A Quasi-Newton Algorithm for Nonconvex, Nonsmooth Optimization with Global Convergence Guarantees

A line search algorithm for minimizing nonconvex and/or nonsmooth objective functions is presented. The algorithm is a hybrid between a standard Broyden–Fletcher–Goldfarb–Shanno (BFGS) and an adaptive gradient sampling (GS) method. The BFGS strategy is employed because it typically yields fast convergence to the vicinity of a stationary point, and together with the adaptive GS strategy … Read more

A modified limited-memory BNS method for unconstrained minimization based on the conjugate directions idea

A modification of the limited-memory variable metric BNS method for large scale unconstrained optimization is proposed, which consist in corrections (derived from the idea of conjugate directions) of the used difference vectors for better satisfaction of previous quasi-Newton conditions. In comparison with [16], where a similar approach is used, correction vectors from more previous iterations … Read more

A globally convergent trust-region algorithm for unconstrained derivative-free optimization

In this work we explicit a derivative-free trust-region algorithm for unconstrained optimization based on the paper (Computational Optimization and Applications 53: 527-555, 2012) proposed by Powell. The objective function is approximated by quadratic models obtained by polynomial interpolation. The number of points of the interpolation set is fixed. In each iteration only one interpolation point … Read more

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

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

Efficient tridiagonal preconditioner for the matrix-free truncated Newton method

In this report, we study an efficient tridiagonal preconditioner, based on numerical differentiation, applied to the matrix-free truncated Newton method for unconstrained optimization. It is proved that this preconditioner is positive definite for many practical problems. The efficiency of the resulting matrix-free truncated Newton method is demonstrated by results of extensive numerical experiments. Citation Technical … Read more

Convergence of trust-region methods based on probabilistic models

In this paper we consider the use of probabilistic or random models within a classical trust-region framework for optimization of deterministic smooth general nonlinear functions. Our method and setting differs from many stochastic optimization approaches in two principal ways. Firstly, we assume that the value of the function itself can be computed without noise, in … Read more

A class of derivative-free nonmonotone optimization algorithms employing coordinate rotations and gradient approximations

In this paper we study a class of derivative-free unconstrained minimization algorithms employing nonmonotone inexact linesearch techniques along a set of suitable search directions. In particular, we define globally convergent nonmonotone versions of some well-known derivative-free methods and we propose a new algorithm combining coordinate rotations with approximate simplex gradients. Through extensive numerical experimentation, we … Read more

Modifications of the limited-memory BNS method for better satisfaction of previous quasi-Newton conditions

Several modifications of the limited-memory variable metric BNS method for large scale unconstrained optimization are proposed, which consist in corrections (derived from the idea of conjugate directions) of the used di®erence vectors to improve satisfaction of previous quasi-Newton conditions, utilizing information from previous or subsequent iterations. In case of quadratic objective functions, conjugacy of all … Read more

An Adaptive Gradient Sampling Algorithm for Nonsmooth Optimization

We present an algorithm for the minimization of f : Rn → R, assumed to be locally Lipschitz and continuously differentiable in an open dense subset D of Rn. The objective f may be non-smooth and/or non-convex. The method is based on the gradient sampling (GS) algorithm of Burke et al. [A robust gradient sampling … Read more

Conjugate gradient methods based on secant conditions that generate descent search directions for unconstrained optimization

Conjugate gradient methods have been paid attention to, because they can be directly applied to large-scale unconstrained optimization problems. In order to incorporate second order information of the objective function into conjugate gradient methods, Dai and Liao (2001) proposed a conjugate gradient method based on the secant condition. However, their method does not necessarily generate … Read more