polynomial optimization

Global optimality in minimum compliance topology optimization of frames and shells by moment-sum-of-squares hierarchy

  • Didier Henrion
  • Martin Kružík
  • Marek Tyburec
  • Jan Zeman
    • Categories Civil and Environmental Engineering, Constrained Nonlinear Optimization, Semi-definite Programming Tags , , , , ,

    The design of minimum-compliance bending-resistant structures with continuous cross-section parameters is a challenging task because of its inherent non-convexity. Our contribution develops a strategy that facilitates computing all guaranteed globally optimal solutions for frame and shell structures under multiple load cases and self-weight. To this purpose, we exploit the fact that the stiffness matrix is … Read more

    Complexity Aspects of Fundamental Questions in Polynomial Optimization

  • Jeffrey Zhang
    • Categories Game Theory, Global Optimization Theory, Nonlinear Optimization Tags , , , ,

    In this thesis, we settle the computational complexity of some fundamental questions in polynomial optimization. These include the questions of (i) finding a local minimum, (ii) testing local minimality of a candidate point, and (iii) deciding attainment of the optimal value. Our results characterize the complexity of these three questions for all degrees of the … Read more

    On the Complexity of Finding a Local Minimizer of a Quadratic Function over a Polytope

  • Amir Ali Ahmadi
  • Jeffrey Zhang
    • Categories Quadratic Programming Tags , , ,

    We show that unless P=NP, there cannot be a polynomial-time algorithm that finds a point within Euclidean distance $c^n$ (for any constant $c \ge 0$) of a local minimizer of an $n$-variate quadratic function over a polytope. This result (even with $c=0$) answers a question of Pardalos and Vavasis that appeared in 1992 on a … Read more

    Complexity Aspects of Local Minima and Related Notions

  • Amir Ali Ahmadi
  • Jeffrey Zhang
    • Categories Global Optimization Theory, Nonlinear Optimization, Semi-definite Programming Tags , , , , ,

    We consider the notions of (i) critical points, (ii) second-order points, (iii) local minima, and (iv) strict local minima for multivariate polynomials. For each type of point, and as a function of the degree of the polynomial, we study the complexity of deciding (1) if a given point is of that type, and (2) if … Read more

    A note on the Lasserre hierarchy for different formulations of the maximum independent set problem

  • Miguel F. Anjos
  • Youssouf Emine
  • Andrea Lodi
  • Zhao Sun
    • Categories Integer Programming, Linear, Cone and Semidefinite Programming, Network Optimization Tags , , ,

    In this note, we consider several polynomial optimization formulations of the max- imum independent set problem and the use of the Lasserre hierarchy with these different formulations. We demonstrate using computational experiments that the choice of formulation may have a significant impact on the resulting bounds. We also provide theoretical justifications for the observed behavior. … Read more

    Integrality of Linearizations of Polynomials over Binary Variables using Additional Monomials

  • Christopher Hojny
  • Marc E. Pfetsch
  • Matthias Walter
    • Categories (Mixed) Integer Nonlinear Programming, Polyhedra Tags , ,

    Polynomial optimization problems over binary variables can be expressed as integer programs using a linearization with extra monomials in addition to those arising in the given polynomial. We characterize when such a linearization yields an integral relaxation polytope, generalizing work by Del Pia and Khajavirad (SIAM Journal on Optimization, 2018) and Buchheim, Crama and Rodríguez-Heck … Read more

    Random projections for quadratic programs

  • Claudia D'Ambrosio
  • Leo Liberti
  • Pierre-Louis Poirion
  • Ky Vu
    • Categories Quadratic Programming Tags , , , ,

    Random projections map a set of points in a high dimensional space to a lower dimen- sional one while approximately preserving all pairwise Euclidean distances. While random projections are usually applied to numerical data, we show they can be successfully applied to quadratic programming formulations over a set of linear inequality constraints. Instead of solving … Read more

    On semi-infinite systems of convex polynomial inequalities and polynomial optimization problems

  • Feng Guo
  • Xiaoxia Sun
    • Categories Semi-definite Programming, Semi-infinite Programming Tags , , , , ,

    We consider the semi-infinite system of polynomial inequalities of the form \[ \mathbf{K}:=\{x\in\mathbb{R}^m\mid p(x,y)\ge 0,\ \ \forall y\in S\subseteq\mathbb{R}^n\}, \] where $p(X,Y)$ is a real polynomial in the variables $X$ and the parameters $Y$, the index set $S$ is a basic semialgebraic set in $\mathbb{R}^n$, $-p(X,y)$ is convex in $X$ for every $y\in S$. We … Read more

    Markov inequalities, Dubiner distance, norming meshes and polynomial optimization on convex bodies

  • Federico Piazzon
  • Marco Vianello
    • Categories Global Optimization Tags , , , , ,

    We construct norming meshes for polynomial optimization by the classical Markov inequality on general convex bodies in R^d, and by a tangential Markov inequality via an estimate of Dubiner distance on smooth convex bodies. These allow to compute a (1−eps)-approximation to the minimum of any polynomial of degree not exceeding n by O((n/sqrt(eps))^(ad)) samples, with … Read more

    Polynomial Optimization on Chebyshev-Dubiner Webs of Starlike Polygons

  • Marco Vianello
    • Categories Global Optimization Tags , , , ,

    We construct web-shaped norming meshes on starlike polygons, by radial and boundary Chebyshev points. Via the approximation theoretic notion of Dubiner distance, we get a (1-eps)-approximation to the minimum of an arbitrary polynomial of degree n by O(n^2/eps) sampling points. Citation Preprint, July 2018 Article Download View Polynomial Optimization on Chebyshev-Dubiner Webs of Starlike Polygons