Global Optimization

Relaxing nonconvex quadratic functions by multiple adaptive diagonal perturbations

  • Hongbo Dong
    • Categories Global Optimization Theory, Quadratic Programming, Semi-definite Programming

    The current bottleneck of globally solving mixed-integer (nonconvex) quadratically constrained problem (MIQCP) is still to construct strong but computationally cheap convex relaxations, especially when dense quadratic functions are present. We pro- pose a cutting surface procedure based on multiple diagonal perturbations to derive strong convex quadratic relaxations for nonconvex quadratic problem with separable constraints. Our … Read more

    A search for quantum coin-flipping protocols using optimization techniques

  • Ashwin Nayak
  • Jamie Sikora
  • Levent Tunçel
    • Categories Global Optimization Applications, Linear, Cone and Semidefinite Programming, Optimization Software Design Principles Tags , ,

    Coin-flipping is a cryptographic task in which two physically separated, mistrustful parties wish to generate a fair coin-flip by communicating with each other. Chailloux and Kerenidis (2009) designed quantum protocols that guarantee coin-flips with near optimal bias away from uniform, even when one party deviates arbitrarily from the protocol. The probability of any outcome in … Read more

    Deriving the convex hull of a polynomial partitioning set through lifting and projection

  • Trang T. Nguyen
  • Jean-Philippe P. Richard
  • Mohit Tawarmalani
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Theory Tags , ,

    Relaxations of the bilinear term, $x_1x_2=x_3$, play a central role in constructing relaxations of factorable functions. This is because they can be used directly to relax products of functions with known relaxations. In this paper, we provide a compact, closed-form description of the convex hull of this and other more general bivariate monomial terms (which … Read more

    Exploiting derivative-free local searches in DIRECT-type algorithms for global optimization

  • Giampaolo Liuzzi
  • Stefano Lucidi
  • Veronica Piccialli
    • Categories Biomedical Applications, Global Optimization

    In this paper we consider bound constrained global optimization problems where first-order derivatives of the objective function can be neither computed nor approximated explicitly. For the solution of such problems the DIRECT Algorithm has been proposed which has strong convergence properties and a good ability to locate promising regions of the feasible domain. However, the … Read more

    Global optimization on the torus, the sphere and the rotation group

  • Manuel Gräf
  • Ralf Hielscher
    • Categories Global Optimization Tags , , , , ,

    Detecting all local extrema or the global extremum of a polynomial on the torus, the sphere or the rotation group is a tough yet often requested numerical problem. We present a heuristic approach that applies common descent methods like nonlinear conjugated gradients or Newtons methods simultaneously to a large number of starting points. The corner … Read more

    Sparsity Optimization in Design of Multidimensional Filter Networks

  • Mats Andersson
  • Oleg Burdakov
  • Hans Knutsson
  • Spartak Zikrin
    • Categories Applications - Science and Engineering, Global Optimization Applications, Multi-Criteria Optimization

    Filter networks are used as a powerful tool aimed at reducing the image processing time and maintaining high image quality. They are composed of sparse sub-filters whose high sparsity ensures fast image processing. The filter network design is related to solving a sparse optimization problem where a cardinality constraint bounds above the sparsity level. In … Read more

    Cutting Planes for RLT Relaxations of Mixed 0-1 Polynomial Programs

  • Franklin Djeumou Fomeni
  • Konstantinos Kaparis
  • Adam N. Letchford
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Tags , ,

    The Reformulation-Linearization Technique (RLT), due to Sherali and Adams, can be used to construct hierarchies of linear programming relaxations of mixed 0-1 polynomial programs. As one moves up the hierarchy, the relaxations grow stronger, but the number of variables increases exponentially. We present a procedure that generates cutting planes at any given level of the … Read more

    A refined error analysis for fixed-degree polynomial optimization over the simplex

  • Zhao Sun
    • Categories Global Optimization Theory Tags ,

    We consider fixed-degree polynomial optimization over the simplex. This problem is well known to be NP-hard, since it contains the maximum stable set problem in combinatorial optimization as a special case. In this paper, we consider a known upper bound by taking the minimum value on a regular grid, and a known lower bound based … Read more

    Equivalence and Strong Equivalence between Sparsest and Least l1-Norm Nonnegative Solutions of Linear Systems and Their Application

  • Yun-Bin Zhao
    • Categories Convex Optimization, Global Optimization, Linear Programming Tags , , , ,

    Many practical problems can be formulated as $\ell_0$-minimization problems with nonnegativity constraints, which seek the sparsest nonnegative solutions to underdetermined linear systems. Recent study indicates that $\ell_1$-minimization is efficient for solving some classes of $\ell_0$-minimization problems. From a mathematical point of view, however, the understanding of the relationship between $\ell_0$- and $\ell_1$-minimization remains incomplete. In … Read more

    Reactive Power Management using Firefly and Spiral Optimization under Static and Dynamic Loading Conditions

  • Ripunjoy Phukan
    • Categories Applications - Science and Engineering, Global Optimization Applications, Stochastic Programming Tags , , , ,

    Power System planning encompasses the concept of minimization of transmission losses keeping in mind the voltage stability and system reliability. Voltage profile decides the state of a system and its control is dependent on Generator source voltage, shunt/series injection, transformer taps etc. Optimal parameter setting in system level is needed for managing the available resources … Read more