This work presents a study of Liu-Storey (LS) nonlinear conjugate gradient (CG) methods to solve vector optimization problems. Three variants of the LS-CG method originally designed to solve single-objective problems are extended to the vector setting. The first algorithm restricts the LS conjugate parameter to be nonnegative and use a sufficiently accurate line search satisfying … 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
We study convergence rates of the classic proximal bundle method for a variety of nonsmooth convex optimization problems. We show that, without any modification, this algorithm adapts to converge faster in the presence of smoothness or a Hölder growth condition. Our analysis reveals that with a constant stepsize, the bundle method is adaptive, yet it … Read more
Difference of Convex (DC) optimization problems have objective functions that are differences between two convex functions. Representative ways of solving these problems are the proximal DC algorithms, which require that the convex part of the objective function have L-smoothness. In this article, we propose the Bregman Proximal DC Algorithm (BPDCA) for solving large-scale DC optimization … Read more
In this article we address the problem of minimizing a strictly convex quadratic function using a novel iterative method. The new algorithm is based on the well–known Nesterov’s accelerated gradient method. At each iteration of our scheme, the new point is computed by performing a line–search scheme using a search direction given by a linear … Read more
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
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
Adaptive cubic regularization (ARC) methods for unconstrained optimization compute steps from linear systems involving a shifted Hessian in the spirit of the Levenberg-Marquardt and trust-region methods. The standard approach consists in performing an iterative search for the shift akin to solving the secular equation in trust-region methods. Such search requires computing the Cholesky factorization of … Read more
This paper continues to address one of its author’s obsession with the well- known problem of dealing with non-convexity during the minimization of a nonlinear function f(x) by Newton-like methods. It builds on some proposals made by the present authors in “A Comparison of methods for traversing regions of non-convexity in optimization problems”. (Numerical Algorithms … Read more
Minimax optimization has become a central tool for modern machine learning with applications in generative adversarial networks, robust optimization, reinforcement learning, etc. These applications are often nonconvex-nonconcave, but the existing theory is unable to identify and deal with the fundamental difficulties posed by nonconvex-nonconcave structures. In this paper, we study the classic proximal point method … Read more