Global Optimization

Monomial-wise Optimal Separable Underestimators for Mixed-Integer Polynomial Optimization

  • Christoph Buchheim
  • Claudia D'Ambrosio
    • Categories (Mixed) Integer Nonlinear Programming, Bound-constrained Optimization, Global Optimization

    In this paper we introduce a new method for solving box-constrained mixed-integer polynomial problems to global optimality. The approach, a specialized branch-and-bound algorithm, is based on the computation of lower bounds provided by the minimization of separable underestimators of the polynomial objective function. The underestimators are the novelty of the approach because the standard approaches … Read more

    Application of the Moment-SOS Approach to Global Optimization of the OPF Problem

  • Jean Charles Gilbert
  • Cedric Josz
  • Jean Maeght
  • Patrick Panciatici
    • Categories Applications - Science and Engineering, Global Optimization Applications, Network Optimization Tags , , , ,

    Finding a global solution to the optimal power flow (OPF) problem is difficult due to its nonconvexity. A convex relaxation in the form of semidefinite programming (SDP) has attracted much attention lately as it yields a global solution in several practical cases. However, it does not in all cases, and such cases have been documented … Read more

    Copositive relaxation beats Lagrangian dual bounds in quadratically and linearly constrained QPs

  • Immanuel M. Bomze
    • Categories Global Optimization Theory, Linear, Cone and Semidefinite Programming, Quadratic Programming Tags , , , , ,

    We study non-convex quadratic minimization problems under (possibly non-convex) quadratic and linear constraints, and characterize both Lagrangian and Semi-Lagrangian dual bounds in terms of conic optimization. While the Lagrangian dual is equivalent to the SDP relaxation (which has been known for quite a while, although the presented form, incorporating explicitly linear constraints, seems to be … Read more

    Narrowing the difficulty gap for the Celis-Dennis-Tapia problem

  • Immanuel M. Bomze
  • Michael L. Overton
    • Categories Constrained Nonlinear Optimization, Global Optimization Theory, Quadratic Programming Tags , , , ,

    We study the {\em Celis-Dennis-Tapia (CDT) problem}: minimize a non-convex quadratic function over the intersection of two ellipsoids. In contrast to the well-studied trust region problem where the feasible set is just one ellipsoid, the CDT problem is not yet fully understood. Our main objective in this paper is to narrow the difficulty gap that … Read more

    A Derivative-Free Algorithm for Constrained Global Optimization based on Exact Penalty Functions

  • G. Di Pillo
  • Stefano Lucidi
  • Francesco Rinaldi
    • Categories Global Optimization, Nonlinear Optimization

    Constrained global optimization problems can be tackled by using exact penalty approaches. In a preceding paper, we proposed an exact penalty algorithm for constrained problems which combines an unconstrained global minimization technique for minimizing a non-differentiable exact penalty func- tion for given values of the penalty parameter, and an automatic updating of the penalty parameter … Read more

    Efficient upper and lower bounds for global mixed-integer optimal control

  • Mathieu Claeys
  • Sebastian Sager
  • Frédéric Messine
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Applications, Optimization of Simulated Systems

    We present a control problem for an electrical vehicle. Its motor can be operated in two discrete modes, leading either to acceleration and energy consumption, or to a recharging of the battery. Mathematically, this leads to a mixed-integer optimal control problem (MIOCP) with a discrete feasible set for the controls taking into account the electrical … Read more

    GLODS: Global and Local Optimization using Direct Search

  • Ana Luisa Custódio
  • J. F. A. Madeira
    • Categories Global Optimization Theory Tags , , , ,

    Locating and identifying points as global minimizers is, in general, a hard and time-consuming task. Difficulties increase when the derivatives of the functions defining the problem are not available for use. In this work, we propose a new class of methods suited for global derivative-free constrained optimization. Using direct search of directional type, the algorithm … Read more

    Completely Positive Reformulations for Polynomial Optimization

  • Javier F. Peña
  • Juan C. Vera
  • Luis F. Zuluaga
    • Categories Convex Optimization, Global Optimization Theory Tags , , ,

    Polynomial optimization encompasses a very rich class of problems in which both the objective and constraints can be written in terms of polynomials on the decision variables. There is a well stablished body of research on quadratic polynomial optimization problems based on reformulations of the original problem as a conic program over the cone of … Read more

    Mathematical Programming: Turing completeness and applications to software analysis

  • Leo Liberti
  • Fabrizio Marinelli
    • Categories (Mixed) Integer Nonlinear Programming, Applications - Science and Engineering, Global Optimization Applications Tags , , ,

    Mathematical Programming is Turing complete, and can be used as a general-purpose declarative language. We present a new constructive proof of this fact, and showcase its usefulness by discussing an application to finding the hardest input of any given program running on a Minsky Register Machine. We also discuss an application of Mathematical Programming to … Read more

    Convex Quadratic Relaxations for Mixed-Integer Nonlinear Programs in Power Systems

  • Carleton Coffrin
  • Hassan L. Hijazi
  • Pascal Van Hentenryck
    • Categories Basic Sciences Applications, Convex Optimization, Global Optimization Applications Tags , , , , ,

    This paper presents a set of new convex quadratic relaxations for nonlinear and mixed-integer nonlinear programs arising in power systems. The considered models are motivated by hybrid discrete/continuous applications where existing approximations do not provide optimality guarantees. The new relaxations offer computational efficiency along with minimal optimality gaps, providing an interesting alternative to state-of-the-art semi-definite … Read more