A generalization of Lowner-John's ellipsoid theorem

• 成果类型: Article
• 署名作者: Lasserre, Jean B.
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Universite de Toulouse; Universite de Toulouse
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-014-0798-5
• 发表日期: 2015
• 页码: 559-591
• 关键词: positive polynomials; volume; optimization; approximation; DECOMPOSITION; moments; CURVES

摘要

We address the following generalization of the Lowner-John ellipsoid problem. Given a (non necessarily convex) compact set and an even integer , find an homogeneous polynomial of degree such that and has minimum volume among all such sets. We show that is a convex optimization problem even if neither nor are convex! We next show that has a unique optimal solution and a characterization with at most contacts points in is also provided. This is the analogue for of Lowner-John's theorem in the quadratic case , but importantly, we neither require the set nor the sublevel set to be convex. More generally, there is also an homogeneous polynomial of even degree and a point such that and has minimum volume among all such sets (but uniqueness is not guaranteed). Finally, we also outline a numerical scheme to approximate as closely as desired the optimal value and an optimal solution. It consists of solving a hierarchy of convex optimization problems with strictly convex objective function and Linear Matrix Inequality constraints.