An algorithm for semi-infinite polynomial optimization

AbstractWe consider the semi-infinite optimization problem: $$f^*:=\min_{\mathbf{x}\in\mathbf{X}} \bigl\{f(\mathbf{x}):g(\mathbf{x},\mathbf{y}) \leq 0, \forall\mathbf{y}\in\mathbf {Y}_\mathbf{x}\bigr\},$$ where f,g are polynomials and X⊂ℝn as well as Yx⊂ℝp, x∈X, are compact basic semi-algebraic sets. To approximate f∗ we proceed in two steps. First, we use the “joint+marginal” approach of Lasserre (SIAM J. Optim. 20:1995–2022, 2010) to approximate from above the function x↦Φ(x)=sup {g(x,y):y∈Yx} by a polynomial Φd≥Φ, of degree at most 2d, with the strong property that Φd converges to Φ for the L1-norm, as d→∞ (and in particular, almost uniformly for some subsequence (dℓ), ℓ∈ℕ). Therefore, to approximate f∗ one may wish to solve the polynomial optimization problem $f^{0}_{d}=\min_{\mathbf{x}\in\mathbf{X}} \{f(\mathbf{x}): \varPhi _{d}(\mathbf{x})\leq0\}$ via a (by now standard) hierarchy of semidefinite relaxations, and for increasing values of d. In practice d is fixed, small, and one relaxes the constraint Φd≤0 to Φd(x)≤ε with ε>0, allowing us to change ε dynamically. As d increases, the limit of the optimal value $f^{\epsilon}_{d}$ is bounded above by f∗+ε.