Greedy expansions in convex optimization

This paper is a follow-up to the author’s previous paper on convex optimization. In that paper we began the process of adjusting greedy-type algorithms from nonlinear approximation for finding sparse solutions of convex optimization problems. We modified there the three most popular greedy algorithms in nonlinear approximation in Banach spaces-Weak Chebyshev Greedy Algorithm, Weak Greedy Algorithm with Free Relaxation, and Weak Relaxed Greedy Algorithm-for solving convex optimization problems. We continue to study sparse approximate solutions to convex optimization problems. It is known that in many engineering applications researchers are interested in an approximate solution of an optimization problem as a linear combination of elements from a given system of elements. There is an increasing interest in building such sparse approximate solutions using different greedy-type algorithms. In this paper we concentrate on greedy algorithms that provide expansions, which means that the approximant at the mth iteration is equal to the sum of the approximant from the previous, (m ? 1)th, iteration and one element from the dictionary with an appropriate coefficient. The problem of greedy expansions of elements of a Banach space is well studied in nonlinear approximation theory. At first glance the setting of a problem of expansion of a given element and the setting of the problem of expansion in an optimization problem are very different. However, it turns out that the same technique can be used for solving both problems. We show how the technique developed in nonlinear approximation theory, in particular, the greedy expansions technique, can be adjusted for finding a sparse solution of an optimization problem given by an expansion with respect to a given dictionary.     

Truncated Trust Region Methods Based on Preconditioned Iterative Subalgorithms for Large Sparse Systems of Nonlinear Equations

This paper is devoted to globally convergent methods for solving large sparse systems of nonlinear equations with an inexact approximation of the Jacobian matrix. These methods include difference versions of the Newton method and various quasi-Newton methods. We propose a class of trust region methods together with a proof of their global convergence and describe an implementable globally convergent algorithm which can be used as a realization of these methods. Considerable attention is concentrated on the application of conjugate gradient-type iterative methods to the solution of linear subproblems. We prove that both the GMRES and the smoothed COS well-preconditioned methods can be used for the construction of globally convergent trust region methods. The efficiency of our algorithm is demonstrated computationally by using a large collection of sparse test problems.     

Performance comparisons of greedy algorithms in compressed sensing

Compressed sensing has motivated the development of numerous sparse approximation algorithms designed to return a solution to an underdetermined system of linear equations where the solution has the fewest number of nonzeros possible, referred to as the sparsest solution. In the compressed sensing setting, greedy sparse approximation algorithms have been observed to be both able to recover the sparsest solution for similar problem sizes as other algorithms and to be computationally efficient; however, little theory is known for their average case behavior. We conduct a large‐scale empirical investigation into the behavior of three of the state of the art greedy algorithms: Normalized Iterative Hard Thresholding (NIHT), Hard Thresholding Pursuit (HTP), and CSMPSP. The investigation considers a variety of random classes of linear systems. The regions of the problem size in which each algorithm is able to reliably recover the sparsest solution is accurately determined, and throughout this region, additional performance characteristics are presented. Contrasting the recovery regions and the average computational time for each algorithm, we present algorithm selection maps, which indicate, for each problem size, which algorithm is able to reliably recover the sparsest vector in the least amount of time. Although no algorithm is observed to be uniformly superior, NIHT is observed to have an advantageous balance of large recovery region, absolute recovery time, and robustness of these properties to additive noise across a variety of problem classes. A principle difference between the NIHT and the more sophisticated HTP and CSMPSP is the balance of asymptotic convergence rate against computational cost prior to potential support set updates. The data suggest that NIHT is typically faster than HTP and CSMPSP because of greater flexibility in updating the support that limits unnecessary computation on incorrect support sets. The algorithm selection maps presented here are the first of their kind for compressed sensing. Copyright © 2014 John Wiley & Sons, Ltd.     

Tensor methods for large sparse systems of nonlinear equations

This paper introduces tensor methods for solving large sparse systems of nonlinear equations. Tensor methods for nonlinear equations were developed in the context of solving small to medium-sized dense problems. They base each iteration on a quadratic model of the nonlinear equations, where the second-order term is selected so that the model requires no more derivative or function information per iteration than standard linear model-based methods, and hardly more storage or arithmetic operations per iteration. Computational experiments on small to medium-sized problems have shown tensor methods to be considerably more efficient than standard Newton-based methods, with a particularly large advantage on singular problems. This paper considers the extension of this approach to solve large sparse problems. The key issue considered is how to make efficient use of sparsity in forming and solving the tensor model problem at each iteration. Accomplishing this turns out to require an entirely new way of solving the tensor model that successfully exploits the sparsity of the Jacobian, whether the Jacobian is nonsingular or singular. We develop such an approach and, based upon it, an efficient tensor method for solving large sparse systems of nonlinear equations. Test results indicate that this tensor method is significantly more efficient and robust than an efficient sparse Newton-based method, in terms of iterations, function evaluations, and execution time. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Work supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, US Department of Energy, under Contract W-31-109-Eng-38, by the National Aerospace Agency under Purchase Order L25935D, and by the National Science Foundation, through the Center for Research on Parallel Computation, under Cooperative Agreement No. CCR-9120008.Research supported by AFOSR Grants No. AFOSR-90-0109 and F49620-94-1-0101, ARO Grants No. DAAL03-91-G-0151 and DAAH04-94-G-0228, and NSF Grant No. CCR-9101795.     

Analytical solution of a near-equidiffusional flame problem

The near-equidiffusional flame problem for determining the effectof acoustic disturbances having a time scale comparable withthe diffusion time is reduced to a set of nonlinear Volterraequations for a pair of functions related to the burning rateand the flame temperature perturbation. Further analysis yieldsa small-time approximation of general validity and an explicitsolution for arbitrary small-amplitude disturbances. A numericalmethod for solving the Volterra-equation problem is described.Numerical examples are exhibited to compare the behaviour ofthe linear approximation to the solution of the full model.Slowly decaying oscillatory behaviour for flames with Lewisnumber near the right stability boundary is predicted by thelinear approximation and observed in numerical examples forthe nonlinear model.     

A Rayleigh–Ritz preconditioner for the iterative solution to large scale nonlinear problems

The approximation to the solution of large sparse symmetric linear problems arising from nonlinear systems of equations is considered. We are focusing herein on reusing information from previous processes while solving a succession of linear problems with a Conjugate Gradient algorithm. We present a new Rayleigh–Ritz preconditioner that is based on the Krylov subspaces and superconvergence properties, and consists of a suitable reuse of a given set of Ritz vectors. The relevance and the mathematical foundations of the current approach are detailed and the construction of the preconditioner is presented either for the unconstrained or the constrained problems. A corresponding practical preconditioner for iterative domain decomposition methods applied to nonlinear elasticity is addressed, and numerical validation is performed on a poorly-conditioned large-scale practical problem. This revised version was published online in June 2006 with corrections to the Cover Date.     

Mitigating the curse of dimensionality: sparse grid characteristics method for optimal feedback control and HJB equations

We address finding the semi-global solutions to optimal feedback control and the Hamilton–Jacobi–Bellman (HJB) equation. Using the solution of an HJB equation, a feedback optimal control law can be implemented in real-time with minimum computational load. However, except for systems with two or three state variables, using traditional techniques for numerically finding a semi-global solution to an HJB equation for general nonlinear systems is infeasible due to the curse of dimensionality. Here we present a new computational method for finding feedback optimal control and solving HJB equations which is able to mitigate the curse of dimensionality. We do not discretize the HJB equation directly, instead we introduce a sparse grid in the state space and use the Pontryagin’s maximum principle to derive a set of necessary conditions in the form of a boundary value problem, also known as the characteristic equations, for each grid point. Using this approach, the method is spatially causality free, which enjoys the advantage of perfect parallelism on a sparse grid. Compared with dense grids, a sparse grid has a significantly reduced size which is feasible for systems with relatively high dimensions, such as the 6-D system shown in the examples. Once the solution obtained at each grid point, high-order accurate polynomial interpolation is used to approximate the feedback control at arbitrary points. We prove an upper bound for the approximation error and approximate it numerically. This sparse grid characteristics method is demonstrated with three examples of rigid body attitude control using momentum wheels.     

Approximate factoring of the inverse

Computation of approximate factors for the inverse constitutes an algebraic approach to preconditioning large and sparse linear systems. In this paper, the aim is to combine standard preconditioning ideas with sparse approximate inverse approximation, to have dense approximate inverse approximations (implicitly). For optimality, the approximate factoring problem is associated with a minimization problem involving two matrix subspaces. This task can be converted into an eigenvalue problem for a Hermitian positive semidefinite operator whose smallest eigenpairs are of interest. Because of storage and complexity constraints, the power method appears to be the only admissible algorithm for devising sparse–sparse iterations. The subtle issue of choosing the matrix subspaces is addressed. Numerical experiments are presented.     

Conjugate Gradient Method for Rank Deficient Saddle Point Problems

We propose an alternative iterative method to solve rank deficient problems arising in many real applications such as the finite element approximation to the Stokes equation and computational genetics. Our main contribution is to transform the rank deficient problem into a smaller full rank problem, with structure as sparse as possible. The new system improves the condition number greatly. Numerical experiments suggest that the new iterative method works very well for large sparse rank deficient saddle point problems.