We consider the problem of minimizing a continuous function that may be nonsmooth and nonconvex, subject to bound constraints. We propose an algorithm that uses the L-BFGS quasi-Newton approximation of the problem’s curvature together with a variant of the weak Wolfe line search. The key ingredient of the method is an active-set selection strategy that … Read more
Recent years have witnessed the surge of asynchronous parallel (async-parallel) iterative algorithms due to problems involving very large-scale data and a large number of decision variables. Because of asynchrony, the iterates are computed with outdated information, and the age of the outdated information, which we call \emph{delay}, is the number of times it has been … Read more
This paper describes a unified approach for solving Box-Constrained Optimization Problems (BCOP) in Euclidian spaces. The variables may be either continuous or discrete; in which case, they range on a grid of isolated points regularly spaced. For the continuous case, convergence is shown under standard assumptions; for the discrete case, slight modifications ensure that the … Read more
Augmented Lagrangian methods with convergence to second-order stationary points in which any constraint can be penalized or carried out to the subproblems are considered in this work. The resolution of each subproblem can be done by any numerical algorithm able to return approximate second-order stationary points. The developed global convergence theory is stronger than the … Read more
Adaptive cubic regularization methods have emerged as a credible alternative to linesearch and trust-region for smooth nonconvex optimization, with optimal complexity amongst second-order methods. Here we consider a general/new class of adaptive regularization methods, that use first- or higher-order local Taylor models of the objective regularized by a(ny) power of the step size and applied … Read more
We describe the fundamentals of algorithms for minimizing a smooth nonlinear function, and extensions of these methods to the sum of a smooth function and a convex nonsmooth function. Such objective functions are ubiquitous in data analysis applications, as we illustrate using several examples. We discuss methods that make use of gradient (first-order) information about … Read more
We propose an asynchronous, decentralized algorithm for consensus optimization. The algorithm runs over a network in which the agents communicate with their neighbors and perform local computation. In the proposed algorithm, each agent can compute and communicate independently at different times, for different durations, with the information it has even if the latest information from … Read more
PDE-constrained optimization aims at finding optimal setups for partial differential equations so that relevant quantities are minimized. Including sparsity promoting terms in the formulation of such problems results in more practically relevant computed controls but adds more challenges to the numerical solution of these problems. The needed L^1-terms as well as additional inclusion of box … Read more
We are interested in the existence of Pareto solutions to the vector optimization problem $$\text{\rm Min}_{\,\mathbb{R}^m_+} \{f(x) \,|\, x\in \mathbb{R}^n\},$$ where $f\colon\mathbb{R}^n\to \mathbb{R}^m$ is a polynomial map. By using the {\em tangency variety} of $f$ we first construct a semi-algebraic set of dimension at most $m – 1$ containing the set of Pareto values of … Read more
We contribute improvements to a Lagrangian dual solution approach applied to large-scale optimization problems whose objective functions are convex, continuously differentiable and possibly nonlinear, while the non-relaxed constraint set is compact but not necessarily convex. Such problems arise, for example, in the split-variable deterministic reformulation of stochastic mixed-integer optimization problems. The dual solution approach needs … Read more