Linear, Cone and Semidefinite Programming

Optimal Low-Rank Matrix Completion: Semidefinite Relaxations and Eigenvector Disjunctions

  • Dimitris Bertsimas
  • Ryan Cory-Wright
  • Sean Lo
  • Jean Pauphilet
    • Categories Data-Mining, Global Optimization Theory, Semi-definite Programming

    Low-rank matrix completion consists of computing a matrix of minimal complexity that recovers a given set of observations as accurately as possible. Unfortunately, existing methods for matrix completion are heuristics that, while highly scalable and often identifying high-quality solutions, do not possess any optimality guarantees. We reexamine matrix completion with an optimality-oriented eye. We reformulate … Read more

    On the Number of Pivots of Dantzig’s Simplex Methods for Linear and Convex Quadratic Programs

  • Shaoning Han
    • Categories Complementarity and Variational Inequalities, Linear Programming, Quadratic Programming Tags , , , , ,

    Refining and extending works by Ye and Kitahara-Mizuno, this paper presents new results on the number of pivots of simplex-type methods for solving linear programs of the Leontief kind, certain linear complementarity problems of the P kind, and nonnegative constrained convex quadratic programs. Our results contribute to the further understanding of the complexity and efficiency … Read more

    On the longest chain of faces of the completely positive and copositive cones

  • Mitsuhiro Nishijima
    • Categories Cone Programming, Linear, Cone and Semidefinite Programming Tags , , , , ,

    We consider a wide class of closed convex cones K in the space of real n*n symmetric matrices and establish the existence of a chain of faces of K, the length of which is maximized at n(n+1)/2 + 1. Examples of such cones include, but are not limited to, the completely positive and the copositive … Read more

    Optimized Dimensionality Reduction for Moment-based Distributionally Robust Optimization

  • ShiyiJiang
  • Jianqiang Cheng
  • Kai Pan
  • Z.J. Max Shen
    • Categories Optimization in Data Science, Robust Optimization, Semi-definite Programming Tags , , , ,

    Moment-based distributionally robust optimization (DRO) provides an optimization framework to integrate statistical information with traditional optimization approaches. Under this framework, one assumes that the underlying joint distribution of random parameters runs in a distributional ambiguity set constructed by moment information and makes decisions against the worst-case distribution within the set. Although most moment-based DRO problems … Read more

    A minimal face constant rank constraint qualification for reducible conic programming

  • Roberto Andreani
  • Gabriel Haeser
  • leomakoto
  • Héctor Ramírez
    • Categories Cone Programming, Linear, Cone and Semidefinite Programming, Nonlinear Optimization Tags , , , ,

    \(\) In a previous paper [R. Andreani, G. Haeser, L. M. Mito, H. Ramírez, T. P. Silveira. First- and second-order optimality conditions for second-order cone and semidefinite programming under a constant rank condition. Mathematical Programming, 2023. DOI: 10.1007/s10107-023-01942-8] we introduced a constant rank constraint qualification for nonlinear semidefinite and second-order cone programming by considering all … Read more

    Hidden convexity, optimization, and algorithms on rotation matrices

  • Kevin Shu
  • Alex L. Wang
  • Akshay Ramachandran
    • Categories Constrained Nonlinear Optimization, Other Topics, Semi-definite Programming Tags , ,

    \(\) This paper studies hidden convexity properties associated with constrained optimization problems over the set of rotation matrices \(\text{SO}(n)\). Such problems are nonconvex due to the constraint\(X\in\text{SO}(n)\). Nonetheless, we show that certain linear images of \(\text{SO}(n)\) are convex, opening up the possibility for convex optimization algorithms with provable guarantees for these problems. Our main technical … Read more

    Jordan automorphisms and derivatives of symmetric cones

  • Michael Orlitzky
    • Categories Cone Programming Tags , , , ,

    Hyperbolicity cones, and in particular symmetric cones, are of great interest in optimization. Renegar showed that every hyperbolicity cone has a family of derivative cones that approximate it. Ito and Lourenço found the automorphisms of those derivatives when the original cone is generated by rank-one elements, as symmetric cones happen to be. We show that … Read more

    Application of a Gas Market Model with Linear Programming. The Influence of the Dollar Exchange Rate on the Wholesale Price of Natural Gas in Northwest Europe until 2040

  • Maik Günther
    • Categories Finance and Economics, Linear Programming, Optimization of Simulated Systems Tags , , , , , , , , ,

    The price of natural gas at wholesale markets in Northwest Europe is influenced by numerous parameters. The USD to EUR exchange rate is one of these parameters. Using the LP-based gas market model WEGA, this paper will examine the impact of USD exchange rates on wholesale natural gas prices in Northwest Europe from 2025 to … Read more

    Closing Duality Gaps of SDPs through Perturbation

  • Takashi Tsuchiya
  • Bruno F. Lourenço
  • Masakazu Muramatsu
  • Takayuki Okuno
    • Categories Linear, Cone and Semidefinite Programming, Semi-definite Programming Tags , , , ,

    \(\) Let \(({\bf P},{\bf D})\) be a primal-dual pair of SDPs with a nonzero finite duality gap. Under such circumstances, \({\bf P}\) and \({\bf D}\) are weakly feasible and if we perturb the problem data to recover strong feasibility, the (common) optimal value function \(v\) as a function of the perturbation is not well-defined at … Read more

    Polyhedral Properties of RLT Relaxations of Nonconvex Quadratic Programs and Their Implications on Exact Relaxations

  • Yuzhou Qiu
  • E. Alper Yildirim
    • Categories Global Optimization, Linear Programming, Quadratic Programming Tags , , ,

    We study linear programming relaxations of nonconvex quadratic programs given by the reformulation-linearization technique (RLT), referred to as RLT relaxations. We investigate the relations between the polyhedral properties of the feasible regions of a quadratic program and its RLT relaxation. We establish various connections between recession directions, boundedness, and vertices of the two feasible regions. … Read more