Naohiko Arima

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

    Further Development in Convex Conic Reformulation of Geometric Nonconvex Conic Optimization Problems

  • Naohiko Arima
  • Sunyoung Kim
  • Masakazu Kojima
    • Categories Global Optimization, Linear, Cone and Semidefinite Programming, Semi-definite Programming Tags , , , , ,

    \(\) A geometric nonconvex conic optimization problem (COP) was recently proposed by Kim, Kojima and Toh asa unified framework for convex conic reformulation of a class of quadratic optimization problems and polynomial optimization problems. The nonconvex COP minimizes a linear function over the intersection of a nonconvex cone K, a convex subcone J of the … Read more

    A robust Lagrangian-DNN method for a class of quadratic optimization problems

  • Naohiko Arima
  • Sunyoung Kim
  • Masakazu Kojima
  • Kim-Chuan Toh
    • Categories Linear, Cone and Semidefinite Programming, Quadratic Programming Tags , , ,

    The Lagrangian-doubly nonnegative (DNN) relaxation has recently been shown to provide effective lower bounds for a large class of nonconvex quadratic optimization problems (QOPs) using the bisection method combined with first-order methods by Kim, Kojima and Toh in 2016. While the bisection method has demonstrated the computational efficiency, determining the validity of a computed lower … Read more

    Lagrangian-Conic Relaxations, Part II: Applications to Polynomial Optimization Problems

  • Naohiko Arima
  • Sunyoung Kim
  • Masakazu Kojima
  • Kim-Chuan Toh
    • Categories Convex Optimization, Semi-definite Programming Tags , , , ,

    We present the moment cone (MC) relaxation and a hierarchy of sparse Lagrangian-SDP relaxations of polynomial optimization problems (POPs) using the unified framework established in Part I. The MC relaxation is derived for a POP of minimizing a polynomial subject to a nonconvex cone constraint and polynomial equality constraints. It is an extension of the … Read more

    Lagrangian-Conic Relaxations, Part I: A Unified Framework and Its Applications to Quadratic Optimization Problems

  • Naohiko Arima
  • Sunyoung Kim
  • Masakazu Kojima
  • Kim-Chuan Toh
    • Categories Convex Optimization, Linear, Cone and Semidefinite Programming Tags , , , , ,

    In Part I of a series of study on Lagrangian-conic relaxations, we introduce a unified framework for conic and Lagrangian-conic relaxations of quadratic optimization problems (QOPs) and polynomial optimization problems (POPs). The framework is constructed with a linear conic optimization problem (COP) in a finite dimensional vector space endowed with an inner product, where the … Read more

    Extension of Completely Positive Cone Relaxation to Polynomial Optimization

  • Naohiko Arima
  • Sunyoung Kim
  • Masakazu Kojima
    • Categories Linear, Cone and Semidefinite Programming Tags , , , ,

    We propose the moment cone relaxation for a class of polynomial optimization problems (POPs) to extend the results on the completely positive cone programming relaxation for the quadratic optimization (QOP) model by Arima, Kim and Kojima. The moment cone relaxation is constructed to take advantage of sparsity of the POPs, so that efficient numerical methods … Read more

    Simplified Copositive and Lagrangian Relaxations for Linearly Constrained Quadratic Optimization Problems in Continuous and Binary Variables

  • Naohiko Arima
  • Sunyoung Kim
  • Masakazu Kojima
    • Categories Linear, Cone and Semidefinite Programming Tags , , , ,

    For a quadratic optimization problem (QOP) with linear equality constraints in continuous nonnegative variables and binary variables, we propose three relaxations in simplified forms with a parameter $\lambda$: Lagrangian, completely positive, and copositive relaxations. These relaxations are obtained by reducing the QOP to an equivalent QOP with a single quadratic equality constraint in nonnegative variables, … Read more

    A Quadratically Constrained Quadratic Optimization Model for Completely Positive Cone Programming

  • Naohiko Arima
  • Sunyoung Kim
  • Masakazu Kojima
    • Categories Linear, Cone and Semidefinite Programming Tags , ,

    We propose a class of quadratic optimization problems whose exact optimal objective values can be computed by their completely positive cone programming relaxations. The objective function can be any quadratic form. The constraints of each problem are described in terms of quadratic forms with no linear terms, and all constraints are homogeneous equalities, except one … Read more