## An optimally fast objective-function-free minimization algorithm using random subspaces

Article Download View An optimally fast objective-function-free minimization algorithm using random subspaces

Article Download View An optimally fast objective-function-free minimization algorithm using random subspaces

\(\) A second-order algorithm is proposed for minimizing smooth nonconvex functions that alternates between regularized Newton and negative curvature steps. In most cases, the Hessian matrix is regularized with the square root of the current gradient and an additional term taking moderate negative curvature into account, a negative curvature step being taken only exceptionnally. As … Read more

A class of multi-level algorithms for unconstrained nonlinear optimization is presented which does not require the evaluation of the objective function. The class contains the momentum-less AdaGrad method as a particular (single-level) instance. The choice of avoiding the evaluation of the objective function is intended to make the algorithms of the class less sensitive to … Read more

An adaptive regularization algorithm for unconstrained nonconvex optimization is presented in which the objective function is never evaluated, but only derivatives are used. This algorithm belongs to the class of adaptive regularization methods, for which optimal worst-case complexity results are known for the standard framework where the objective function is evaluated. It is shown in … Read more

An Adagrad-inspired class of algorithms for smooth unconstrained optimization is presented in which the objective function is never evaluated and yet the gradient norms decrease at least as fast as O(1/\sqrt{k+1}) while second-order optimality measures converge to zero at least as fast as O(1/(k+1)^{1/3}). This latter rate of convergence is shown to be essentially sharp … Read more

A class of algorithms for optimization in the presence of noise is presented, that does not require the evaluation of the objective function. This class generalizes the well-known Adagrad method. The complexity of this class is then analyzed as a function of its parameters, and it is shown that some methods of the class enjoy … Read more

A class of algorithms for unconstrained nonconvex optimization is considered where the value of the objective function is never computed. The class contains a deterministic version of the first-order Adagrad method typically used for minimization of noisy function, but also allows the use of second-order information when available. The rate of convergence of methods in … Read more

An adaptive regularization algorithm for unconstrained nonconvex optimization is proposed that is capable of handling inexact objective-function and derivative values, and also of providing approximate minimizer of arbitrary order. In comparison with a similar algorithm proposed in Cartis, Gould, Toint (2022), its distinguishing feature is that it is based on controlling the relative error between … Read more

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

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