Concise Complexity Analyses for Trust-Region Methods

Concise complexity analyses are presented for simple trust region algorithms for solving unconstrained optimization problems. In contrast to a traditional trust region algorithm, the algorithms considered in this paper require certain control over the choice of trust region radius after any successful iteration. The analyses highlight the essential algorithm components required to obtain certain complexity … Read more

Regional Complexity Analysis of Algorithms for Nonconvex Smooth Optimization

A strategy is proposed for characterizing the worst-case performance of algorithms for solving nonconvex smooth optimization problems. Contemporary analyses characterize worst-case performance by providing, under certain assumptions on an objective function, an upper bound on the number of iterations (or function or derivative evaluations) required until a pth-order stationarity condition is approximately satisfied. This arguably … Read more

A Shifted Primal-Dual Interior Method for Nonlinear Optimization

Interior methods provide an effective approach for the treatment of inequality constraints in nonlinearly constrained optimization. A new primal-dual interior method is proposed based on minimizing a sequence of shifted primal-dual penalty-barrier functions. Certain global convergence properties are established. In particular, it is shown that every limit point is either an infeasible stationary point, or … Read more

A Self-Correcting Variable-Metric Algorithm Framework for Nonsmooth Optimization

An algorithm framework is proposed for minimizing nonsmooth functions. The framework is variable-metric in that, in each iteration, a step is computed using a symmetric positive definite matrix whose value is updated as in a quasi-Newton scheme. However, unlike previously proposed variable-metric algorithms for minimizing nonsmooth functions, the framework exploits self-correcting properties made possible through … Read more

An Inexact Regularized Newton Framework with a Worst-Case Iteration Complexity of $\mathcal{O}(\epsilon^{-3/2})$ for Nonconvex Optimization

An algorithm for solving smooth nonconvex optimization problems is proposed that, in the worst-case, takes $\mathcal{O}(\epsilon^{-3/2})$ iterations to drive the norm of the gradient of the objective function below a prescribed positive real number $\epsilon$ and can take $\mathcal{O}(\epsilon^{-3})$ iterations to drive the leftmost eigenvalue of the Hessian of the objective above $-\epsilon$. The proposed … Read more

Exploiting Negative Curvature in Deterministic and Stochastic Optimization

This paper addresses the question of whether it can be beneficial for an optimization algorithm to follow directions of negative curvature. Although some prior work has established convergence results for algorithms that integrate both descent and negative curvature directions, there has not yet been numerical evidence showing that such methods offer significant performance improvements. In … Read more

Complexity Analysis of a Trust Funnel Algorithm for Equality Constrained Optimization

A method is proposed for solving equality constrained nonlinear optimization problems involving twice continuously differentiable functions. The method employs a trust funnel approach consisting of two phases: a first phase to locate an $\epsilon$-feasible point and a second phase to seek optimality while maintaining at least $\epsilon$-feasibility. A two-phase approach of this kind based on … Read more

A Reduced-Space Algorithm for Minimizing $\ell_1hBcRegularized Convex Functions

We present a new method for minimizing the sum of a differentiable convex function and an $\ell_1$-norm regularizer. The main features of the new method include: $(i)$ an evolving set of indices corresponding to variables that are predicted to be nonzero at a solution (i.e., the support); $(ii)$ a reduced-space subproblem defined in terms of … Read more

A Dual Gradient-Projection Method for Large-Scale Strictly Convex Quadratic Problems

The details of a solver for minimizing a strictly convex quadratic objective function subject to general linear constraints is presented. The method uses a gradient projection algorithm enhanced with subspace acceleration to solve the bound-constrained dual optimization problem. Such gradient projection methods are well-known, but are typically employed to solve the primal problem when only … Read more

A Flexible ADMM Algorithm for Big Data Applications

We present a flexible Alternating Direction Method of Multipliers (F-ADMM) algorithm for solving optimization problems involving a strongly convex objective function that is separable into $n \geq 2$ blocks, subject to (non-separable) linear equality constraints. The F-ADMM algorithm uses a \emph{Gauss-Seidel} scheme to update blocks of variables, and a regularization term is added to each … Read more