Nonlinear Optimization

FrankWolfe.jl: a high-performance and flexible toolbox for Frank-Wolfe algorithms and Conditional Gradients

  • Mathieu Besançon
  • Alejandro Carderera
  • Sebastian Pokutta
    • Categories Constrained Nonlinear Optimization, Convex Optimization, Optimization Software and Modeling Systems Tags , ,

    We present FrankWolfe.jl, an open-source implementation of several popular Frank-Wolfe and Conditional Gradients variants for first-order constrained optimization. The package is designed with flexibility and high-performance in mind, allowing for easy extension and relying on few assumptions regarding the user-provided functions. It supports Julia’s unique multiple dispatch feature, and interfaces smoothly with generic linear optimization … Read more

    Solving Bang-Bang Problems Using The Immersed Interface Method and Integer Programming

  • Sarah Strikwerda
  • Ryan Vogt
    • Categories Combinatorial Optimization, Nonlinear Optimization Tags , , ,

    In this paper we study numerically solving optimal control problems with bang-bang control functions. We present a formal Lagrangian approach for solving the optimal control problem, and address difficulties encountered when numerically solving the state and adjoint equations by using the immersed interface method. We note that our numerical approach does not approximate the discontinuous … Read more

    Smoothing fast iterative hard thresholding algorithm for $\ell_0$ regularized nonsmooth convex regression problem

  • Wei Bian
  • Fan Wu
  • Xiaoping Xue
    • Categories Nonlinear Optimization Tags , , , , ,

    We investigate a class of constrained sparse regression problem with cardinality penalty, where the feasible set is defined by box constraint, and the loss function is convex, but not necessarily smooth. First, we put forward a smoothing fast iterative hard thresholding (SFIHT) algorithm for solving such optimization problems, which combines smoothing approximations, extrapolation techniques and … Read more

    A Riemannian smoothing steepest descent method for non-Lipschitz optimization on submanifolds

  • Xiaojun Chen
  • Shiqian Ma
  • Chao Zhang
    • Categories Nonlinear Optimization Tags , , , ,

    In this paper, we propose a Riemannian smoothing steepest descent method to minimize a nonconvex and non-Lipschitz function on submanifolds. The generalized subdifferentials on Riemannian manifold and the Riemannian gradient sub-consistency are defined and discussed. We prove that any accumulation point of the sequence generated by the Riemannian smoothing steepest descent method is a stationary … Read more

    Accelerated derivative-free nonlinear least-squares applied to the estimation of Manning coefficients

  • Ernesto G. Birgin
  • José Mario Martínez
    • Categories Nonlinear Systems and Least-Squares Tags , , ,

    A general framework for solving nonlinear least squares problems without the employment of derivatives is proposed in the present paper together with a new general global convergence theory. With the aim to cope with the case in which the number of variables is big (for the standards of derivative-free optimization), two dimension-reduction procedures are introduced. … Read more

    Hölder Gradient Descent and Adaptive Regularization Methods in Banach Spaces for First-Order Points

  • Serge Gratton
  • Philippe L. Toint
  • Sadok Jerad
    • Categories Unconstrained Optimization Tags , ,

    This paper considers optimization of smooth nonconvex functionals in smooth infinite dimensional spaces. A Hölder gradient descent algorithm is first proposed for finding approximate first-order points of regularized polynomial functionals. This method is then applied to analyze the evaluation complexity of an adaptive regularization method which searches for approximate first-order points of functionals with $\beta$-H\”older … Read more

    The Impact of Noise on Evaluation Complexity: The Deterministic Trust-Region Case

  • Stefania Bellavia
  • Gianmarco Gurioli
  • Benedetta Morini
  • Philippe L. Toint
    • Categories Unconstrained Optimization Tags , , ,

    Intrinsic noise in objective function and derivatives evaluations may cause premature termination of optimization algorithms. Evaluation complexity bounds taking this situation into account are presented in the framework of a deterministic trust-region method. The results show that the presence of intrinsic noise may dominate these bounds, in contrast with what is known for methods in … Read more

    Robust Interior Point Method for Quantum Key Distribution Rate Computation

  • Hao Hu
  • Haesol Im
  • Jie Lin
  • Norbert Lutkenhaus
  • Henry Wolkowicz
    • Categories Applications - OR and Management Sciences, Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , , , , ,

    While the security proof method for quantum key distribution, QKD, based on the numerical key rate calculation problem, is powerful in principle, the practicality of the method is limited by computational resources and the efficiency of the underlying algorithm for convex optimization. We derive a stable reformulation of the convex nonlinear semidefinite programming, SDP, model … Read more

    Algorithms for Difference-of-Convex (DC) Programs Based on Difference-of-Moreau-Envelopes Smoothing

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

    In this paper we consider minimization of a difference-of-convex (DC) function with and without linear constraints. We first study a smooth approximation of a generic DC function, termed difference-of-Moreau-envelopes (DME) smoothing, where both components of the DC function are replaced by their respective Moreau envelopes. The resulting smooth approximation is shown to be Lipschitz differentiable, … Read more

    Penetration depth between two convex polyhedra: An efficient global optimization approach

  • Mark A. Abramson
  • Griffin D. Kent
  • Gavin W. Smith
    • Categories Applications - Science and Engineering, Constrained Nonlinear Optimization, Global Optimization Applications Tags , , , ,

    During the detailed design phase of an aerospace program, one of the most important consistency checks is to ensure that no two distinct objects occupy the same physical space. Since exact geometrical modeling is usually intractable, geometry models are discretized, which often introduces small interferences not present in the fully detailed model. In this paper, … Read more