我们研究了图 G 的稳定数 α(G) 的半定界 θ^(r)(G) 层级，该层级基于其协正规划表述，由 de Klerk 和 Pasechnik [《SIAM J. Optim.》12 (2002)，第875–892页] 引入，他们猜想在 r=α(G)-1 步时收敛到 α(G)。即使是声称有限收敛的较弱猜想也仍未解决。我们建立了该层级与基于 α(G) 的 Motzkin-Straus 表述的平方和层级之间的联系，并利用这些联系证明当 G 为非临界图时，即对 G 的所有边 e 都有 α(G\e)=α(G) 时，其有限收敛性。这尤其依赖于理解 Motzkin-Straus 表述极小解的结构，并证明恰好在 G 为非临界图时其数量有限。此外，我们证明这些结果在赋予正节点权重的图的加权稳定集问题这一一般设定下成立。另外，作为副产品，我们证明判定一个标准二次规划是否具有有限多个极小解不存在多项式时间算法，除非 P=NP。
英文摘要：
We investigate a hierarchy of semidefinite bounds $\vartheta^{(r)}(G)$ for the stability number $α(G)$ of a graph $G$, based on its copositive programming formulation and introduced by de Klerk and Pasechnik [{\em SIAM J. Optim.} 12 (2002), pp.875--892], who conjectured convergence to $α(G)$ in $r=α(G)-1$ steps. Even the weaker conjecture claiming finite convergence is still open. We establish links between this hierarchy and sum-of-squares hierarchies based on the Motzkin-Straus formulation of $α(G)$, which we use to show finite convergence when $G$ is acritical, i.e., when $α(G\setminus e)=α(G)$ for all edges $e$ of $G$. This relies, in particular, on understanding the structure of the minimizers of the Motzkin-Straus formulation and showing that their number is finite precisely when $G$ is acritical. Moreover we show that these results hold in the general setting of the weighted stable set problem for graphs equipped with positive node weights. In addition, as a byproduct we show that deciding whether a standard quadratic program has finitely many minimizers does not admit a polynomial-time algorithm unless P=NP.
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A central question in optimization is to maximize (or minimize) a linear function over a given polytope P. To solve such a problem in practice one needs a concise description of the polytope P. In this paper we are interested in representations of P using the positive semidefinite cone: a positive semidefinite lift (psd lift) of a polytope P is a representation of P as the projection of an affine slice of the positive semidefinite cone $\mathbf{S}^d_+$. Such a representation allows linear optimization problems over P to be written as semidefinite programs of size d. Such representations can be…
The problem of optimizing over the cone of nonnegative polynomials is a fundamental problem in computational mathematics, with applications to polynomial optimization, control, machine learning, game theory, and combinatorics, among others. A number of breakthrough papers in the early 2000s showed that this problem, long thought to be out of reach, could be tackled by using sum of squares programming. This technique however has proved to be expensive for large-scale problems, as it involves solving large semidefinite programs (SDPs). In the first part of this thesis, we present two methods for…
We propose an iterative algorithm for the numerical computation of sums of squares of polynomials approximating given data at prescribed interpolation points. The method is based on the definition of a convex functional $G$ arising from the dualization of a quadratic regression over the Cholesky factors of the sum of squares decomposition. In order to justify the construction, the domain of $G$, the boundary of the domain and the behavior at infinity are analyzed in details. When the data interpolate a positive univariate polynomial, we show that in the context of the Lukacs sum of squares rep…
We prove decomposition theorems for sparse positive (semi)definite polynomial matrices that can be viewed as sparsity-exploiting versions of the Hilbert--Artin, Reznick, Putinar, and Putinar--Vasilescu Positivstellensätze. First, we establish that a polynomial matrix $P(x)$ with chordal sparsity is positive semidefinite for all $x\in \mathbb{R}^n$ if and only if there exists a sum-of-squares (SOS) polynomial $σ(x)$ such that $σP$ is a sum of sparse SOS matrices. Second, we show that setting $σ(x)=(x_1^2 + \cdots + x_n^2)^ν$ for some integer $ν$ suffices if $P$ is homogeneous and positive defin…
