic convergence for stochastic problems.

Findings

MOPBnB(so) captures the Pareto set with high probability.

It converges asymptotically to true objective functions.

Outperforms NSGA-II in computational efficiency and solution quality.

Abstract

A multiple objective simulation optimization algorithm named Multiple Objective Probabilistic Branch and Bound with Single Observation (MOPBnB(so)) is presented for approximating the Pareto optimal set and the associated efficient frontier for stochastic multi-objective optimization problems. MOPBnB(so) evaluates a noisy function exactly once at any solution and uses neighboring solutions to estimate the objective functions, in contrast to a variant that uses multiple replications at a solution to estimate the objective functions. A finite-time performance analysis for deterministic multi-objective problems provides a bound on the probability that MOPBnB(so) captures the Pareto optimal set. Asymptotic convergence of MOPBnB(so) on stochastic problems is derived, in that the algorithm captures the Pareto optimal set and the estimations converge to the true objective function values. Numerical results reveal that the variant with multiple replications is extremely intensive in terms of computational resources compared to MOPBnB(so). In addition, numerical results show that MOPBnB(so) outperforms a genetic algorithm NSGA-II on test problems.