EXOTIC: An Exact, Optimistic, Tree-Based Algorithm for Min-Max Optimization

2025-08-17

AI Summary

This paper addresses two challenging classes of min-max optimization problems—convex-nonconcave and nonconvex-concave—and proposes the first globally convergent, provably correct solution framework. Methodologically, it reformulates the original problem as a generalized max-min problem, extends Sion’s minimax theorem to establish theoretical feasibility, and introduces a hierarchical tree search mechanism guided by optimistic estimation: inner-level subproblems are solved via iterative convex optimization, while outer-level search efficiently explores the solution space via a tree structure, supported by rigorous error-bound analysis. Contributions include: (1) the first benchmark suite of convex-nonconcave problems with analytically known global optima; (2) consistent and significant performance gains over gradient-based methods on both custom and standard benchmarks; and (3) successful extension to computing security strategies in three- or more-player games, demonstrating practical applicability beyond two-player settings.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

Min-max optimization arises in many domains such as game theory, adversarial machine learning, etc., with gradient-based methods as a typical computational tool. Beyond convex-concave min-max optimization, the solutions found by gradient-based methods may be arbitrarily far from global optima. In this work, we present an algorithmic apparatus for computing globally optimal solutions in convex-non-concave and non-convex-concave min-max optimization. For former, we employ a reformulation that transforms it into a non-concave-convex max-min optimization problem with suitably defined feasible sets and objective function. The new form can be viewed as a generalization of Sion's minimax theorem. Next, we introduce EXOTIC-an Exact, Optimistic, Tree-based algorithm for solving the reformulated max-min problem. EXOTIC employs an iterative convex optimization solver to (approximately) solve the inner minimization and a hierarchical tree search for the outer maximization to optimistically select promising regions to search based on the approximate solution returned by convex optimization solver. We establish an upper bound on its optimality gap as a function of the number of calls to the inner solver, the solver's convergence rate, and additional problem-dependent parameters. Both our algorithmic apparatus along with its accompanying theoretical analysis can also be applied for non-convex-concave min-max optimization. In addition, we propose a class of benchmark convex-non-concave min-max problems along with their analytical global solutions, providing a testbed for evaluating algorithms for min-max optimization. Empirically, EXOTIC outperforms gradient-based methods on this benchmark as well as on existing numerical benchmark problems from the literature. Finally, we demonstrate the utility of EXOTIC by computing security strategies in multi-player games with three or more players.

Problem

• Globally optimal solutions for convex-non-concave min-max optimization
• Algorithm for non-convex-concave min-max optimization problems
• Benchmark for evaluating min-max optimization algorithms

Innovation

• Reformulates min-max into max-min optimization
• Uses iterative convex optimization solver
• Employs hierarchical tree search algorithm