Lasserre's moment-SOS hierarchy consists of approximating instances of the generalized moment problem (GMP) with moment relaxations and sums-of-squares (SOS) strenghtenings that boil down to convex semidefinite programming (SDP) 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 (POP), the optimal control problem (OCP), the volume computation problem, stability sets approximation problems, and solving nonlinear partial differential equations (PDE). 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.
翻译:Lasserre 的瞬间 SOS 等级包括一般时点问题(GMP) 的近似近似实例, 包括瞬间松动和瞬间宽度( SOS) 平方平方平方平方平方平( SDP) 问题。 由于最初的GMOP 的普遍性, 这一技术的应用无穷无穷, 其中可以列举多位优化问题( POPP)、 最佳控制问题( OCP)、 量计算问题( OCP)、 近似问题( OPE) 和解决非线性部分方程( PDE) 。 最初的GMP 的解决方案随后近似于其时序的有限短径。 对于每一种应用, 证明这些短径与最佳时序的趋同, 提供了对问题的宝贵洞察力, 包括宽松值与原GMP 最佳值的趋同。 本说明提出了这种趋同的一般证据, 不论在简单的标准假设下遇到的问题如何。 作为这一证据的副产品, 一次还获得了无限的软度GMP 和极限的软度 。