Nonmonotone spectral projected gradient methods on convex sets

Nonmonotone projected gradient techniques are considered for the minimization of differentiable functions on closed convex sets. The classical projected gradient schemes are extended to include a nonmonotone steplength strategy that is based on the Grippo-Lampariello-Lucidi nonmonotone line search. In particular, the nonmonotone strategy is combined with the spectral gradient choice of steplength to accelerate the convergence process. In addition to the classical projected gradient nonlinear path, the feasible spectral projected gradient is used as a search direction to avoid additional trial projections during the one-dimensional search process. Convergence properties and extensive numerical results are presented.1041196121

Convergence properties of nonmonotone spectral projected gradient methods

AbstractIn a recent paper, a nonmonotone spectral projected gradient (SPG) method was introduced by Birgin et al. for the minimization of differentiable functions on closed convex sets and extensive presented results showed that this method was very efficient. In this paper, we give a more comprehensive theoretical analysis of the SPG method. In doing so, we remove various boundedness conditions that are assumed in existing results, such as boundedness from below of f, boundedness of xk or existence of accumulation point of {xk}. If ∇f(·) is uniformly continuous, we establish the convergence theory of this method and prove that the SPG method forces the sequence of projected gradients to zero. Moreover, we show under appropriate conditions that the SPG method has some encouraging convergence properties, such as the global convergence of the sequence of iterates generated by this method and the finite termination, etc. Therefore, these results show that the SPG method is attractive in theory

An optimal subgradient algorithm for large-scale convex optimization in simple domains

This paper shows that the optimal subgradient algorithm, OSGA, proposed in \cite{NeuO} can be used for solving structured large-scale convex constrained optimization problems. Only first-order information is required, and the optimal complexity bounds for both smooth and nonsmooth problems are attained. More specifically, we consider two classes of problems: (i) a convex objective with a simple closed convex domain, where the orthogonal projection on this feasible domain is efficiently available; (ii) a convex objective with a simple convex functional constraint. If we equip OSGA with an appropriate prox-function, the OSGA subproblem can be solved either in a closed form or by a simple iterative scheme, which is especially important for large-scale problems. We report numerical results for some applications to show the efficiency of the proposed scheme. A software package implementing OSGA for above domains is available

An Augmented Lagrangian Approach for Sparse Principal Component Analysis

Principal component analysis (PCA) is a widely used technique for data analysis and dimension reduction with numerous applications in science and engineering. However, the standard PCA suffers from the fact that the principal components (PCs) are usually linear combinations of all the original variables, and it is thus often difficult to interpret the PCs. To alleviate this drawback, various sparse PCA approaches were proposed in literature [15, 6, 17, 28, 8, 25, 18, 7, 16]. Despite success in achieving sparsity, some important properties enjoyed by the standard PCA are lost in these methods such as uncorrelation of PCs and orthogonality of loading vectors. Also, the total explained variance that they attempt to maximize can be too optimistic. In this paper we propose a new formulation for sparse PCA, aiming at finding sparse and nearly uncorrelated PCs with orthogonal loading vectors while explaining as much of the total variance as possible. We also develop a novel augmented Lagrangian method for solving a class of nonsmooth constrained optimization problems, which is well suited for our formulation of sparse PCA. We show that it converges to a feasible point, and moreover under some regularity assumptions, it converges to a stationary point. Additionally, we propose two nonmonotone gradient methods for solving the augmented Lagrangian subproblems, and establish their global and local convergence. Finally, we compare our sparse PCA approach with several existing methods on synthetic, random, and real data, respectively. The computational results demonstrate that the sparse PCs produced by our approach substantially outperform those by other methods in terms of total explained variance, correlation of PCs, and orthogonality of loading vectors.Comment: 42 page

Adaptive First-Order Methods for General Sparse Inverse Covariance Selection

In this paper, we consider estimating sparse inverse covariance of a Gaussian graphical model whose conditional independence is assumed to be partially known. Similarly as in [5], we formulate it as an $l_1$-norm penalized maximum likelihood estimation problem. Further, we propose an algorithm framework, and develop two first-order methods, that is, the adaptive spectral projected gradient (ASPG) method and the adaptive Nesterov's smooth (ANS) method, for solving this estimation problem. Finally, we compare the performance of these two methods on a set of randomly generated instances. Our computational results demonstrate that both methods are able to solve problems of size at least a thousand and number of constraints of nearly a half million within a reasonable amount of time, and the ASPG method generally outperforms the ANS method.Comment: 19 pages, 1 figur