Nonlinear Optimization

The Sound of APALM Clapping: Faster Nonsmooth Nonconvex Optimization with Stochastic Asynchronous PALM

  • Damek Davis
  • Brent Edmunds
  • Madeleine Udell
    • Categories Nonlinear Optimization, Nonsmooth Optimization Tags , , , , ,

    We introduce the Stochastic Asynchronous Proximal Alternating Linearized Minimization (SAPALM) method, a block coordinate stochastic proximal-gradient method for solving nonconvex, nonsmooth optimization problems. SAPALM is the first asynchronous parallel optimization method that provably converges on a large class of nonconvex, nonsmooth problems. We prove that SAPALM matches the best known rates of convergence — among … Read more

    Efficient Symmetric Hessian Propagation for Direct Optimal Control

  • Moritz Diehl
  • Boris Houska
  • Rien Quirynen
    • Categories Systems governed by Differential Equations Optimization Tags , , ,

    Direct optimal control algorithms first discretize the continuous-time optimal control problem and then solve the resulting finite dimensional optimization problem. If Newton type optimization algorithms are used for solving the discretized problem, accurate first as well as second order sensitivity information needs to be computed. This article develops a novel approach for computing Hessian matrices … Read more

    Lifted Collocation Integrators for Direct Optimal Control in ACADO Toolkit

  • Moritz Diehl
  • Sebastien Gros
  • Boris Houska
  • Rien Quirynen
    • Categories Systems governed by Differential Equations Optimization Tags , , ,

    This paper presents a class of efficient Newton-type algorithms for solving the nonlinear programs (NLPs) arising from applying a direct collocation approach to continuous time optimal control. The idea is based on an implicit lifting technique including a condensing and expansion step, such that the structure of each subproblem corresponds to that of the multiple … Read more

    A note on the squared slack variables technique for nonlinear optimization

  • Ellen H. Fukuda
  • Masao Fukushima
    • Categories Constrained Nonlinear Optimization Tags , , ,

    In constrained nonlinear optimization, the squared slack variables can be used to transform a problem with inequality constraints into a problem containing only equality constraints. This reformulation is usually not considered in the modern literature, mainly because of possible numerical instabilities. However, this argument only concerns the development of algorithms, and nothing stops us in … Read more

    pyomo.dae: A Modeling and Automatic Discretization Framework for Optimization with Differential and Algebraic Equations

  • Lorenz T. Biegler
  • Jean-Paul Watson
  • Victor M. Zavala
  • Bethany Nicholson
  • John D. Siirola
    • Categories Modeling Languages and Systems, Optimization of Systems modeled by PDEs, Systems governed by Differential Equations Optimization Tags , , , ,

    We describe pyomo.dae, an open source Python-based modeling framework that enables high-level abstract specification of optimization problems with differential and algebraic equations. The pyomo.dae framework is integrated with the Pyomo open source algebraic modeling language, and is available at http: //www.pyomo.org. One key feature of pyomo.dae is that it does not restrict users to standard, … Read more

    Piecewise Parametric Structure in the Pooling Problem – from Sparse Strongly-Polynomial Solutions to NP-Hardness

  • Radu Baltean-Lugojan
  • Ruth Misener
    • Categories Combinatorial Optimization, Quadratic Programming Tags , , , , , ,

    The standard pooling problem is a NP-hard sub-class of non-convex quadratically-constrained optimization problems that commonly arises in process systems engineering applications. We take a parametric approach to uncovering topological structure and sparsity of the standard pooling problem in its p-formulation. The structure uncovered in this approach validates Professor Christodoulos A. Floudas’ intuition that pooling problems … Read more

    A fresh CP look at mixed-binary QPs: New formulations and relaxations

  • Immanuel M. Bomze
  • Peter J.C. Dickinson
  • Abdel Lisser
  • Jianqiang Cheng
    • Categories Constrained Nonlinear Optimization, Quadratic Programming Tags , , , , , ,

    Triggered by Burer’s seminal characterization from 2009, many copositive (CP) reformulations of mixed-binary QPs have been discussed by now. Most of them can be used as proper relaxations, if the intractable co(mpletely )positive cones are replaced by tractable approximations. While the widely used approximation hierarchies have the disadvantage to use positive-semidefinite (psd) matrices of orders … Read more

    Randomized Primal-Dual Proximal Block Coordinate Updates

  • Xiang Gao
  • Yangyang Xu
  • Shuzhong Zhang
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , , ,

    In this paper we propose a randomized primal-dual proximal block coordinate updating framework for a general multi-block convex optimization model with coupled objective function and linear constraints. Assuming mere convexity, we establish its $O(1/t)$ convergence rate in terms of the objective value and feasibility measure. The framework includes several existing algorithms as special cases such … Read more

    The complexity of simple models – a study of worst and typical hard cases for the Standard Quadratic Optimization Problem

  • Immanuel M. Bomze
  • Werner Schachinger
  • Reinhard Ullrich
    • Categories Game Theory, Global Optimization Theory, Quadratic Programming

    In a Standard Quadratic Optimization Problem (StQP), a possibly indefinite quadratic form (the simplest nonlinear function) is extremized over the standard simplex, the simplest polytope. Despite this simplicity, the nonconvex instances of this problem class allow for remarkably rich patterns of coexisting local solutions, which are closely related to practical difficulties in solving StQPs globally. … Read more

    Solving PhaseLift by low-rank Riemannian optimization methods for complex semidefinite constraints

  • Kyle A. Gallivan
  • Wen Huang
  • Xiangxiong Zhang
    • Categories Constrained Nonlinear Optimization, Semi-definite Programming

    A framework, PhaseLift, was recently proposed to solve the phase retrieval problem. In this framework, the problem is solved by optimizing a cost function over the set of complex Hermitian positive semidefinite matrices. This approach to phase retrieval motivates a more general consideration of optimizing cost functions on semidefinite Hermitian matrices where the desired minimizers … Read more