No free lunch theorems for optimization

TLDR

No‑free‑lunch theorems demonstrate that any algorithm’s superior performance on one class of problems is offset by poorer performance on another, providing a geometric view of algorithm suitability. The paper develops a framework to investigate how effective optimization algorithms relate to the problems they solve. The framework applies NFL theorems to information‑theoretic optimization, benchmark performance, time‑varying problems, and head‑to‑head minimax distinctions among algorithms.

Abstract

A framework is developed to explore the connection between effective optimization algorithms and the problems they are solving. A number of "no free lunch" (NFL) theorems are presented which establish that for any algorithm, any elevated performance over one class of problems is offset by performance over another class. These theorems result in a geometric interpretation of what it means for an algorithm to be well suited to an optimization problem. Applications of the NFL theorems to information-theoretic aspects of optimization and benchmark measures of performance are also presented. Other issues addressed include time-varying optimization problems and a priori "head-to-head" minimax distinctions between optimization algorithms, distinctions that result despite the NFL theorems' enforcing of a type of uniformity over all algorithms.