Nonlinear Optimization

ACCELERATING CONVERGENCE OF A GLOBALIZED SEQUENTIAL QUADRATIC PROGRAMMING METHOD TO CRITICAL LAGRANGE MULTIPLIERS

  • Alexey F. Izmailov
    • Categories Constrained Nonlinear Optimization

    This paper concerns the issue of asymptotic acceptance of the true Hessian and the full step by the sequential quadratic programming algorithm for equality-constrained optimization problems. In order to enforce global convergence, the algorithm is equipped with a standard Armijo linesearch procedure for a nonsmooth exact penalty function. The specificity of considerations here is that … Read more

    Average Curvature FISTA for Nonconvex Smooth Composite Optimization Problems

  • Jiaming Liang
  • Renato D.C. Monteiro
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , , , ,

    A previous authors’ paper introduces an accelerated composite gradient (ACG) variant, namely AC-ACG, for solving nonconvex smooth composite optimization (N-SCO) problems. In contrast to other ACG variants, AC-ACG estimates the local upper curvature of the N-SCO problem by using the average of the observed upper-Lipschitz curvatures obtained during the previous iterations, and uses this estimation … Read more

    Stochastic Variance-Reduced Prox-Linear Algorithms for Nonconvex Composite Optimization

  • Lin Xiao
  • Junyu Zhang
    • Categories Nonlinear Optimization, Stochastic Programming Tags , , ,

    We consider the problem of minimizing composite functions of the form $f(g(x))+h(x)$, where~$f$ and~$h$ are convex functions (which can be nonsmooth) and $g$ is a smooth vector mapping. In addition, we assume that $g$ is the average of finite number of component mappings or the expectation over a family of random component mappings. We propose … Read more

    New Bregman proximal type algorithms for solving DC optimization problems

  • Shota Takahashi
  • Mituhiro Fukuda
  • Mirai Tanaka
    • Categories Nonlinear Optimization, Unconstrained Optimization Tags , , , , ,

    Difference of Convex (DC) optimization problems have objective functions that are differences between two convex functions. Representative ways of solving these problems are the proximal DC algorithms, which require that the convex part of the objective function have L-smoothness. In this article, we propose the Bregman Proximal DC Algorithm (BPDCA) for solving large-scale DC optimization … Read more

    Sums of Separable and Quadratic Polynomials

  • Amir Ali Ahmadi
  • Cemil Dibek
  • Georgina Hall
    • Categories Linear, Cone and Semidefinite Programming, Nonlinear Optimization, Statistics Tags , ,

    We study separable plus quadratic (SPQ) polynomials, i.e., polynomials that are the sum of univariate polynomials in different variables and a quadratic polynomial. Motivated by the fact that nonnegative separable and nonnegative quadratic polynomials are sums of squares, we study whether nonnegative SPQ polynomials are (i) the sum of a nonnegative separable and a nonnegative … Read more

    Radial Duality Part I: Foundations

  • Benjamin Grimmer
    • Categories Global Optimization Theory, Nonlinear Optimization

    (Renegar, 2016) introduced a novel approach to transforming generic conic optimization problems into unconstrained, uniformly Lipschitz continuous minimization. We introduce radial transformations generalizing these ideas, equipped with an entirely new motivation and development that avoids any reliance on convex cones or functions. Perhaps of greatest practical importance, this facilitates the development of new families of … Read more

    Radial Duality Part II: Applications and Algorithms

  • Benjamin Grimmer
    • Categories Global Optimization Theory, Nonlinear Optimization, Nonsmooth Optimization

    The first part of this work established the foundations of a radial duality between nonnegative optimization problems, inspired by the work of (Renegar, 2016). Here we utilize our radial duality theory to design and analyze projection-free optimization algorithms that operate by solving a radially dual problem. In particular, we consider radial subgradient, smoothing, and accelerated … Read more

    Parallel Strategies for Direct Multisearch

  • C. P. Brás
  • Ana Luisa Custódio
  • S. Tavares
  • V. Duarte
  • P. Medeiros
    • Categories Bound-constrained Optimization, Multi-Criteria Optimization, Parallel Algorithms

    Direct Multisearch (DMS) is a Derivative-free Optimization class of algorithms suited for computing approximations to the complete Pareto front of a given Multiobjective Optimization problem. It has a well-supported convergence analysis and simple implementations present a good numerical performance, both in academic test sets and in real applications. Recently, this numerical performance was improved with … Read more

    An Accelerated Minimal Gradient Method with Momentum for Convex Quadratic Optimization

  • Oscar S. Dalmau
  • Rafael Herrera
  • Harry Oviedo
    • Categories Convex Optimization, Unconstrained Optimization Tags , ,

    In this article we address the problem of minimizing a strictly convex quadratic function using a novel iterative method. The new algorithm is based on the well–known Nesterov’s accelerated gradient method. At each iteration of our scheme, the new point is computed by performing a line–search scheme using a search direction given by a linear … Read more

    Homogeneous polynomials and spurious local minima on the unit sphere

  • Jean-Bernard Lasserre
    • Categories Constrained Nonlinear Optimization, Global Optimization Theory, Nonlinear Optimization

    We consider degree-d forms on the Euclidean unit sphere. We specialize to our setting a genericity result by Nie obtained in a more general framework. We exhibit an homogeneous polynomial Res in the coefficients of f, such that if Res(f) is not zero then all points that satisfy first- and second-order necessary optimality conditions are … Read more