Cone Programming

On exact copositive representation of simplicial quadratic optimization problems, their strong conic duality and a new proof of the Frank-Wolfe theorem

  • Immanuel M. Bomze
  • E. Alper Yildirim
    • Categories Cone Programming, Global Optimization, Global Optimization Theory

    We are interested in exactness, strong conic duality and dual attainability in copositive relaxations of quadratic optimization problems (QPs) of a special form, in which any (feasible) QP can be recast. By using our results, the celebrated Frank-Wolfe theorem on the attainability of any bounded QP even over unbounded polyhedra, regardless of whether the objective … Read more

    Disjunctive Sum of Squares

  • Amir Ali Ahmadi
  • Sanjeeb Dash
  • Yixuan Hua
  • Bartolomeo Stellato
    • Categories Cone Programming, Convex Optimization, Semi-definite Programming Tags , , , , , , ,

    We introduce the concept of disjunctive sum of squares for certifying nonnegativity of polynomials. Unlike the popular sum of squares approach where nonnegativity is certified by a single algebraic identity, the disjunctive sum of squares approach certifies nonnegativity with multiple algebraic identities which can be found in parallel. Our main result is a disjunctive Positivstellensatz … Read more

    On the Convexification of a Class of Mixed-Integer Conic Sets

  • GuxinDU
  • Rui Chen
  • Linchuan Wei
    • Categories (Mixed) Integer Nonlinear Programming, Cone Programming

    We investigate mixed-integer second-order conic (SOC) sets with a nonlinear right-hand side in the SOC constraint, a structure frequently arising in mixed-integer quadratically constrained programming (MIQCP). Under mild assumptions, we show that the convex hull can be exactly described by replacing the right-hand side with its concave envelope. This characterization enables strong relaxations for MIQCPs … Read more

    Convexification of a Separable Function over a Polyhedral Ground Set

  • Santanu S. Dey
  • Burak Kocuk
    • Categories Cone Programming, Convex Optimization, Global Optimization Tags , , , , ,

    In this paper, we study the set \(\mathcal{S}^\kappa = \{ (x,y)\in\mathcal{G}\times\mathbb{R}^n : y_j = x_j^\kappa , j=1,\dots,n\}\), where \(\kappa > 1\) and the ground set \(\mathcal{G}\) is a nonempty polytope contained in \( [0,1]^n\). This nonconvex set is closely related to separable standard quadratic programming and appears as a substructure in potential-based network flow problems … Read more

    Dual certificates of primal cone membership

  • Joonyeob Lee
  • Dávid Papp
  • Anita Varga
    • Categories Cone Programming, Convex Optimization, Semi-definite Programming Tags , , , , ,

    We discuss optimization problems over convex cones in which membership is difficult to verify directly. In the standard theory of duality, vectors in the dual cone \(K^*\) are associated with separating hyperplanes and interpreted as certificates of non-membership in the primal cone \(K\). Complementing this perspective, we develop easily verifiable certificates of membership in \(K\) … Read more

    A Practical GPU-Enhanced Matrix-Free Primal-Dual Method for Large-Scale Conic Programs

  • Zikai Xiong
    • Categories Cone Programming, Linear, Cone and Semidefinite Programming, Second-Order Cone Programming

    In this paper, we introduce a practical GPU-enhanced matrix-free first-order method for solving large-scale conic programming problems, which we refer to as PDCS, standing for the Primal-Dual Conic Programming Solver. Problems that it solves include linear programs, second-order cone programs, convex quadratic programs, and exponential cone programs. The method avoids matrix factorizations and leverages sparse … Read more

    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