First Order Methods for LargeScale Sparse Optimization

Title:

First Order Methods for LargeScale Sparse Optimization

Author(s):

Aybat, Necdet Serhat

Date:

2011

Type:

Dissertations

Department:

Industrial Engineering and Operations Research

Permanent URL:

http://hdl.handle.net/10022/AC:P:10735

Notes:

Ph.D., Columbia University.

Abstract:

In today's digital world, improvements in acquisition and storage technology are allowing us to acquire more accurate and finer applicationspecific data, whether it be tickbytick price data from the stock market or framebyframe high resolution images and videos from surveillance systems, remote sensing satellites and biomedical imaging systems. Many important largescale applications can be modeled as optimization problems with millions of decision variables. Very often, the desired solution is sparse in some form, either because the optimal solution is indeed sparse, or because a sparse solution has some desirable properties. Sparse and lowrank solutions to large scale optimization problems are typically obtained by regularizing the objective function with L1 and nuclear norms, respectively. Practical instances of these problems are very high dimensional (~ million variables) and typically have dense and illconditioned data matrices. Therefore, interior point based methods are illsuited for solving these problems. The large scale of these problems forces one to use the socalled firstorder methods that only use gradient information at each iterate. These methods are efficient for problems with a "simple" feasible set such that Euclidean projections onto the set can be computed very efficiently, e.g. the positive orthant, the ndimensional hypercube, the simplex, and the Euclidean ball. When the feasible set is "simple", the subproblems used to compute the iterates can be solved efficiently. Unfortunately, most applications do not have "simple" feasible sets. A commonly used technique to handle general constraints is to relax them so that the resulting problem has only "simple" constraints, and then to solve a single penalty or Lagrangian problem. However, these methods generally do not guarantee convergence to feasibility. The focus of this thesis is on developing new fast firstorder iterative algorithms for computing sparse and lowrank solutions to largescale optimization problems with very mild restrictions on the feasible set  we allow linear equalities, normball and conic inequalities, and also certain nonsmooth convex inequalities to define the constraint set. The proposed algorithms guarantee that the sequence of iterates converges to an optimal feasible solution of the original problem, and each subproblem is an optimization problem with a "simple" feasible set. In addition, for any eps > 0, by relaxing the feasibility requirement of each iteration, the proposed algorithms can compute an epsoptimal and epsfeasible solution within O(log(1/eps)) iterations which requires O(1/eps) basic operations in the worst case. Algorithm parameters do not depend on eps > 0. Thus, these new methods compute iterates arbitrarily close to feasibility and optimality as they continue to run. Moreover, the computational complexity of each basic operation for these new algorithms is the same as that of existing firstorder algorithms running on "simple" feasible sets. Our numerical studies showed that only O(log(1/eps)) basic operations, as opposed to O(1/eps) worst case theoretical bound, are needed for obtaining epsfeasible and epsoptimal solutions. We have implemented these new firstorder methods for the following problem classes: Basis Pursuit (BP) in compressed sensing, Matrix Rank Minimization, Principal Component Pursuit (PCP) and Stable Principal Component Pursuit (SPCP) in principal component analysis. These problems have applications in signal and image processing, video surveillance, face recognition, latent semantic indexing, and ranking and collaborative filtering. To best of our knowledge, an algorithm for the SPCP problem that has O(1/eps) iteration complexity and has a per iteration complexity equal to that of a singular value decomposition is given for the first time.

Subject(s):

Operations research
Applied mathematics
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