Convergence Analysis and Improvements for Projection Algorithms and Splitting Methods

Non-smooth convex optimization problems occur in all fields of engineering. A common approach to solving this class of problems is proximal algorithms, or splitting methods. These first-order optimization algorithms are often simple, well suited to solve large-scale problems and have a low computational cost per iteration. Essentially, they encode the solution to an optimization problem as a fixed point of some operator, and iterating this operator eventually results in convergence to an optimal point. However, as for other first order methods, the convergence rate is heavily dependent on the conditioning of the problem. Even though the per-iteration cost is usually low, the number of iterations can become prohibitively large for ill-conditioned problems, especially if a high accuracy solution is sought.In this thesis, a few methods for alleviating this slow convergence are studied, which can be divided into two main approaches. The first are heuristic methods that can be applied to a range of fixed-point algorithms. They are based on understanding typical behavior of these algorithms. While these methods are shown to converge, they come with no guarantees on improved convergence rates.The other approach studies the theoretical rates of a class of projection methods that are used to solve convex feasibility problems. These are problems where the goal is to find a point in the intersection of two, or possibly more, convex sets. A study of how the parameters in the algorithm affect the theoretical convergence rate is presented, as well as how they can be chosen to optimize this rate

Projection methods in conic optimization

There exist efficient algorithms to project a point onto the intersection of a convex cone and an affine subspace. Those conic projections are in turn the work-horse of a range of algorithms in conic optimization, having a variety of applications in science, finance and engineering. This chapter reviews some of these algorithms, emphasizing the so-called regularization algorithms for linear conic optimization, and applications in polynomial optimization. This is a presentation of the material of several recent research articles; we aim here at clarifying the ideas, presenting them in a general framework, and pointing out important techniques

A Primal-Dual Algorithmic Framework for Constrained Convex Minimization

We present a primal-dual algorithmic framework to obtain approximate solutions to a prototypical constrained convex optimization problem, and rigorously characterize how common structural assumptions affect the numerical efficiency. Our main analysis technique provides a fresh perspective on Nesterov's excessive gap technique in a structured fashion and unifies it with smoothing and primal-dual methods. For instance, through the choices of a dual smoothing strategy and a center point, our framework subsumes decomposition algorithms, augmented Lagrangian as well as the alternating direction method-of-multipliers methods as its special cases, and provides optimal convergence rates on the primal objective residual as well as the primal feasibility gap of the iterates for all.Comment: This paper consists of 54 pages with 7 tables and 12 figure

Convex optimization over intersection of simple sets: improved convergence rate guarantees via an exact penalty approach

We consider the problem of minimizing a convex function over the intersection of finitely many simple sets which are easy to project onto. This is an important problem arising in various domains such as machine learning. The main difficulty lies in finding the projection of a point in the intersection of many sets. Existing approaches yield an infeasible point with an iteration-complexity of $O(1/\varepsilon^2)$ for nonsmooth problems with no guarantees on the in-feasibility. By reformulating the problem through exact penalty functions, we derive first-order algorithms which not only guarantees that the distance to the intersection is small but also improve the complexity to $O(1/\varepsilon)$ and $O(1/\sqrt{\varepsilon})$ for smooth functions. For composite and smooth problems, this is achieved through a saddle-point reformulation where the proximal operators required by the primal-dual algorithms can be computed in closed form. We illustrate the benefits of our approach on a graph transduction problem and on graph matching

A Smooth Primal-Dual Optimization Framework for Nonsmooth Composite Convex Minimization

We propose a new first-order primal-dual optimization framework for a convex optimization template with broad applications. Our optimization algorithms feature optimal convergence guarantees under a variety of common structure assumptions on the problem template. Our analysis relies on a novel combination of three classic ideas applied to the primal-dual gap function: smoothing, acceleration, and homotopy. The algorithms due to the new approach achieve the best known convergence rate results, in particular when the template consists of only non-smooth functions. We also outline a restart strategy for the acceleration to significantly enhance the practical performance. We demonstrate relations with the augmented Lagrangian method and show how to exploit the strongly convex objectives with rigorous convergence rate guarantees. We provide numerical evidence with two examples and illustrate that the new methods can outperform the state-of-the-art, including Chambolle-Pock, and the alternating direction method-of-multipliers algorithms.Comment: 35 pages, accepted for publication on SIAM J. Optimization. Tech. Report, Oct. 2015 (last update Sept. 2016