nonconvex optimization

A Sequential Quadratic Programming Algorithm for Nonconvex, Nonsmooth Constrained Optimization

  • Frank E. Curtis
  • Michael L. Overton
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , , ,

    We consider optimization problems with objective and constraint functions that may be nonconvex and nonsmooth. Problems of this type arise in important applications, many having solutions at points of nondifferentiability of the problem functions. We present a line search algorithm for situations when the objective and constraint functions are locally Lipschitz and continuously differentiable on … Read more

    Identifying Active Manifolds in Regularization Problems

  • Warren Hare
    • Categories Generalized Convexity/Monoticity, Nonsmooth Optimization Tags , , , ,

    In this work we consider the problem $\min_x \{ f(x) + P(x) \}$, where $f$ is $\mathcal{C}^2$ and $P$ is nonsmooth, but contains an underlying smooth substructure. Specifically, we assume the function $P$ is prox-regular partly smooth with respect to a active manifold $\M$. Recent work by Tseng and Yun \cite{tseng-yun-2009}, showed that such a … Read more

    Nonconvex Robust Optimization

  • Kwong Meng Teo
    • Categories Applications - Science and Engineering, Constrained Nonlinear Optimization, Robust Optimization Tags , , , , , ,

    We propose a novel robust optimization technique, which is applicable to nonconvex and simulation-based problems. Robust optimization finds decisions with the best worst-case performance under uncertainty. If constraints are present, decisions should also be feasible under perturbations. In the real-world, many problems are nonconvex and involve computer-based simulations. In these applications, the relationship between decision … Read more

    An algorithmic framework for MINLP with separable non-convexity

  • Claudia D'Ambrosio
  • Jon Lee
  • Andreas Wächter
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Tags , , , ,

    Global optimization algorithms, e.g., spatial branch-and-bound approaches like those implemented in codes such as BARON and COUENNE, have had substantial success in tackling complicated, but generally small scale, non-convex MINLPs (i.e., mixed-integer nonlinear programs having non-convex continuous relaxations). Because they are aimed at a rather general class of problems, the possibility remains that larger instances … Read more

    Eigenvalue techniques for proving bounds for convex objective, nonconvex programs

  • Daniel Bienstock
    • Categories Combinatorial Optimization, Convex and Nonsmooth Optimization Tags ,

    We describe techniques combining the S-lemma and computation of projected quadratics which experimentally yield strong bounds on the value of convex quadratic programs with nonconvex constraints Citationunpublished report, Columbia University, March 2009ArticleDownload View PDF

    A Redistributed Proximal Bundle Method for Nonconvex Optimization

  • Warren Hare
  • Claudia Sagastizábal
    • Categories Generalized Convexity/Monoticity, Nonsmooth Optimization Tags , , , , ,

    Proximal bundle methods have been shown to be highly successful optimization methods for unconstrained convex problems with discontinuous first derivatives. This naturally leads to the question of whether proximal variants of bundle methods can be extended to a nonconvex setting. This work proposes an approach based on generating cutting-planes models, not of the objective function … Read more

    A quasisecant method for minimizing nonsmooth functions

  • Adil Bagirov
  • Asef Nazari
    • Categories Nonlinear Optimization, Nonsmooth Optimization Tags , , ,

    In this paper a new algorithm to locally minimize nonsmooth, nonconvex functions is developed. We introduce the notion of secants and quasisecants for nonsmooth functions. The quasisecants are applied to find descent directions of locally Lipschitz functions. We design a minimization algorithm which uses quasisecants to find descent directions. We prove that this algorithm converges … Read more

    An Interior-Point Algorithm for Large-Scale Nonlinear Optimization with Inexact Step Computations

  • Frank E. Curtis
  • Olaf Schenk
    • Categories Nonlinear Optimization, Optimization of Systems modeled by PDEs Tags , , , , , ,

    We present a line-search algorithm for large-scale continuous optimization. The algorithm is matrix-free in that it does not require the factorization of derivative matrices. Instead, it uses iterative linear system solvers. Inexact step computations are supported in order to save computational expense during each iteration. The algorithm is an interior-point approach derived from an inexact … Read more

    Branching and bounds tightening techniques for non-convex MINLP

  • Pietro Belotti
  • Jon Lee
  • Leo Liberti
  • François Margot
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Tags , , ,

    Many industrial problems can be naturally formulated using Mixed Integer Nonlinear Programming (MINLP). Motivated by the demand for Open-Source solvers for real-world MINLP problems, we have developed a spatial Branch-and-Bound software package named COUENNE (Convex Over- and Under-ENvelopes for Nonlinear Estimation). In this paper, we present the structure of couenne and discuss in detail our … Read more

    A Matrix-free Algorithm for Equality Constrained Optimization Problems with Rank-deficient Jacobians

  • Frank E. Curtis
  • Jorge Nocedal
  • Andreas Wächter
    • Categories Constrained Nonlinear Optimization, Infinite Dimensional Optimization Tags , , , , ,

    We present a line search algorithm for large-scale constrained optimization that is robust and efficient even for problems with (nearly) rank-deficient Jacobian matrices. The method is matrix-free (i.e., it does not require explicit storage or factorizations of derivative matrices), allows for inexact step computations, and is applicable for nonconvex problems. The main components of the … Read more