Constrained Nonlinear Optimization

A sequential adaptive regularisation using cubics algorithm for solving nonlinear equality constrained optimization

  • Yonggang Pei
  • Shaofang Song
  • Zhu Detong
    • Categories Constrained Nonlinear Optimization Tags , , ,

    The adaptive regularisation algorithm using cubics (ARC) is initially proposed for unconstrained optimization. ARC has excellent convergence properties and complexity. In this paper, we extend ARC to solve nonlinear equality constrained optimization and propose a sequential adaptive regularisation using cubics algorithm inspired by sequential quadratic programming (SQP) methods. In each iteration of our method, the … Read more

    A Globally Convergent Distributed Jacobi Scheme for Block-Structured Nonconvex Constrained Optimization Problems

  • Anirudh Subramanyam
  • Youngdae Kim
  • Michel Schanen
  • François Pacaud
  • Mihai Anitescu
    • Categories Constrained Nonlinear Optimization Tags , ,

    Motivated by the increasing availability of high-performance parallel computing, we design a distributed parallel algorithm for linearly-coupled block-structured nonconvex constrained optimization problems. Our algorithm performs Jacobi-type proximal updates of the augmented Lagrangian function, requiring only local solutions of separable block nonlinear programming (NLP) problems. We provide a cheap and explicitly computable Lyapunov function that allows … Read more

    Model-Based Derivative-Free Methods for Convex-Constrained Optimization

  • Matthew Hough
  • Lindon Roberts
    • Categories Constrained Nonlinear Optimization Tags , , ,

    We present a model-based derivative-free method for optimization subject to general convex constraints, which we assume are unrelaxable and accessed only through a projection operator that is cheap to evaluate. We prove global convergence and a worst-case complexity of $O(\epsilon^{-2})$ iterations and objective evaluations for nonconvex functions, matching results for the unconstrained case. We introduce … Read more

    Global Convergence of Algorithms Under Constant Rank Conditions for Nonlinear Second-Order Cone Programming

  • Roberto Andreani
  • Gabriel Haeser
  • Leonardo M. Mito
  • Héctor Ramírez
  • Thiago P. Silveira
    • Categories Constrained Nonlinear Optimization, Second-Order Cone Programming Tags , , , ,

    In [R. Andreani, G. Haeser, L. M. Mito, H. Ramírez C., Weak notions of nondegeneracy in nonlinear semidefinite programming, arXiv:2012.14810, 2020] the classical notion of nondegeneracy (or transversality) and Robinson’s constraint qualification have been revisited in the context of nonlinear semidefinite programming exploiting the structure of the problem, namely, its eigendecomposition. This allows formulating the … Read more

    An Accelerated Inexact Dampened Augmented Lagrangian Method for Linearly-Constrained Nonconvex Composite Optimization Problems

  • Weiwei Kong
  • Renato D.C. Monteiro
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , ,

    This paper proposes and analyzes an accelerated inexact dampened augmented Lagrangian (AIDAL) method for solving linearly-constrained nonconvex composite optimization problems. Each iteration of the AIDAL method consists of: (i) inexactly solving a dampened proximal augmented Lagrangian (AL) subproblem by calling an accelerated composite gradient (ACG) subroutine; (ii) applying a dampened and under-relaxed Lagrange multiplier update; … Read more

    Constrained Optimization in the Presence of Noise

  • Jorge Nocedal
  • Richard H. Byrd
  • Figen Oztoprak
    • Categories Constrained Nonlinear Optimization Tags , ,

    The problem of interest is the minimization of a nonlinear function subject to nonlinear equality constraints using a sequential quadratic programming (SQP) method. The minimization must be performed while observing only noisy evaluations of the objective and constraint functions. In order to obtain stability, the classical SQP method is modified by relaxing the standard Armijo … Read more

    Dual descent ALM and ADMM

  • Kaizhao Sun
  • Xu Andy Sun
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags ,

    Classical primal-dual algorithms attempt to solve $\max_{\mu}\min_{x} \mathcal{L}(x,\mu)$ by alternatively minimizing over the primal variable $x$ through primal descent and maximizing the dual variable $\mu$ through dual ascent. However, when $\mathcal{L}(x,\mu)$ is highly nonconvex with complex constraints in $x$, the minimization over $x$ may not achieve global optimality, and hence the dual ascent step loses … Read more

    A Local MM Subspace Method for Solving Constrained Variational Problems in Image Recovery

  • Émilie Chouzenoux
  • Ségolène Martin
  • Jean-Christophe Pesquet
    • Categories Constrained Nonlinear Optimization, Convex Optimization, Nonsmooth Optimization Tags , , , , , ,

    This article introduces a new Penalized Majorization-Minimization Subspace algorithm (P-MMS) for solving smooth, constrained optimization problems. In short, our approach consists of embedding a subspace algorithm in an inexact exterior penalty procedure. The subspace strategy, combined with a Majoration-Minimization step-size search, takes great advantage of the smoothness of the penalized cost function, while the penalty … Read more

    Nonlinear matrix recovery using optimization on the Grassmann manifold

  • Coralia Cartis
  • Armin Eftekhari
  • Florentin Goyens
    • Categories Applications - Science and Engineering, Constrained Nonlinear Optimization Tags , , ,

    We investigate the problem of recovering a partially observed high-rank matrix whose columns obey a nonlinear structure such as a union of subspaces, an algebraic variety or grouped in clusters. The recovery problem is formulated as the rank minimization of a nonlinear feature map applied to the original matrix, which is then further approximated by … Read more

    Developments in mathematical algorithms and computational tools for proton CT and particle therapy treatment planning

  • Yair Censor
  • Keith E. Schubert
  • Reinhard W. Schulte
    • Categories Basic Sciences Applications, Biomedical Applications, Constrained Nonlinear Optimization Tags , , , , , , , ,

    We summarize recent results and ongoing activities in mathematical algorithms and computer science methods related to proton computed tomography (pCT) and intensitymodulated particle therapy (IMPT) treatment planning. Proton therapy necessitates a high level of delivery accuracy to exploit the selective targeting imparted by the Bragg peak. For this purpose, pCT utilizes the proton beam itself … Read more