Efficient Trust Region Methods for Nonconvex Optimization

For decades, a great deal of nonlinear optimization research has focused on modeling and solving convex problems. This has been due to the fact that convex objects typically represent satisfactory estimates of real-world phenomenon, and since convex objects have very nice mathematical properties that makes analyses of them relatively straightforward. However, this focus has been changing. In various important applications, such as large-scale data fitting and learning problems, researchers are starting to turn away from simple, convex models toward more challenging nonconvex models that better represent real-world behaviors and can offer more useful solutions.To contribute to this new focus on nonconvex optimization models, we discuss and present new techniques for solving nonconvex optimization problems that possess attractive theoretical and practical properties. First, we propose a trust region algorithm that, in the worst case, is able to drive the norm of the gradient of the objective function below a prescribed threshold of $\epsilon \in (0,\infty)$ after at most $\mathcal{O}(\epsilon^{-3/2})$ iterations, function evaluations, and derivative evaluations. This improves upon the $\mathcal{O}(\epsilon^{-2})$ bound known to hold for some other trust region algorithms and matches the $\mathcal{O}(\epsilon^{-3/2})$ bound for the recently proposed Adaptive Regularisation framework using Cubics, also known as the ARC algorithm. Our algorithm, entitled TRACE, follows a trust region framework, but employs modified step acceptance criteria and a novel trust region update mechanism that allow the algorithm to achieve such a worst-case global complexity bound. Importantly, we prove that our algorithm also attains global and fast local convergence guarantees under similar assumptions as for other trust region algorithms. We also prove a worst-case upper bound on the number of iterations the algorithm requires to obtain an approximate second-order stationary point.The aforementioned algorithm is based on techniques that require an exact subproblem solution in every iteration. This is a reasonable assumption for small- to medium-scale problems, but is intractable for large-scale optimization. To address this issue, the second project of this thesis involves a proposal of a general \emph{inexact} framework, which contains a wide range of algorithms with optimal complexity bounds, through defining a novel primal-dual subproblem and a set of loose conditions for an inexact solution of it. The proposed framework enjoys the same worst-case iteration complexity bounds for locating approximate first- and second-order stationary points as \RACE. However, it does not require one to solve subproblems exactly. In addition, the framework allows one to use inexact Newton steps whenever possible, a feature which allows the algorithm to use Hessian matrix-free approaches such as the \emph{conjugate gradient} method. This improves the practical performance of the algorithm, as our numerical experiments show.We close by proposing a globally convergent trust funnel algorithm for equality constrained optimization. The proposed algorithm, under some standard assumptions, is able to find a relative first-order stationary point after at most $\mathcal{O}(\epsilon^{-3/2})$ iterations. This matches the complexity bound of the recently proposed Short-Step ARC algorithm. Our proposed algorithm uses the step decomposition and feasibility control mechanism of a trust funnel algorithm, but incorporates ideas from our TRACE framework in order to achieve good complexity bounds