Nonlinear Optimization

Enhancements of Discretization Approaches for Non-Convex Mixed-Integer Quadratically Constraint Quadratic Programming: Part II

  • Ben Beach
  • Robert Burlacu
  • Andreas Bärmann
  • Lukas Hager
  • Robert Hildebrand
    • Categories (Mixed) Integer Nonlinear Programming, Nonlinear Optimization, Quadratic Programming Tags , , , ,

    Abstract. This is Part II of a study on mixed-integer programming (MIP) relaxation techniques for the solution of non-convex mixed-integer quadratically constrained quadratic programs (MIQCQPs). We set the focus on MIP relaxation methods for non-convex continuous variable products and extend the well-known MIP relaxation normalized multiparametric disaggregation technique (NMDT), applying a sophisticated discretization to both … Read more

    On an iteratively reweighted linesearch based algorithm for nonconvex composite optimization

  • Silvia Bonettini
  • Marco Prato
  • Simone Rebegoldi
    • Categories Nonlinear Optimization, Nonsmooth Optimization Tags , ,

    In this paper we propose a new algorithm for solving a class of nonsmooth nonconvex problems, which is obtained by combining the iteratively reweighted scheme with a finite number of forward–backward iterations based on a linesearch procedure. The new method overcomes some limitations of linesearch forward–backward methods, since it can be applied also to minimize … Read more

    A Novel Stepsize for Gradient Descent Method

  • Pham Thi Hoai
    • Categories Constrained Nonlinear Optimization, Convex Optimization, Nonlinear Optimization Tags , , , ,

    In this paper, we propose a novel stepsize for the classical gradient descent scheme to solve unconstrained nonlinear optimization problems. We are concerned with the convex and smooth objective without the globally Lipschitz gradient condition. Our new method just needs the locally Lipschitz gradient but still gets the rate $O(\frac{1}{k})$ of $f(x^k)-f_*$ at most. By … Read more

    Gas Transport Network Optimization: PDE-Constrained Models

  • Falk M. Hante
  • Martin Schmidt
    • Categories Applications - Science and Engineering, Infinite Dimensional Optimization, Systems governed by Differential Equations Optimization Tags , , , ,

    The optimal control of gas transport networks was and still is a very important topic for modern economies and societies. Accordingly, a lot of research has been carried out on this topic during the last years and decades. Besides mixed-integer aspects in gas transport network optimization, one of the main challenges is that a physically … Read more

    Barzilai-Borwein-like rules in proximal gradient schemes for ℓ1−regularized problems

  • Serena Crisci
  • Simone Rebegoldi
  • Gerardo Toraldo
  • Marco Viola
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , ,

    We propose a novel steplength selection rule in proximal gradient methods for minimizing the sum of a differentiable function plus an ℓ1-norm penalty term. The proposed rule modifies one of the classical Barzilai-Borwein steplength, extending analogous results obtained in the context of gradient projection methods for constrained optimization. We analyze the spectral properties of the … Read more

    The Jordan algebraic structure of the rotated quadratic cone

  • Baha Alzalg
  • Karima Tamsaouete
    • Categories Convex and Nonsmooth Optimization, Convex Optimization, Quadratic Programming

    In this paper, we look into the rotated quadratic cone and analyze its algebraic structure. We construct an algebra associated with this cone and show that this algebra is a Euclidean Jordan algebra (EJA) with a certain inner product. We also demonstrate some spectral and algebraic characteristics of this EJA. The rotated quadratic cone is … Read more

    Minimizing the difference of convex and weakly convex functions via bundle method

  • Ksenia Syrtseva
  • Welington de Oliveira
  • Sophie Demassey
  • Wim van Ackooij
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , ,

    We consider optimization problems with objective and constraint being the difference of convex and weakly convex functions. This framework covers a vast family of nonsmooth and nonconvex optimization problems, particularly those involving certain classes of composite and nonconvex value functions. We investigate several stationary conditions and extend the proximal bundle algorithm of [van Ackooij et … Read more

    A Newton-CG based barrier-augmented Lagrangian method for general nonconvex conic optimization

  • Chuan He
  • Zhaosong Lu
    • Categories Cone Programming, Nonlinear Optimization Tags , , , , , ,

    In this paper we consider finding an approximate second-order stationary point (SOSP) of general nonconvex conic optimization that minimizes a twice differentiable function subject to nonlinear equality constraints and also a convex conic constraint. In particular, we propose a Newton-conjugate gradient (Newton-CG) based barrier-augmented Lagrangian method for finding an approximate SOSP of this problem. Under … Read more

    Analyzing Inexact Hypergradients for Bilevel Learning

  • Matthias J. Ehrhardt
  • Lindon Roberts
    • Categories Nonlinear Optimization, Optimization in Data Science Tags , ,

    Estimating hyperparameters has been a long-standing problem in machine learning. We consider the case where the task at hand is modeled as the solution to an optimization problem. Here the exact gradient with respect to the hyperparameters cannot be feasibly computed and approximate strategies are required. We introduce a unified framework for computing hypergradients that … Read more

    Splitted Levenberg-Marquardt Method for Large-Scale Sparse Problems

  • Greta Malaspina
  • Nataša Krejić
    • Categories Nonlinear Optimization, Nonlinear Systems and Least-Squares Tags , , ,

    We consider large-scale nonlinear least squares problems with sparse residuals, each of them depending on a small number of variables. A decoupling procedure which results in a splitting of the original problems into a sequence of independent problems of smaller sizes is proposed and analysed. The smaller size problems are modified in a way that … Read more