Min-max and robust polynomial optimization

Abstract

We consider the robust (or min-max) optimization problem

$$J^*:=\max_{\mathbf{y}\in{\Omega}}\min_{\mathbf{x}}\{f(\mathbf{x},\mathbf{y}): (\mathbf{x},\mathbf{y})\in\mathbf{\Delta}\}$$

where f is a polynomial and ${\mathbf{\Delta}\subset\mathbb{R}^n\times\mathbb{R}^p}$ as well as ${{\Omega}\subset\mathbb{R}^p}$ are compact basic semi-algebraic sets. We first provide a sequence of polynomial lower approximations ${(J_i)\subset\mathbb{R}[\mathbf{y}]}$ of the optimal value function ${J(\mathbf{y}):=\min_\mathbf{x}\{f(\mathbf{x},\mathbf{y}): (\mathbf{x},\mathbf{y})\in \mathbf{\Delta}\}}$. The polynomial ${J_i\in\mathbb{R}[\mathbf{y}]}$ is obtained from an optimal (or nearly optimal) solution of a semidefinite program, the ith in the “joint + marginal” hierarchy of semidefinite relaxations associated with the parametric optimization problem ${\mathbf{y}\mapsto J(\mathbf{y})}$, recently proposed in Lasserre (SIAM J Optim 20, 1995-2022, 2010). Then for fixed i, we consider the polynomial optimization problem ${J^*_i:=\max\nolimits_{\mathbf{y}}\{J_i(\mathbf{y}):\mathbf{y}\in{\Omega}\}}$ and prove that ${\hat{J}^*_i(:=\displaystyle\max\nolimits_{\ell=1,\ldots,i}J^*_\ell)}$ converges to J* as i → ∞. Finally, for fixed ℓ ≤ i, each ${J^*_\ell}$ (and hence ${\hat{J}^*_i}$) can be approximated by solving a hierarchy of semidefinite relaxations as already described in Lasserre (SIAM J Optim 11, 796–817, 2001; Moments, Positive Polynomials and Their Applications. Imperial College Press, London 2009).