Worst-case complexity analysis of derivative-free methods for multi-objective optimization
Giampaolo Liuzzi, Stefano Lucidi
TL;DR
This work addresses the worst-case complexity of derivative-free a posteriori methods for unconstrained multi-objective optimization, where objective evaluations are black-box. It introduces two main algorithms, DFMOstrong (full exploration of the current non-dominated set) and DFMOlight (single-point exploration), each equipped with a linesearch expansion mechanism and analyzed through a hypervolume-based progress framework, including Pareto-criticality measures and a stationarity metric $\mu(x)$. The paper derives explicit iteration and evaluation bounds, including $\,|K_\epsilon| = \mathcal{O}(n^q \epsilon^{-2q})$ and related complexity for both DFMOstrong and DFMOlight, and extends the analysis to special variants DFMOmin and DFMOmax with distinct robustness guarantees. The results provide theoretical guarantees for the efficiency of exploring the Pareto front with derivative-free methods and offer guidance on trade-offs between exhaustive versus lightweight exploration in practice.
Abstract
In this work, we are concerned with the worst case complexity analysis of "a posteriori" methods for unconstrained multi-objective optimization problems where objective function values can only be obtained by querying a black box. We present two main algorithms, namely DFMOnew and DFMOlight which are based on a linesearch expansion technique. In particular, \DFMOnew, requires a complete exploration of the points in the current set of non-dominated solutions, whereas DFMOlight only requires the exploration around a single point in the set of non-dominated solutions. For these algorithms, we derive worst case iteration and evaluation complexity results. In particular, the complexity results for DFMOlight aligns with those recently proved in the literature for a directional multisearch method. Furthermore, exploiting an expansion technique of the step, we are also able to give further complexity results concerning the number of iterations with a measure of stationarity above a prefixed tolerance.