多项式优化中的精确矩表示
Exact Moment Representation in Polynomial Optimization
AI总结：
针对多项式优化中实半代数集上多项式下确界问题，研究矩矩阵（MoM）层次结构的精确性，揭示其与扩展实根的平方和（SoS）层次结构的对偶性，证明平坦截断与支撑集零维的等价性，并通过边界黑塞条件等验证精确性。
摘要：
我们研究多项式优化问题中矩序列的测度表示问题，该问题旨在求实数多项式在由多项式不等式定义的实半代数集上的下确界。我们分析了矩矩阵（MoM）层次结构的精确性，该层次结构与平方和（SoS）层次结构对偶，是拉塞尔（Lasserre）引入的凸锥序列，用于逼近测度和正多项式。我们特别研究了平坦截断性质，这些性质能有效检验MoM精确性何时成立并恢复极小值点。我们证明MoM层次结构的对偶与扩展了定义二次模Q支撑集实根的SoS层次结构一致；由此推导出平坦截断成立当且仅当极小值点相关二次模的支撑集维数为零；还对平坦截断成立时的层次结构阶数进行了界定。作为推论，我们表明当边界黑塞条件（已知的正则性条件）成立时（因此MoM精确性一般成立），以及当二次模Q的支撑集为零维时，平坦截断和MoM精确性均成立；有效数值计算验证了这些平坦截断性质。
英文摘要：
We investigate the problem of representing moment sequences by measures in the context ofPolynomial Optimization Problems, that consist in finding the infimum of a real polynomial ona real semialgebraic set defined by polynomial inequalities. We analyze the exactness of MomentMatrix (MoM) hierarchies, dual to the Sum of Squares (SoS) hierarchies, which are sequences ofconvex cones introduced by Lasserre to approximate measures and positive polynomials. Weinvestigate in particular flat truncation properties, which allow testing effectively when MoMexactness holds and recovering the minimizers.We show that the dual of the MoM hierarchy coincides with the SoS hierarchy extendedwith the real radical of the support of the defining quadratic module Q. We deduce thatflat truncation happens if and only if the support of the quadratic module associated withthe minimizers is of dimension zero. We also bound the order of the hierarchy at which flattruncation holds.As corollaries, we show that flat truncation and MoM exactness hold when regularityconditions, known as Boundary Hessian Conditions, hold (and thus that MoM exactness holdsgenerically); and when the support of the quadratic module Q is zero-dimensional. Effectivenumerical computations illustrate these flat truncation properties.
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