Mixed Strategy May Outperform Pure Strategy: An Initial Study

TL;DR

This study compares mixed and pure strategy meta-heuristics, demonstrating that mixed strategies can outperform pure strategies in certain problems and providing a theoretical foundation for their effectiveness.

Contribution

It provides empirical evidence and a rigorous theorem showing mixed strategies can outperform pure strategies and introduces the Complementary Strategy Theorem.

Findings

Mixed strategies outperform pure strategies in up to 77.8% of 0-1 knapsack problem instances.

Experimental results support the advantage of mixed strategies over pure strategies.

The Complementary Strategy Theorem establishes conditions under which mixed strategies are more efficient.

Abstract

In pure strategy meta-heuristics, only one search strategy is applied for all time. In mixed strategy meta-heuristics, each time one search strategy is chosen from a strategy pool with a probability and then is applied. An example is classical genetic algorithms, where either a mutation or crossover operator is chosen with a probability each time. The aim of this paper is to compare the performance between mixed strategy and pure strategy meta-heuristic algorithms. First an experimental study is implemented and results demonstrate that mixed strategy evolutionary algorithms may outperform pure strategy evolutionary algorithms on the 0-1 knapsack problem in up to 77.8% instances. Then Complementary Strategy Theorem is rigorously proven for applying mixed strategy at the population level. The theorem asserts that given two meta-heuristic algorithms where one uses pure strategy 1 and another uses pure strategy 2, the condition of pure strategy 2 being complementary to pure strategy 1 is sufficient and necessary if there exists a mixed strategy meta-heuristics derived from these two pure strategies and its expected number of generations to find an optimal solution is no more than that of using pure strategy 1 for any initial population, and less than that of using pure strategy 1 for some initial population.