Convergence of Lasserre’s hierarchy: the general case
Convergence of Lasserre’s hierarchy: the general case
复制标题
拉塞尔层次结构的收敛：一般情况
作者：
M. Tacchi
Lasserre的矩-SOS层次结构包含了具有矩松弛和平方和强度的广义矩问题（GMP）的近似实例，可归结为凸半定规划问题。由于初始GMP的通用性，该技术的应用数不胜数，其中包括多项式优化问题、最优控制问题、体积计算问题、稳定集逼近问题、求解非线性偏微分方程等。然后用其矩序列的有限截断来近似原始GMP的解。对于每个应用，证明这些截断点向最优矩序列收敛，可以对问题有价值的见解，包括松弛值收敛到原始GMP的最优值。本文在简单的标准假设下，提出了这种收敛性的一般证明，而不考虑所面临的问题。作为这个证明的副产品，我们也得到了无限维GMP及其有限维弛豫的强对偶性质。
Lasserre’s moment-SOS hierarchy consists in approximating instances of the generalized moment problem (GMP) with moment relaxations and sums-of-squares (SOS) strenghtenings that boil down to convex semidefinite programming problems. Due to the generality of the initial GMP, applications of this technology are countless, and one can cite among them the polynomial optimization problem, the optimal control problem, the volume computation problem, stability sets approximation problems, and solving nonlinear partial differential equations. The solution to the original GMP is then approximated with finite truncatures of its moment sequence. For each application, proving convergence of these truncatures towards the optimal moment sequence gives valuable insight on the problem, including convergence of the relaxed values to the original GMP’s optimal value. This note proposes a general proof of such convergence, regardless the problem one is faced with, under simple standard assumptions. As a byproduct of this proof, one also obtains strong duality properties both in the infinite dimensional GMP and its finite dimensional relaxations.