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We consider the problem of the semidefinite representation of a class of non-compact basic semialgebraic sets. We introduce the conditions of pointedness and closedness at infinity of a semialgebraic set and show that under these conditions our modified hierarchies of nested theta bodies and Lasserres relaxations converge to the closure of the convex hull of $S$. Moreover, if the PP-BDR property is satisfied, our theta body and Lasserres relaxation are exact when the order is large enough; if the PP-BDR property does not hold, our hierarchies convergent uniformly to the closure of the convex hull of $S$ restricted to every fixed ball centered at the origin. We illustrate through a set of examples that the conditions of pointedness and closedness are essential to ensure the convergence. Finally, we provide some strategies to deal with cases where the conditions of pointedness and closedness are violated.
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In this paper we generalize the factorization theorem of Gouveia, Parrilo and Thomas to a broader class of convex sets. Given a general convex set, we define a slack operator associated to the set and its polar according to whether the convex set is full dimensional, whether it is a translated cone and whether it contains lines. We strengthen the condition of a cone lift by requiring not only the convex set is the image of an affine slice of a given closed convex cone, but also its recession cone is the image of the linear slice of the closed convex cone. We show that the generalized lift of a convex set can also be characterized by the cone factorization of a properly defined slack operator.
We propose an algorithm for solving nonlinear convex programs defined in terms of a symmetric positive semidefinite matrix variable $X$. This algorithm rests on the factorization $X=Y Y^T$, where the number of columns of Y fixes the rank of $X$. It is thus very effective for solving programs that have a low rank solution. The factorization $X=Y Y^T$ evokes a reformulation of the original problem as an optimization on a particular quotient manifold. The present paper discusses the geometry of that manifold and derives a second order optimization method. It furthermore provides some conditions on the rank of the factorization to ensure equivalence with the original problem. The efficiency of the proposed algorithm is illustrated on two applications: the maximal cut of a graph and the sparse principal component analysis problem.
For an infinite cardinal $kappa$ let $ell_2(kappa)$ be the linear hull of the standard othonormal base of the Hilbert space $ell_2(kappa)$ of density $kappa$. We prove that a non-separable convex subset $X$ of density $kappa$ in a locally convex linear metric space if homeomorphic to the space (i) $ell_2^f(kappa)$ if and only if $X$ can be written as countable union of finite-dimensional locally compact subspaces, (ii) $[0,1]^omegatimes ell_2^f(kappa)$ if and only if $X$ contains a topological copy of the Hilbert cube and $X$ can be written as a countable union of locally compact subspaces.
The Euclidean space notion of convex sets (and functions) generalizes to Riemannian manifolds in a natural sense and is called geodesic convexity. Extensively studied computational problems such as convex optimization and sampling in convex sets also have meaningful counterparts in the manifold setting. Geodesically convex optimization is a well-studied problem with ongoing research and considerable recent interest in machine learning and theoretical computer science. In this paper, we study sampling and convex optimization problems over manifolds of non-negative curvature proving polynomial running time in the dimension and other relevant parameters. Our algorithms assume a warm start. We first present a random walk based sampling algorithm and then combine it with simulated annealing for solving convex optimization problems. To our knowledge, these are the first algorithms in the general setting of positively curved manifolds with provable polynomial guarantees under reasonable assumptions, and the first study of the connection between sampling and optimization in this setting.
Given a subset $mathbf{S}={A_1, dots, A_m}$ of $mathbb{S}^n$, the set of $n times n$ real symmetric matrices, we define its {it spectrahull} as the set $SH(mathbf{S}) = {p(X) equiv (Tr(A_1 X), dots, Tr(A_m X))^T : X in mathbf{Delta}_n}$, where ${bf Delta}_n$ is the {it spectraplex}, ${ X in mathbb{S}^n : Tr(X)=1, X succeq 0 }$. We let {it spectrahull membership} (SHM) to be the problem of testing if a given $b in mathbb{R}^m$ lies in $SH(mathbf{S})$. On the one hand when $A_i$s are diagonal matrices, SHM reduces to the {it convex hull membership} (CHM), a fundamental problem in LP. On the other hand, a bounded SDP feasibility is reducible to SHM. By building on the {it Triangle Algorithm} (TA) cite{kalchar,kalsep}, developed for CHM and its generalization, we design a TA for SHM, where given $varepsilon$, in $O(1/varepsilon^2)$ iterations it either computes a hyperplane separating $b$ from $SH(mathbf{S})$, or $X_varepsilon in mathbf{Delta}_n$ such that $Vert p(X_varepsilon) - b Vert leq varepsilon R$, $R$ maximum error over $mathbf{Delta}_n$. Under certain conditions iteration complexity improves to $O(1/varepsilon)$ or even $O(ln 1/varepsilon)$. The worst-case complexity of each iteration is $O(mn^2)$, plus testing the existence of a pivot, shown to be equivalent to estimating the least eigenvalue of a symmetric matrix. This together with a semidefinite version of Caratheodory theorem allow implementing TA as if solving a CHM, resorting to the {it power method} only as needed, thereby improving the complexity of iterations. The proposed Triangle Algorithm for SHM is simple, practical and applicable to general SDP feasibility and optimization. Also, it extends to a spectral analogue of SVM for separation of two spectrahulls.
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