cubic regularization

A filter sequential adaptive cubic regularisation algorithm for nonlinear constrained optimization

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

    In this paper, we propose a filter sequential adaptive regularisation algorithm using cubics (ARC) for solving nonlinear equality constrained optimization. Similar to sequential quadratic programming methods, an ARC subproblem with linearized constraints is considered to obtain a trial step in each iteration. Composite step methods and reduced Hessian methods are employed to tackle the linearized … Read more

    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

    Derivative-free separable quadratic modeling and cubic regularization for unconstrained optimization

  • Ana Luisa Custódio
  • R. Garmanjani
  • Marcos Raydan
    • Categories Unconstrained Optimization Tags , , , ,

    We present a derivative-free separable quadratic modeling and cubic regularization technique for solving smooth unconstrained minimization problems. The derivative-free approach is mainly concerned with building a quadratic model that could be generated by numerical interpolation or using a minimum Frobenious norm approach, when the number of points available does not allow to build a complete … Read more

    Solving Large Scale Cubic Regularization by a Generalized Eigenvalue Problem

  • Felix Lieder
    • Categories Nonlinear Optimization, Optimization Software and Modeling Systems, Unconstrained Optimization Tags , ,

    Cubic Regularization methods have several favorable properties. In particular under mild assumptions, they are globally convergent towards critical points with second order necessary conditions satisfied. Their adoption among practitioners, however, does not yet match the strong theoretical results. One of the reasons for this discrepancy may be additional implementation complexity needed to solve the occurring … Read more

    On the use of iterative methods in cubic regularization for unconstrained optimization

  • Tommaso Bianconcini
  • Giampaolo Liuzzi
  • Benedetta Morini
  • Marco Sciandrone
    • Categories Nonlinear Optimization Tags , , ,

    In this paper we consider the problem of minimizing a smooth function by using the Adaptive Cubic Regularized framework (ARC). We focus on the computation of the trial step as a suitable approximate minimizer of the cubic model and discuss the use of matrix-free iterative methods. Our approach is alternative to the implementation proposed in … Read more

    Interior-Point Methods for Nonconvex Nonlinear Programming: Primal-Dual Methods and Cubic Regularization

  • Hande Benson
  • David F. Shanno
    • Categories Constrained Nonlinear Optimization Tags , ,

    In this paper, we present a primal-dual interior-point method for solving nonlinear programming problems. It employs a Levenberg-Marquardt (LM) perturbation to the Karush-Kuhn-Tucker (KKT) matrix to handle indefinite Hessians and a line search to obtain sufficient descent at each iteration. We show that the LM perturbation is equivalent to replacing the Newton step by a … Read more

    Interior-Point Methods for Nonconvex Nonlinear Programming: Cubic Regularization

  • Hande Benson
  • David F. Shanno
    • Categories Constrained Nonlinear Optimization Tags , ,

    In this paper, we present a barrier method for solving nonlinear programming problems. It employs a Levenberg-Marquardt perturbation to the Karush-Kuhn-Tucker (KKT) matrix to handle indefinite Hessians and a line search to obtain sufficient descent at each iteration. We show that the Levenberg-Marquardt perturbation is equivalent to replacing the Newton step by a cubic regularization … Read more

    A branch and bound algorithm for the global optimization of Hessian Lipschitz continuous functions

  • Chris Farmer
  • Nicholas I. M. Gould
  • Jaroslav M. Fowkes
    • Categories Global Optimization Tags , ,

    We present a branch and bound algorithm for the global optimization of a twice differentiable nonconvex objective function with a Lipschitz continuous Hessian over a compact, convex set. The algorithm is based on applying cubic regularisation techniques to the objective function within an overlapping branch and bound algorithm for convex constrained global optimization. Unlike other … Read more

    Updating the regularization parameter in the adaptive cubic regularization algorithm

  • Nicholas I. M. Gould
  • Margherita Porcelli
  • Philippe L. Toint
    • Categories Nonlinear Optimization, Unconstrained Optimization Tags , ,

    The adaptive cubic regularization method [Cartis, Gould, Toint, 2009-2010] has been recently proposed for solving unconstrained minimization problems. At each iteration of this method, the objective function is replaced by a cubic approximation which comprises an adaptive regularization parameter whose role is related to the local Lipschitz constant of the objective’s Hessian. We present new … Read more

    Evaluation complexity of adaptive cubic regularization methods for convex unconstrained optimization

  • Coralia Cartis
  • Nicholas I. M. Gould
  • Philippe L. Toint
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags ,

    The adaptive cubic regularization algorithms described in Cartis, Gould & Toint (2009, 2010) for unconstrained (nonconvex) optimization are shown to have improved worst-case efficiency in terms of the function- and gradient-evaluation count when applied to convex and strongly convex objectives. In particular, our complexity upper bounds match in order (as a function of the accuracy … Read more