参数多项式optimization的“联合+边缘”方法
A "joint+marginal" approach to parametric polynomial optimization
摘要：
给定一个紧参数集$Y\subset R^p$，我们考虑$R^n$上的多项式优化问题（$(P_y$），其描述依赖于参数$y\inY$ .我们假设可以在$Y$上计算某个概率测度$ϕ$的所有矩，相对于勒贝格测度绝对连续（例如$Y$是一个盒子或单纯形，$ϕ$是均匀分布的）。然后，我们提供了一个层次的半定松弛，其相关的序列的最优解收敛到矩向量的概率测度，编码的所有信息的所有全球最优解$x^*(y)$的$P_y$。特别地，可以尽可能接近地近似最优解的任何多项式泛函，例如它们的$ϕ$ -均值。此外，使用关于矩的这种知识，可以例如通过最大熵方法来估计最优解的第$k$个坐标的可测量函数$y\mapsto x^*_k(y)$。此外，对于布尔变量$x_k$，可以尽可能接近地近似其持久性$ϕ(\{y:x^*_k(y)=1\})$，即在最优解$x^*(y)$中坐标$x^*_k(y)$取值1的概率。最后，从对偶半定松弛的最优解出发，给出了一个多项式序列（分别为：分段多项式）下近似与$L_1(ϕ)$（分别）。几乎一致）收敛到最优值函数。
英文摘要：
Given a compact parameter set $Y\subset R^p$, we consider polynomial optimization problems $(P_y$) on $R^n$ whose description depends on the parameter $y\inY$. We assume that one can compute all moments of some probability measure $ϕ$ on $Y$, absolutely continuous with respect to the Lebesgue measure (e.g. $Y$ is a box or a simplex and $ϕ$ is uniformly distributed). We then provide a hierarchy of semidefinite relaxations whose associated sequence of optimal solutions converges to the moment vector of a probability measure that encodes all information about all global optimal solutions $x^*(y)$ of $P_y$. In particular, one may approximate as closely as desired any polynomial functional of the optimal solutions, like e.g. their $ϕ$-mean. In addition, using this knowledge on moments, the measurable function $y\mapsto x^*_k(y)$ of the $k$-th coordinate of optimal solutions, can be estimated, e.g. by maximum entropy methods. Also, for a boolean variable $x_k$, one may approximate as closely as desired its persistency $ϕ(\{y:x^*_k(y)=1\})$, i.e. the probability that in an optimal solution $x^*(y)$, the coordinate $x^*_k(y)$ takes the value 1. At last but not least, from an optimal solution of the dual semidefinite relaxations, one provides a sequence of polynomial (resp. piecewise polynomial) lower approximations with $L_1(ϕ)$ (resp. almost uniform) convergence to the optimal value function.
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