Semi-definite Programming

A Polynomial Time Constraint-Reduced Algorithm for Semidefinite Optimization Problems, with Convergence Proofs

  • Dianne P. O'Leary
  • Sungwoo Park
    • Categories Semi-definite Programming Tags , , , , , , , , ,

    We present an infeasible primal-dual interior point method for semidefinite optimization problems, making use of constraint reduction. We show that the algorithm is globally convergent and has polynomial complexity, the first such complexity result for primal-dual constraint reduction algorithms for any class of problems. Our algorithm is a modification of one with no constraint reduction … Read more

    A Semidefinite Hierarchy for Containment of Spectrahedra

  • Kai Kellner
  • Thorsten Theobald
  • Christian Trabandt
    • Categories Semi-definite Programming Tags , , , , ,

    A spectrahedron is the positivity region of a linear matrix pencil, thus defining the feasible set of a semidefinite program. We propose and study a hierarchy of sufficient semidefinite conditions to certify the containment of a spectrahedron in another one. This approach comes from applying a moment relaxation to a suitable polynomial optimization formulation. The … Read more

    Worst-Case Results For Positive Semidefinite Rank

  • João Gouveia
  • Richard Z. Robinson
  • Rekha R. Thomas
    • Categories Linear, Cone and Semidefinite Programming, Semi-definite Programming Tags , ,

    This paper presents various worst-case results on the positive semidefinite (psd) rank of a nonnegative matrix, primarily in the context of polytopes. We prove that the psd rank of a generic n-dimensional polytope with v vertices is at least (nv)^(1/4) improving on previous lower bounds. For polygons with v vertices, we show that psd rank … Read more

    A Perturbed Sums of Squares Theorem for Polynomial Optimization and its Applications

  • Masakazu Muramatsu
  • Levent Tunçel
  • Hayato Waki
    • Categories Convex Optimization, Quadratic Programming, Semi-definite Programming

    We consider a property of positive polynomials on a compact set with a small perturbation. When applied to a Polynomial Optimization Problem (POP), the property implies that the optimal value of the corresponding SemiDefinite Programming (SDP) relaxation with sufficiently large relaxation order is bounded from below by $(f^¥ast – ¥epsilon)$ and from above by $f^¥ast … Read more

    A generalization of the Lowner-John’s ellipsoid theorem

  • Jean-Bernard Lasserre
    • Categories Convex Optimization, Nonlinear Optimization, Semi-definite Programming

    We address the following generalization $P$ of the Lowner-John’s ellipsoid problem. Given a (non necessarily convex) compact set $K\subset R^n$ and an even integer $d, find an homogeneous polynomial $g$ of degree $d$ such that $K\subset G:=\{x:g(x)\leq1\}$ and $G$ has minimum volume among all such sets. We show that $P$ is a convex optimization problem … Read more

    Positive Semidefinite Matrix Completion, Universal Rigidity and the Strong Arnold Property

  • Monique Laurent
  • Antonios Varvitsiotis
    • Categories Semi-definite Programming Tags , , , , ,

    This paper addresses the following three topics: positive semidefinite (psd) matrix completions, universal rigidity of frameworks, and the Strong Arnold Property (SAP). We show some strong connections among these topics, using semidefinite programming as unifying theme. Our main contribution is a sufficient condition for constructing partial psd matrices which admit a unique completion to a … Read more

    Strong duality in conic linear programming: facial reduction and extended duals

  • Gabor Pataki
    • Categories Convex Optimization, Linear, Cone and Semidefinite Programming, Semi-definite Programming

    The facial reduction algorithm of Borwein and Wolkowicz and the extended dual of Ramana provide a strong dual for the conic linear program (P) \sup { | Ax \leq_K b} in the absence of any constraint qualification. The facial reduction algorithm solves a sequence of auxiliary optimization problems to obtain such a dual. Ramana’s dual … Read more

    S-semigoodness for Low-Rank Semidefinite Matrix Recovery

  • Lingchen Kong
  • Jie Sun
  • Naihua Xiu
    • Categories Convex Optimization, Semi-definite Programming Tags , , , ,

    We extend and characterize the concept of $s$-semigoodness for a sensing matrix in sparse nonnegative recovery (proposed by Juditsky , Karzan and Nemirovski [Math Program, 2011]) to the linear transformations in low-rank semidefinite matrix recovery. We show that s-semigoodness is not only a necessary and sufficient condition for exact $s$-rank semidefinite matrix recovery by a … Read more

    On metric regularity for weakly almost piecewise smooth functions and some applications in nonlinear semidefinite programming

  • Peter Fusek
    • Categories Semi-definite Programming Tags , , ,

    The one-to-one relation between the points fulfilling the KKT conditions of an optimization problem and the zeros of the corresponding Kojima function is well-known. In the present paper we study the interplay between metric regularity and strong regularity of this a priori nonsmooth function in the context of semidefinite programming. Having in mind the topological … Read more

    A Primal-Dual Regularized Interior-Point Method for Semidefinite Programming

  • Ahad Dehghani
  • Jean-Louis Goffin
  • Dominique Orban
    • Categories Semi-definite Programming Tags , , , , ,

    Interior-point methods in semidefinite programming (SDP) require the solution of a sequence of linear systems which are used to derive the search directions. Safeguards are typically required in order to handle rank-deficient Jacobians and free variables. We generalize the primal-dual regularization of \cite{friedlander-orban-2012} to SDP and show that it is possible to recover an optimal … Read more