Runtime Analysis of Evolutionary Algorithms for Multi-party Multi-objective Optimization

TL;DR

This work addresses the runtime analysis of evolutionary algorithms for bi-party multi-objective optimization, revealing that standard MOEAs struggle to produce consensus and proposing dedicated EMPMO schemes. It introduces BPAOAZ and BPMOSP as synthetic benchmarks to derive rigorous bounds for simple, random, payoff-based, and consensus-based EMPMO variants, establishing both upper and lower performance guarantees. Theoretical results show that randomized and payoff-driven approaches can significantly reduce runtime, while consensus-based strategies improve efficiency and precision in shortest-path problems. Empirical experiments on BPAOAZ and a two-party UAV path-planning scenario corroborate the theory, illustrating practical benefits of maintaining a common solution set and negotiating across parties during evolution.

Abstract

In scenarios where multiple decision-makers operate within a common decision space, each focusing on their own multi-objective optimization problem (e.g., bargaining games), the problem can be modeled as a multi-party multi-objective optimization problem (MPMOP). While numerous evolutionary algorithms have been proposed to solve MPMOPs, most results remain empirical. This paper presents the first theoretical analysis of the expected runtime of evolutionary algorithms on bi-party multi-objective optimization problems (BPMOPs). Our findings demonstrate that employing traditional multi-objective optimization algorithms to solve MPMOPs is both time-consuming and inefficient, as the resulting population contains many solutions that fail to achieve consensus among decision-makers. An alternative approach involves decision-makers individually solving their respective optimization problems and seeking consensus only in the final stage. While feasible for pseudo-Boolean optimization problems, this method may fail to guarantee approximate performance for one party in NP-hard problems. Finally, we propose evolutionary multi-party multi-objective optimizers (EMPMO) for pseudo-Boolean optimization and shortest path problems within a multi-party multi-objective context, maintain a common solution set among all parties. Theoretical and experimental results demonstrate that the proposed \( \text{EMPMO}_{\text{random}} \) outperforms previous algorithms in terms of the lower bound on the expected runtime for pseudo-Boolean optimization problems. Additionally, the consensus-based evolutionary multi-party multi-objective optimizer( \( \text{EMPMO}_{\text{cons}}^{\text{SP}} \) ) achieves better efficiency and precision in solving shortest path problems compared to existing algorithms.

Figures (6)

• Figure 1: The objective space of the AORZ and AOFZ problem with $n=8$.
• Figure 2: The weighted directed graph $G$.
• Figure 3: The average runtime of SEMO, $\text{EMPMO}_{\text{simple}}$, $\text{EMPMO}_{\text{random}}$, and $\text{EMPMO}_{\text{payoff}}$ on artificial problem $\mathrm{BPAOAZ}$ and the y-axis is the average runtime in base 10.
• Figure 4: The average runtime of SEMO, $\text{EMPMO}_{\text{simple}}$, $\text{EMPMO}_{\text{random}}$, and $\text{EMPMO}_{\text{payoff}}$ on artificial problem $\mathrm{BPAOAZ}$ and the y-axis is the average runtime in base 10.
• Figure 5: The largest minimum approximate degree of DEMO, $\text{EMPMO}^{\text{SP}}_{\text{simple}}$, and $\text{EMPMO}^{\text{SP}}_{\text{cons}}$ on BPUAVPP.

Theorems & Definitions (42)

• Definition 1: MPMOP liu2020evolutionary
• Definition 2: Domination
• Definition 3: Pareto Optimality
• Definition 4: Common Pareto Optimality liu2020evolutionary
• Definition 5: BPAOAZ
• Lemma 1
• proof
• Lemma 2: General Upper Bound I laumanns2004running
• Lemma 3: General Upper Bound II laumanns2004running
• Theorem 1