多项式optimization的“联合+边缘”算法
A "joint+marginal" algorithm for polynomial optimization
摘要：
我们提出了一个新的算法求解多项式规划P的基础上最近的“联合+边际”的方法，第一作者的参数优化。其思想是首先考虑变量x1作为参数，并求解相关的（n-1）变量（x2，...，xn）问题P（x1），其中参数x1是固定的，并且在某个区间Y1中取值，其中某个概率均匀分布在Y1上。然后考虑我们称之为“联合+边际”半定松弛的层次结构，其解提供一系列一元多项式逼近，随着k的增加，该逼近收敛到问题P（x1）的最优值函数J（x1）。然后，在k先验固定的情况下，计算区间Y 1上的一元多项式pk（x1）的最小值，这简化为求解单个半定规划。一个迭代的过程，现在一个（n-2）-变量的问题P（x2）与参数x2在一些新的间隔Y2等近似的质量取决于如何大k可以选择（一般为显着的大小问题，k=1是唯一的选择）。初步的数值结果提供
英文摘要：
We present a new algorithm for solving a polynomial program P based on the recent "joint + marginal" approach of the first author for, parametric optimization. The idea is to first consider the variable x1 as a parameter and solve the associated (n-1)-variable (x2,...,xn) problem P(x1) where the parameter x1 is fixed and takes values in some interval Y1 with some probability uniformly distributed on Y1. Then one considers the hierarchy of what we call "joint+marginal" semidefinite relaxations, whose duals provide a sequence of univariate polynomial approximations that converges to the optimal value function J(x1) of problem P(x1), as k increases. Then with k fixed a priori, one computes a minimizer of the univariate polynomial pk(x1) on the interval Y1, which reduces to solving a single semidefinite program. One iterates the procedure with now an (n-2)-variable problem P(x2) with parameter x2 in some new interval Y2, etc. The quality of the approximation depends on how large k can be chosen (in general for significant size problems, k=1 is the only choice). Preliminary numerical results are provided
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