Nicholas I. M. Gould

On the complexity of steepest descent, Newton’s and regularized Newton’s methods for nonconvex unconstrained optimization

  • Coralia Cartis
  • Nicholas I. M. Gould
  • Philippe L. Toint
    • Categories Unconstrained Optimization Tags , ,

    It is shown that the steepest descent and Newton’s method for unconstrained nonconvex optimization under standard assumptions may be both require a number of iterations and function evaluations arbitrarily close to O(epsilon^{-2}) to drive the norm of the gradient below epsilon. This shows that the upper bound of O(epsilon^{-2}) evaluations known for the steepest descent … Read more

    An adaptive cubic regularisation algorithm for nonconvex optimization with convex constraints and its function-evaluation complexity

  • Coralia Cartis
  • Nicholas I. M. Gould
  • Philippe L. Toint
    • Categories Bound-constrained Optimization, Constrained Nonlinear Optimization Tags , , , , ,

    The adaptive cubic overestimation algorithm described in Cartis, Gould and Toint (2007) is adapted to the problem of minimizing a nonlinear, possibly nonconvex, smooth objective function over a convex domain. Convergence to first-order critical points is shown under standard assumptions, but without any Lipschitz continuity requirement on the objective’s Hessian. A worst-case complexity analysis in … Read more

    On solving trust-region and other regularised subproblems in optimization

  • Sue Dollar
  • Nicholas I. M. Gould
  • Daniel P. Robinson
    • Categories Optimization Software and Modeling Systems, Quadratic Programming, Unconstrained Optimization Tags , ,

    The solution of trust-region and regularisation subproblems which arise in unconstrained optimization is considered. Building on the pioneering work of Gay, More’ and Sorensen, methods which obtain the solution of a sequence of parametrized linear systems by factorization are used. Enhancements using high-order polynomial approximation and inverse iteration ensure that the resulting method is both … Read more

    A second derivative SQP method: local convergence

  • Nicholas I. M. Gould
  • Daniel P. Robinson
    • Categories Constrained Nonlinear Optimization Tags , , ,

    Gould and Robinson (NAR 08/18, Oxford University Computing Laboratory, 2008) gave global convergence results for a second-derivative SQP method for minimizing the exact $\ell_1$-merit function for a \emph{fixed} value of the penalty parameter. To establish this result, we used the properties of the so-called Cauchy step, which was itself computed from the so-called predictor step. … Read more

    A second derivative SQP method: theoretical issues

  • Nicholas I. M. Gould
  • Daniel P. Robinson
    • Categories Constrained Nonlinear Optimization Tags , , , ,

    Sequential quadratic programming (SQP) methods form a class of highly efficient algorithms for solving nonlinearly constrained optimization problems. Although second derivative information may often be calculated, there is little practical theory that justifies exact-Hessian SQP methods. In particular, the resulting quadratic programming (QP) subproblems are often nonconvex, and thus finding their global solutions may be … Read more

    A SECOND DERIVATIVE SQP METHOD WITH IMPOSED DESCENT

  • Nicholas I. M. Gould
  • Daniel P. Robinson
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , ,

    Sequential quadratic programming (SQP) methods form a class of highly efficient algorithms for solving nonlinearly constrained optimization problems. Although second derivative information may often be calculated, there is little practical theory that justifies exact-Hessian SQP methods. In particular, the resulting quadratic programming (QP) subproblems are often nonconvex, and thus finding their global solutions may be … Read more

    LANCELOt_simple, a simple interface to LANCELOT B

  • Nicholas I. M. Gould
  • Dominique Orban
  • Philippe L. Toint
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization, Optimization Software and Modeling Systems Tags , ,

    We describe LANCELOT_simple, an interface to the LANCELOT B nonlinear optimization package within the GALAHAD} library (Gould, Orban, Toint, 2003) which ignores problem structure. The result is an easy-to-use Fortran 90 subroutine, with a small number of intuitively interpretable arguments. However, since structure is ignored, the means of presenting problems to the solver limited and … Read more

    Adaptive cubic overestimation methods for unconstrained optimization

  • Coralia Cartis
  • Nicholas I. M. Gould
  • Philippe L. Toint
    • Categories Unconstrained Optimization Tags , , ,

    An Adaptive Cubic Overestimation (ACO) algorithm for unconstrained optimization is proposed, generalizing at the same time an unpublished method due to Griewank (Technical Report NA/12, 1981, DAMTP, Univ. of Cambridge), an algorithm by Nesterov & Polyak (Math. Programming 108(1), 2006, pp 177-205) and a proposal by Weiser, Deuflhard & Erdmann (Optim. Methods Softw. 22(3), 2007, … Read more

    A Filter Active-Set Trust-Region Method

  • Michael P. Friedlander
  • Nicholas I. M. Gould
  • Sven Leyffer
  • Todd S. Munson
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , ,

    We develop a new active-set method for nonlinear programming problems that solves a regularized linear program to predict the active set and then fixes the active constraints to solve an equality-constrained quadratic program for fast convergence. Global convergence is promoted through the use of a filter. We show that the regularization parameter fulfills the same … Read more