In this lecture, we start with a review of equilibrium and efficiency/dominance concepts from game theory – specifically the Nash equilibrium, Pareto efficiency, and payoff and risk dominance. We apply these for both a discrete game (the Stag Hunt) and a generic continuous game. That allows us to introduce Variational Inequalities as a more general numerical problem set that includes the Nash equilibrium as a member (for continuous games). We then pivot to Multi-Objective Optimization (MOO) and motivate the concept of Pareto improvements, Pareto efficiency, Pareto-efficient sets, and Pareto frontiers/fronts. We close with discussions about scalarization approaches to solve MOO problems, including linear scalarization, targets, satisficing, and Chebyshev/weighted minimax. We discuss problems with these approaches and then hint that we will move forward toward fitness concepts that do not require weighting/scalarization. We will pick up with that point in the next lecture, where we introduce several different forms of Multi-Objective Evolutionary Algorithms (and Pareto ranking).
Whiteboard notes for this lecture can be found at:
https://www.dropbox.com/scl/fi/hqfni3lxa8z09c8kzkfi9/IEE598-Lecture3B-2026-02-24-Multi_Objective_Optimality_and_Intro_to_Multi_Objectivce_Genetic_Algrithms-Notes.pdf?rlkey=si10th7dvglfj25wcv2kyoqrh&dl=0
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