We consider the semi-infinite system of polynomial inequalities of the form$$\begin{aligned} {{\mathbf {K}}}:=\{x\in {{\mathbb {R}}}^m\mid p(x,y)\ge 0,\quad \forall y\in S\subseteq {{\mathbb {R}}}^n\}, \end{aligned}$$where p(x, y) is a real polynomial in the variables x and the parameters y, the index set S is a basic semialgebraic set in \({{\mathbb {R}}}^n\), \(-p(x,y)\) is convex in x for every \(y\in S\). We propose a procedure to construct approximate semidefinite representations of \({{\mathbf {K}}}\). There are two indices to index these approximate semidefinite representations. As two indices increase, these semidefinite representation sets expand and contract, respectively, and can approximate \({{\mathbf {K}}}\) as closely as possible under some assumptions. In some special cases, we can fix one of the two indices or both. Then, we consider the optimization problem of minimizing a convex polynomial over \({{\mathbf {K}}}\). We present an SDP relaxation method for this optimization problem by similar strategies used in constructing approximate semidefinite representations of \({{\mathbf {K}}}\). Under certain assumptions, some approximate minimizers of the optimization problem can also be obtained from the SDP relaxations. In some special cases, we show that the SDP relaxation for the optimization problem is exact and all minimizers can be extracted.

中文翻译：

---

凸多项式不等式的半无限系统和多项式优化问题

我们考虑形式为$$ \ begin {aligned} {{\ mathbf {K}}}：= \ {x \ in {{\ mathbb {R}}} ^ m \ mid p的多项式不等式的半无限系统（x，y）\ ge 0，\ quad \ forall y \ in S \ subseteq {{\ mathbb {R}}} ^ n \}，\ end {aligned} $$其中p（x，  y）是实数变量x和参数y的多项式，索引集S是\（{{\ mathbb {R}}} ^ n \）中的基本半代数集合，\（-p（x，y）\）是凸的每个\（y \ in S \）中的x。我们提出了一个程序来构造\（{{\ mathbf {K}}} \）的近似半定表示。有两个索引来索引这些近似半定表示。随着两个索引的增加，这些半定表示集分别扩展和收缩，并且在某些假设下可以尽可能接近\（{{\ mathbf {K}}} \\）。在某些特殊情况下，我们可以修复两个索引之一或同时修复两个索引。然后，我们考虑使\（{{\ mathbf {K}}} \）上的凸多项式最小的优化问题。我们通过构造\（{{\ mathbf {K}}} \）的近似半定表示中使用的类似策略，针对此优化问题提出了一种SDP松弛方法。。在某些假设下，还可以从SDP松弛中获得一些最优化问题的近似极小值。在某些特殊情况下，我们表明针对优化问题的SDP松弛是精确的，并且可以提取所有最小化器。