Andreas Themelis

Forward---backward quasi-Newton methods for nonsmooth optimization problems

The forward–backward splitting method (FBS) for minimizing a nonsmooth composite function can be interpreted as a (variable-metric) gradient method over a continuously differentiable function which we call forward–backward envelope (FBE). This allows to extend algorithms for smooth unconstrained optimization and apply them to nonsmooth (possibly constrained) problems. Since the FBE can be computed by simply evaluating forward–backward steps, the resulting methods rely on a similar black-box oracle as FBS. We propose an algorithmic scheme that enjoys the same global convergence properties of FBS when the problem is convex, or when the objective function possesses the Kurdyka–Łojasiewicz property at its critical points. Moreover, when using quasi-Newton directions the proposed method achieves superlinear convergence provided that usual second-order sufficiency conditions on the FBE hold at the limit point of the generated sequence. Such conditions translate into milder requirements on the original function involving generalized second-order differentiability. We show that BFGS fits our framework and that the limited-memory variant L-BFGS is well suited for large-scale problems, greatly outperforming FBS or its accelerated version in practice, as well as ADMM and other problem-specific solvers. The analysis of superlinear convergence is based on an extension of the Dennis and Moré theorem for the proposed algorithmic scheme.

Stochastic gradient methods for stochastic model predictive control

We introduce a new stochastic gradient algorithm, SAAGA, and investigate its employment for solving Stochastic MPC problems and multi-stage stochastic optimization programs in general. The method is particularly attractive for scenario-based formulations that involve a large number of scenarios, for which “batch” formulations may become inefficient due to high computational costs. Benefits of the method include cheap computations per iteration and fast convergence due to the sparsity of the proposed problem decomposition.

A primal-dual line search method and applications in image processing

Operator splitting algorithms are enjoying wide acceptance in signal processing for their ability to solve generic convex optimization problems exploiting their structure and leading to efficient implementations. These algorithms are instances of the Krasnoselskil-Mann scheme for finding fixed points of averaged operators. Despite their popularity, however, operator splitting algorithms are sensitive to ill conditioning and often converge slowly. In this paper we propose a line search primal-dual method to accelerate and robustify the Chambolle-Pock algorithm based on SuperMann: a recent extension of the Kras-noselskil-Mann algorithmic scheme. We discuss the convergence properties of this new algorithm and we showcase its strengths on the problem of image denoising using the anisotropic total variation regularization.