Optimization problems show up in every field from physics to biology. Simple optimizations minimize a global function and have single solutions. These usually consist of a single, specific design and are often easy to describe. Some cases, especially in more complex systems, require a Multi-Objective (or Pareto) Optimization (MOO) approach. This minimizes a series of conflicting targets simultaneously, and its solutions encompass optimal tradeoffs (called Pareto fronts) that usually encompass a collection of designs, instead of a single one. Thermodynamics can be written as one such MOO problem with an extra constraint. Its Pareto front is the so-called Gibbs surface, from which phase transitions, critical points, and susceptibilities can be easily read. This connection allows us to generalize, robustly, concepts from Statistical Mechanics to arbitrary MOO problems. It also suggests that Pareto selective forces (evolutionary forces implementing MOO) are a mechanism that can robustly drive systems towards critically self-organized states. We review examples from graph theory, linguistics, and others from the literature that, when studied under the light of our framework, reveal phase transitions and critical points otherwise missed.
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