evaluation complexity

Trust-region algorithms: probabilistic complexity and intrinsic noise with applications to subsampling techniques

  • Stefania Bellavia
  • Gianmarco Gurioli
  • Benedetta Morini
  • Philippe L. Toint
    • Categories Nonlinear Systems and Least-Squares, Unconstrained Optimization Tags , , , , ,

    A trust-region algorithm is presented for finding approximate minimizers of smooth unconstrained functions whose values and derivatives are subject to random noise. It is shown that, under suitable probabilistic assumptions, the new method finds (in expectation) an epsilon-approximate minimizer of arbitrary order q > 0 in at most O(epsilon^{-(q+1)}) inexact evaluations of the function and … Read more

    Adaptive Regularization Minimization Algorithms with Non-Smooth Norms

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

    A regularization algorithm (AR1pGN) for unconstrained nonlinear minimization is considered, which uses a model consisting of a Taylor expansion of arbitrary degree and regularization term involving a possibly non smooth norm. It is shown that the non-smoothness of the norm does not affect the O(\epsilon_1^{-(p+1)/p}) upper bound on evaluation complexity for finding first-order \epsilon_1-approximate minimizers … 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

    Strong Evaluation Complexity of An Inexact Trust-Region Algorithm for Arbitrary-Order Unconstrained Nonconvex Optimization

  • Coralia Cartis
  • Nicholas I. M. Gould
  • Philippe L. Toint
    • Categories Nonlinear Systems and Least-Squares, Unconstrained Optimization Tags , ,

    A trust-region algorithm using inexact function and derivatives values is introduced for solving unconstrained smooth optimization problems. This algorithm uses high-order Taylor models and allows the search of strong approximate minimizers of arbitrary order. The evaluation complexity of finding a $q$-th approximate minimizer using this algorithm is then shown, under standard conditions, to be $\mathcal{O}\big(\min_{j\in\{1,\ldots,q\}}\epsilon_j^{-(q+1)}\big)$ … Read more

    High-order Evaluation Complexity of a Stochastic Adaptive Regularization Algorithm for Nonconvex Optimization Using Inexact Function Evaluations and Randomly Perturbed Derivatives

  • Stefania Bellavia
  • Benedetta Morini
  • Philippe L. Toint
  • Gianmarco Gurioli
    • Categories Bound-constrained Optimization, Constrained Nonlinear Optimization, Unconstrained Optimization Tags , , ,

    A stochastic adaptive regularization algorithm allowing random noise in derivatives and inexact function values is proposed for computing strong approximate minimizers of any order for inexpensively constrained smooth optimization problems. For an objective function with Lipschitz continuous p-th derivative in a convex neighbourhood of the feasible set and given an arbitrary optimality order q, it … Read more

    Strong Evaluation Complexity Bounds for Arbitrary-Order Optimization of Nonconvex Nonsmooth Composite Functions

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

    We introduce the concept of strong high-order approximate minimizers for nonconvex optimization problems. These apply in both standard smooth and composite non-smooth settings, and additionally allow convex or inexpensive constraints. An adaptive regularization algorithm is then proposed to find such approximate minimizers. Under suitable Lipschitz continuity assumptions, whenever the feasible set is convex, it is … Read more

    Minimization of nonsmooth nonconvex functions using inexact evaluations and its worst-case complexity

  • Serge Gratton
  • Philippe L. Toint
  • Ehouarn Simon
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization, Unconstrained Optimization Tags , , , ,

    An adaptive regularization algorithm using inexact function and derivatives evaluations is proposed for the solution of composite nonsmooth nonconvex optimization. It is shown that this algorithm needs at most O(|log(epsilon)|.epsilon^{-2}) evaluations of the problem’s functions and their derivatives for finding an $\epsilon$-approximate first-order stationary point. This complexity bound therefore generalizes that provided by [Bellavia, Gurioli, … Read more

    Convergence and evaluation-complexity analysis of a regularized tensor-Newton method for solving nonlinear least-squares problems subject to convex constraints

  • Nicholas I. M. Gould
  • Tyrone Rees
  • Jennifer Scott
    • Categories Nonlinear Systems and Least-Squares Tags , , ,

    Given a twice-continuously differentiable vector-valued function $r(x)$, a local minimizer of $\|r(x)\|_2$ within a convex set is sought. We propose and analyse tensor-Newton methods, in which $r(x)$ is replaced locally by its second-order Taylor approximation. Convergence is controlled by regularization of various orders. We establish global convergence to a constrained first-order critical point of $\|r(x)\|_2$, … Read more

    Adaptive regularization algorithms with inexact evaluations for nonconvex optimization

  • Stefania Bellavia
  • Benedetta Morini
  • Philippe L. Toint
  • Gianmarco Gurioli
    • Categories Constrained Nonlinear Optimization, Data-Mining, Unconstrained Optimization Tags , , , ,

    A regularization algorithm using inexact function values and inexact derivatives is proposed and its evaluation complexity analyzed. This algorithm is applicable to unconstrained problems and to problems with inexpensive constraints (that is constraints whose evaluation and enforcement has negligible cost) under the assumption that the derivative of highest degree is beta-H\”{o}lder continuous. It features a … Read more