In this note we consider the iteration complexity of solving strongly convex multi objective optimization. We discuss the precise meaning of this problem, and indicate it is loosely defined, but the most natural notion is to find a set of Pareto optimal points across a grid of scalarized problems. We derive that in most cases, … Read more
This work presents the convergence rate analysis of stochastic variants of the broad class of direct-search methods of directional type. It introduces an algorithm designed to optimize differentiable objective functions $f$ whose values can only be computed through a stochastically noisy blackbox. The proposed stochastic directional direct-search (SDDS) algorithm accepts new iterates by imposing a … Read more
The delayed weighted gradient method, recently introduced in [13], is a low-cost gradient-type method that exhibits a surprisingly and perhaps unexpected fast convergence behavior that competes favorably with the well-known conjugate gradient method for the minimization of convex quadratic functions. In this work, we establish several orthogonality properties that add understanding to the practical behavior … Read more
In this paper, a reliable hybrid algorithm for solving convex quadratic minimization problems is presented. At each iteration, two points are computed: first, an auxiliary point $\dot{x}_k$ is generated by performing a gradient step equipped with an optimal steplength, then, the next iterate $x_{k+1}$ is obtained through a weighted sum of $\dot{x}_k$ with the penultimate … Read more
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
In this paper, we suggest two ways of calculating interpolation models for unconstrained smooth nonlinear optimization when Hessian-vector products are available. The main idea is to interpolate the objective function using a quadratic on a set of points around the current one and concurrently using the curvature information from products of the Hessian times appropriate … Read more
Worst-case complexity guarantees for nonconvex optimization algorithms have been a topic of growing interest. Multiple frameworks that achieve the best known complexity bounds among a broad class of first- and second-order strategies have been proposed. These methods have often been designed primarily with complexity guarantees in mind and, as a result, represent a departure from … Read more
A stochastic second-order trust region method is proposed, which can be viewed as a second-order extension of the trust-region-ish (TRish) algorithm proposed by Curtis et al. [INFORMS J. Optim. 1(3) 200–220, 2019]. In each iteration, a search direction is computed by (approximately) solving a trust region subproblem defined by stochastic gradient and Hessian estimates. The … Read more
Cubic Regularization methods have several favorable properties. In particular under mild assumptions, they are globally convergent towards critical points with second order necessary conditions satisfied. Their adoption among practitioners, however, does not yet match the strong theoretical results. One of the reasons for this discrepancy may be additional implementation complexity needed to solve the occurring … Read more
We present a stochastic extension of the mesh adaptive direct search (MADS) algorithm originally developed for deterministic blackbox optimization. The algorithm, called StoMADS, considers the unconstrained optimization of an objective function f whose values can be computed only through a blackbox corrupted by some random noise following an unknown distribution. The proposed method is based … Read more