Semidefinite programming for min–max problems and games

Abstract

Abstract. We consider two min-max problems: (1) minimizing the supremum of finitely many rational functions over a compact basic semi-algebraic set and (2) solving a 2-player zero-sum polynomial game in randomized strategies with compact basic semi-algebraic sets of pure strategies. In both problems the optimal value can be approximated by solving a hierarchy of semidefinite relaxations, in the spirit of the moment approach developed in Lasserre [24,26]. This provides a unified approach and a class of algorithms to compute Nash equilibria and min-max strategies of several static and dynamic games. Each semidefinite relaxation can be solved in time which is polynomial in its input size and practice on a sample of experiments reveals that few relaxations are needed for a good approximation (and sometimes even for finite convergence), a behavior similar to what was observed in polynomial optimization.

Robust Model Predictive Flight Control of Unmanned Rotorcrafts

“…Within this formulation τ is defined as a positive scalar value that bounds the objective function and is employed to formulate the semidefinite relaxation of the minimax optimization problem based on the methodologies and strategies presented in [33,38,43] The peak constraint may be equivalently reformulated as two sets of matrix inequalities:…”

Reconstruction of support of a measure from its moments

“…Putinars property: [12], [15]. Putinar's property holds if the level set {x : P j (x) ≥ 0} is compact for some j, or if all P j are affine and K is compact -see [12].…”

“…Putinars property: [12], [15]. Putinar's property holds if the level set {x : P j (x) ≥ 0} is compact for some j, or if all P j are affine and K is compact -see [12]. Clearly these results imply that if there exits M > 0 such that the polynomial P ℓ+1 (x) := M − x 2 ≥ 0 for all x ∈ K, then K ∩ {x : P ℓ+1 ≥ 0} satisfies Putinar's property.…”

Semidefinite programming methods for min-max robust parameter design

“…Algorithms to compute the optimal strategies in games with polynomial payoff and scalar continuous strategy sets were introduced by [22] and have been recently improved by [2]. The vector version of the continous min-max problem is has recently been shown to be solvable through a hierarchy of semi-definite programs [23], [5], mainly based on the advances in polynomial optimization by [4], [24]. In this work, we apply a variant of the algorithm proposed in [5] to various practical examples.…”