Convex sets with semidefinite representation

• 成果类型: Article
• 署名作者: Lasserre, Jean B.
• 署名单位: Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-008-0222-0
• 发表日期: 2009
• 页码: 457-477
• 关键词: Optimization; POLYNOMIALS; complexity

摘要

We provide a sufficient condition on a class of compact basic semialgebraic sets K subset of R-n for their convex hull co(K) to have a semidefinite representation (SDr). This SDr is explicitly expressed in terms of the polynomials g(j) that define K. Examples are provided. We also provide an approximate SDr; that is, for every fixed epsilon > 0, there is a convex set K-epsilon such that co(K) subset of K-epsilon subset of co(K) + epsilon B (where B is the unit ball of R-n), and K-epsilon has an explicit SDr in terms of the g(j)'s. For convex and compact basic semi-algebraic sets K defined by concave polynomials, we provide a simpler explicit SDr when the nonnegative Lagrangian L-f associated with K and any linear f is an element of R[X] is a sum of squares. We also provide an approximate SDr specific to the convex case.