Cone Programming

Facial approach for constructing stationary points for mathematical programs with cone complementarity constraints

  • Javier I. Madariaga
  • Héctor Ramírez
    • Categories Complementarity and Variational Inequalities, Cone Programming Tags , , ,

    This paper studies stationary points in mathematical programs with cone complementarity constraints (CMPCC). We begin by reviewing various formulations of CMPCC and revisiting definitions for Bouligand, proximal strong, regular strong, Wachsmuth’s strong, L-strong, weak, as well as Mordukhovich and Clarke stationary points, establishing a comprehensive framework for CMPCC. Building on key principles related to cone … Read more

    The uniqueness of Lyapunov rank among symmetric cones

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

    The Lyapunov rank of a cone is the dimension of the Lie algebra of its automorphism group. It is invariant under linear isomorphism and in general not unique—two or more non-isomorphic cones can share the same Lyapunov rank. It is therefore not possible in general to identify cones using Lyapunov rank. But suppose we look … Read more

    Interior-point algorithms with full Newton steps for nonsymmetric convex conic optimization

  • Dávid Papp
  • Anita Varga
    • Categories Cone Programming, Convex Optimization Tags , , ,

    We design and analyze primal-dual, feasible interior-point algorithms (IPAs) employing full Newton steps to solve convex optimization problems in standard conic form. Unlike most nonsymmetric cone programming methods, the algorithms presented in this paper require only a logarithmically homogeneous self-concordant barrier (LHSCB) of the primal cone, but compute feasible and \(\varepsilon\)-optimal solutions to both the … Read more

    Proximity results in convex mixed-integer programming

  • Burak Kocuk
  • Diego Moran Ramirez
    • Categories (Mixed) Integer Linear Programming, (Mixed) Integer Nonlinear Programming, Cone Programming Tags , , , ,

    We study proximity (resp. integrality gap), that is, the distance (resp. difference) between the optimal solutions (resp. optimal values) of convex integer programs (IP) and the optimal solutions (resp. optimal values) of their continuous relaxations. We show that these values can be upper bounded in terms of the recession cone of the feasible region of … Read more

    Faces of homogeneous cones and applications to homogeneous chordality

  • João Gouveia
  • Masaru Ito
  • Bruno F. Lourenço
    • Categories Cone Programming, Linear, Cone and Semidefinite Programming, Semi-definite Programming Tags , , , ,

    A convex cone K is said to be homogeneous if its group of automorphisms acts transitively on its relative interior. Important examples of homogeneous cones include symmetric cones and cones of positive semidefinite (PSD) matrices that follow a sparsity pattern given by a homogeneous chordal graph. Our goal in this paper is to elucidate the … Read more

    Jordan and isometric cone automorphisms in Euclidean Jordan algebras

  • Michael Orlitzky
    • Categories Cone Programming, Convex Optimization, Linear, Cone and Semidefinite Programming Tags , , ,

    Every symmetric cone K arises as the cone of squares in a Euclidean Jordan algebra V. As V is a real inner-product space, we may denote by Isom(V) its group of isometries. The groups JAut(V) of its Jordan-algebra automorphisms and Aut(K) of the linear cone automorphisms are then related. For certain inner products, JAut(V) = … Read more

    Generator Subadditive Functions for Mixed-Integer Programs

  • Gustavo Angulo
  • Burak Kocuk
  • Diego Moran Ramirez
    • Categories (Mixed) Integer Nonlinear Programming, Cone Programming, Integer Programming Tags , , , ,

    For equality-constrained linear mixed-integer programs (MIP) defined by rational data, it is known that the subadditive dual is a strong dual and that there exists an optimal solution of a particular form, termed generator subadditive function. Motivated by these results, we explore the connection between Lagrangian duality, subadditive duality and generator subadditive functions for general … Read more

    Exact SDP relaxations for a class of quadratic programs with finite and infinite quadratic constraints

  • Naohiko Arima
  • Sunyoung Kim
  • Masakazu Kojima
    • Categories Cone Programming, Semi-definite Programming Tags , , ,

    We investigate exact semidefinite programming (SDP) relaxations for the problem of minimizing a nonconvex quadratic objective function over a feasible region defined by both finitely and infinitely many nonconvex quadratic inequality constraints (semi-infinite QCQPs). Specifically, we present two sufficient conditions on the feasible region under which the QCQP, with any quadratic objective function over the … Read more

    Projection onto hyperbolicity cones and beyond: a dual Frank-Wolfe approach

  • Takayuki Nagano
  • Bruno F. Lourenço
  • Akiko Takeda
    • Categories Cone Programming, Convex Optimization, Linear, Cone and Semidefinite Programming Tags , , , ,

    We discuss the problem of projecting a point onto an arbitrary hyperbolicity cone from both theoretical and numerical perspectives. While hyperbolicity cones are furnished with a generalization of the notion of eigenvalues, obtaining closed form expressions for the projection operator as in the case of semidefinite matrices is an elusive endeavour. To address that we … Read more

    Tighter yet more tractable relaxations and nontrivial instance generation for sparse standard quadratic optimization

  • Immanuel M. Bomze
  • Bo Peng
  • Yuzhou Qiu
  • E. Alper Yildirim
    • Categories (Mixed) Integer Nonlinear Programming, Cone Programming, Semi-definite Programming Tags , , , ,

    The Standard Quadratic optimization Problem (StQP), arguably the simplest among all classes of NP-hard optimization problems, consists of extremizing a quadratic form (the simplest nonlinear polynomial) over the standard simplex (the simplest polytope/compact feasible set). As a problem class, StQPs may be nonconvex with an exponential number of inefficient local solutions. StQPs arise in a … Read more