Global minimization of polynomial integral functionals
Giovanni Fantuzzi, Federico Fuentes
TL;DR
We address the global minimization of nonconvex polynomial integral functionals F(u)=∫Ω f(x,u,grad u) dx under Dirichlet boundary conditions. The authors propose a two-step scheme: first a bounded finite-element discretization that yields a polynomial optimization problem over a compact set, then a sparsity-exploiting moment-SOS relaxation that produces a sequence of semidefinite programs whose solutions converge to the global minimum as h→0 and ω→∞. Under standard growth, coercivity and quasiconvexity assumptions and the (strong) uniqueness of the minimizer, they prove convergence of the SDP-derived approximations to u^* in W^{1,p} (and in L^q for q<p^*), and convergence of the discrete minima to F^*. They also show that, when f separates into a convex gradient-term and a lower-order part, the energy values converge from above. Computational experiments on singularly perturbed two-well and Swift–Hohenberg energies illustrate practical convergence, provide guidance on discretization and relaxation choices, and demonstrate that the approach yields good initial guesses for traditional solvers even when some assumptions are mildly violated.
Abstract
We describe a `discretize-then-relax' strategy to globally minimize integral functionals over functions $u$ in a Sobolev space subject to Dirichlet boundary conditions. The strategy applies whenever the integral functional depends polynomially on $u$ and its derivatives, even if it is nonconvex. The `discretize' step uses a bounded finite element scheme to approximate the integral minimization problem with a convergent hierarchy of polynomial optimization problems over a compact feasible set, indexed by the decreasing size $h$ of the finite element mesh. The `relax' step employs sparse moment-sum-of-squares relaxations to approximate each polynomial optimization problem with a hierarchy of convex semidefinite programs, indexed by an increasing relaxation order $ω$. We prove that, as $ω\to\infty$ and $h\to 0$, solutions of such semidefinite programs provide approximate minimizers that converge in a suitable sense (including in certain $L^p$ norms) to the global minimizer of the original integral functional if it is unique. We also report computational experiments showing that our numerical strategy works well even when technical conditions required by our theoretical analysis are not satisfied.