Adaptive Augmented Lagrangian Methods: Algorithms and Practical Numerical Experience

In this paper, we consider augmented Lagrangian (AL) algorithms for solving large-scale nonlinear optimization problems that execute adaptive strategies for updating the penalty parameter. Our work is motivated by the recently proposed adaptive AL trust region method by Curtis et al. [An adaptive augmented Lagrangian method for large-scale constrained optimization, Math. Program. 152 (2015), pp.201–245.]. … Read more

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

Iterative Reweighted Linear Least Squares for Exact Penalty Subproblems on Product Sets

We present two matrix-free methods for solving exact penalty subproblems on product sets that arise when solving large-scale optimization problems. The first approach is a novel iterative reweighting algorithm (IRWA), which iteratively minimizes quadratic models of relaxed subproblems while automatically updating a relaxation vector. The second approach is based on alternating direction augmented Lagrangian (ADAL) … Read more

An Interior-Point Trust-Funnel Algorithm for Nonlinear Optimization

We present an interior-point trust-funnel algorithm for solving large-scale nonlinear optimization problems. The method is based on an approach proposed by Gould and Toint (Math Prog 122(1):155–196, 2010) that focused on solving equality constrained problems. Our method is similar in that it achieves global convergence guarantees by combining a trust-region methodology with a funnel mechanism, … Read more

A Sequential Quadratic Optimization Algorithm with Rapid Infeasibility Detection

We present a sequential quadratic optimization (SQO) algorithm for nonlinear constrained optimization. The method attains all of the strong global and fast local convergence guarantees of classical SQO methods, but has the important additional feature that fast local convergence is guaranteed when the algorithm is employed to solve infeasible instances. A two-phase strategy, carefully constructed … Read more

An Inexact Sequential Quadratic Optimization Algorithm for Nonlinear Optimization

We propose a sequential quadratic optimization method for solving nonlinear optimization problems with equality and inequality constraints. The novel feature of the algorithm is that, during each iteration, the primal-dual search direction is allowed to be an inexact solution of a given quadratic optimization subproblem. We present a set of generic, loose conditions that the … Read more

An Adaptive Augmented Lagrangian Method for Large-Scale Constrained Optimization

We propose an augmented Lagrangian algorithm for solving large-scale constrained optimization problems. The novel feature of the algorithm is an adaptive update for the penalty parameter motivated by recently proposed techniques for exact penalty methods. This adaptive updating scheme greatly improves the overall performance of the algorithm without sacrificing the strengths of the core augmented … Read more

A Globally Convergent Primal-Dual Active-Set Framework for Large-Scale Convex Quadratic Optimization

We present a primal-dual active-set framework for solving large-scale convex quadratic optimization problems (QPs). In contrast to classical active-set methods, our framework allows for multiple simultaneous changes in the active- set estimate, which often leads to rapid identification of the optimal active-set regardless of the initial estimate. The iterates of our framework are the active-set … 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

A Note on the Implementation of an Interior-Point Algorithm for Nonlinear Optimization with Inexact Step Computations

This paper describes an implementation of an interior-point algorithm for large-scale nonlinear optimization. It is based on the algorithm proposed by Curtis et al. (SIAM J Sci Comput 32:3447–3475, 2010), a method that possesses global convergence guarantees to first-order stationary points with the novel feature that inexact search direction calculations are allowed in order to … Read more