Steplength selection in gradient projection methods for box-constrained quadratic programs

The role of the steplength selection strategies in gradient methods has been widely in- vestigated in the last decades. Starting from the work of Barzilai and Borwein (1988), many efficient steplength rules have been designed, that contributed to make the gradient approaches an effective tool for the large-scale optimization problems arising in important real-world applications. Most of these steplength rules have been thought in unconstrained optimization, with the aim of exploiting some second-order information for achieving a fast annihilation of the gradient of the objective function. However, these rules are successfully used also within gradient projection methods for constrained optimization, though, to our knowledge, a detailed analysis of the effects of the constraints on the steplength selections is still not available. In this work we investigate how the presence of the box constraints affects the spectral properties of the Barzilai\u2013Borwein rules in quadratic programming problems. The proposed analysis suggests the introduction of new steplength selection strategies specifically designed for taking account of the active constraints at each iteration. The results of a set of numerical experiments show the effectiveness of the new rules with respect to other state of the art steplength selections and their potential usefulness also in case of box-constrained non-quadratic optimization problems

New bundle methods and U-Lagrangian for generic nonsmooth optimization

Nonsmooth optimization consists of minimizing a continuous function by systematically choosing iterative points from the feasible set via the computation of function values and generalized gradients (called subgradients). Broadly speaking, this thesis contains two research themes: nonsmooth optimization algorithms and theories about the substructure of special nonsmooth functions. Specifically, in terms of algorithms, we develop new bundle methods and bundle trust region methods for generic nonsmooth optimization. For theoretical work, we generalize the notion of U-Lagrangian and investigate its connections with some subsmooth structures. This PhD project develops trust region methods for generic nonsmooth optimization. It assumes the functions are Lipschitz continuous and the optimization problem is not necessarily convex. Currently the project also assumes the objective function is prox-regular but no structural information is given. Trust region methods create a local model of the problem in a neighborhood of the iteration point (called the `Trust Region'). They minimize the model over the Trust Region and consider the minimizer as a trial point for next iteration. If the model is an appropriate approximation of the objective function then the trial point is expected to generate function reduction. The model problem is usually easy to solve. Therefore by comparing the reduction of the model's value and that of the real problem, trust region methods adjust the radius of the trust region to continue to obtain reduction by solving model problems. At the end of this project, it is clear that (1) It is possible to develop a pure bundle method with linear subproblems and without trust region update for convex optimization problems; such method converges to minimizers if it generates an infinite sequence of serious steps; otherwise, it can be shown that the method generates a sequence of minor updates and the last serious step is a minimizer. First, this PhD project develops a bundle trust region algorithm with linear model and linear subproblem for minimizing a prox-regular and Lipschitz function. It adopts a convexification technique from the redistributed bundle method. Global convergence of the algorithm is established in the sense that the sequence of iterations converges to the fixed point of the proximal-point mapping given that convexification is successful. Preliminary numerical tests on standard academic nonsmooth problems show that the algorithm is comparable to bundle methods with quadratic subproblem. Second, following the philosophy behind bundle method of making full use of the previous information of the iteration process and obtaining a flexible understanding of the function structure, the project revises the algorithm developed in the first part by applying the nonmonotone trust region method.We study the performance of numerical implementation and successively refine the algorithm in an effort to improve its practical performance. Such revisions include allowing the convexification parameter to possibly decrease and the algorithm to restart after a finite process determined by various heuristics. The second theme of this project is about the theories of nonsmooth analysis, focusing on U-Lagrangian. When restricted to a subspace, a nonsmooth function can be differentiable within this space. It is known that for a nonsmooth convex function, at a point, the Euclidean space can be decomposed into two subspaces: U, over which a special Lagrangian (called the U-Lagrangian) can be defined and has nice smooth properties and V space, the orthogonal complement subspace of the U space. In this thesis we generalize the definition of UV-decomposition and U-Lagrangian to the context of nonconvex functions, specifically that of a prox-regular function. Similar work in the literature includes a quadratic sub-Lagrangian. It is our interest to study the feasibility of a linear localized U-Lagrangian. We also study the connections of the new U-Lagrangian and other subsmooth structures including fast tracks and partial smooth functions. This part of the project tries to provide answers to the following questions: (1) based on a generalized UV-decomposition, can we develop a linear U-Lagrangian of a prox-regular function that maintains prox-regularity? (2) through the new U-Lagrangian can we show that partial smoothness and fast tracks are equivalent under prox-regularity? At the end of this project, it is clear that for a function f that is properly prox-regular at a point x*, a new linear localized U-Lagrangian can be defined and its value at 0 coincides with f(x*); under some conditions, it can be proved that the U-Lagrangian is also prox-regular at 0; moreover partial smoothness and fast tracks are equivalent under prox-regularity and other mild conditions

Standard Bundle Methods: Untrusted Models and Duality

We review the basic ideas underlying the vast family of algorithms for nonsmooth convex optimization known as "bundle methods|. In a nutshell, these approaches are based on constructing models of the function, but lack of continuity of first-order information implies that these models cannot be trusted, not even close to an optimum. Therefore, many different forms of stabilization have been proposed to try to avoid being led to areas where the model is so inaccurate as to result in almost useless steps. In the development of these methods, duality arguments are useful, if not outright necessary, to better analyze the behaviour of the algorithms. Also, in many relevant applications the function at hand is itself a dual one, so that duality allows to map back algorithmic concepts and results into a "primal space" where they can be exploited; in turn, structure in that space can be exploited to improve the algorithms' behaviour, e.g. by developing better models. We present an updated picture of the many developments around the basic idea along at least three different axes: form of the stabilization, form of the model, and approximate evaluation of the function

A Partially Feasible Distributed SQO Method for Two-block General Linearly Constrained Smooth Optimization

This paper discusses a class of two-block smooth large-scale optimization problems with both linear equality and linear inequality constraints, which have a wide range of applications, such as economic power dispatch, data mining, signal processing, etc.Our goal is to develop a novel partially feasible distributed (PFD) sequential quadratic optimization (SQO) method (PFD-SQO method) for this kind of problems. The design of the method is based on the ideas of SQO method and augmented Lagrangian Jacobian splitting scheme as well as feasible direction method,which decomposes the quadratic optimization (QO) subproblem into two small-scale QOs that can be solved independently and parallelly. A novel disturbance contraction term that can be suitably adjusted is introduced into the inequality constraints so that the feasible step size along the search direction can be increased to 1. The new iteration points are generated by the Armijo line search and the partially augmented Lagrangian function that only contains equality constraints as the merit function. The iteration points always satisfy all the inequality constraints of the problem. The theoretical properties, such as global convergence, iterative complexity, superlinear and quadratic rates of convergence of the proposed PFD-SQO method are analyzed under appropriate assumptions, respectively. Finally, the numerical effectiveness of the method is tested on a class of academic examples and an economic power dispatch problem, which shows that the proposed method is quite promising

Signal eigen-analysis and L1 inversion of seismic data

This thesis covers seismic signal analysis and inversion. It can be divided into two parts. The first part includes principal component analysis (PCA) and singular spectrum analysis (SSA). The objectives of these two eigen-analyses are extracting weak signals and designing optimal spatial sampling interval. The other part is on least squares inverse problems with a L1 norm constraint. The study covers seismic reflectivity inversion in which L1 regularization provides us a sparse solution of reflectivity series, and seismic reverse time migration in which L1 regularization generates high-resolution images. PCA is a well-known eigenvector-based multivariate analysis technique which decomposes a data set into principal components, in order to maximize the information content in the recorded data with fewer dimensions. PCA can be described from two viewpoints, one of which is derived by maximizing the variance of the principal components, and the other draws a connection between the representation of data variance and the representation of data themself by using Singular Value Decomposition (SVD). Each approach has a unique motivation, and thus comparison of these two approaches provides further understanding of the PCA theory. While dominant components contain primary energy of the original seismic data, remaining may be used to reconstruct weak signals, which reflect the geometrical properties of fractures, pores and fluid properties in the reservoirs. When PCA is conducted on time-domain data, Singular Spectrum Analysis (SSA) technology is applied to frequency-domain data, to analyse signal characters related to spatial sampling. For a given frequency, this technique transforms the spatial acquisition data into a Hankel matrix. Ideally, the rank of this matrix is the total number of plane waves within the selected spatial window. However, the existence of noise and absence of seismic traces may increase the rank of Hankel matrix. Thus deflation could be an effective way for noise attenuation and trace exploration. In this thesis, SSA is conducted on seismic data, to find an optimal spatial sampling interval. Seismic reflectivity inversion is a deconvolution process which compresses the seismic wavelet and retrieves the reflectivity series from seismic records. It is a key technique for further inversion, as seismic reflectivity series are required to retrieve impedance and other elastic parameters. Sparseness is an important feature of the reflectivity series. Under the sparseness assumption, the location of a reflectivity indicates the position of an impedance contrast interface, and the amplitude indicates the reflection energy. When using L1 regulation as sparseness constraint, inverse problem becomes nonlinear. Therefore, it is presented as a Basis Pursuit Denosing (BPDN) or Least Absolute Shrinkage and Selection Operator (LASSO) optimal problem and solved by spectral projected gradient (SPG) algorithm. Migration is a key technique to image Earth’s subsurface structures by moving dipping reflections to their true subsurface locations and collapsing diffractions. Reverse time migration (RTM) is a depth migration method which constructs wavefields along the time axis. RTM extrapolates wavefields using a two-way wave equation in the time-space domain, and uses the adjoint operator, instead of the inverse operator, to migrate the record. To improve the signal-to-noise ratio and the resolution of RTM images, RTM may be implemented as a least-squares inverse problem with L1 norm constraint. In this way, the advantages of RTM itself, least-squares RTM, and L1 regularization are utilized to obtain a high-resolution, two-way wave equation-based depth migration image.Open Acces

First-order Convex Optimization Methods for Signal and Image Processing

In this thesis we investigate the use of first-order convex optimization methods applied to problems in signal and image processing. First we make a general introduction to convex optimization, first-order methods and their iteration com-plexity. Then we look at different techniques, which can be used with first-order methods such as smoothing, Lagrange multipliers and proximal gradient meth-ods. We continue by presenting different applications of convex optimization and notable convex formulations with an emphasis on inverse problems and sparse signal processing. We also describe the multiple-description problem. We finally present the contributions of the thesis. The remaining parts of the thesis consist of five research papers. The first paper addresses non-smooth first-order convex optimization and the trade-off between accuracy and smoothness of the approximating smooth function. The second and third papers concern discrete linear inverse problems and reliable numerical reconstruction software. The last two papers present a convex opti-mization formulation of the multiple-description problem and a method to solve it in the case of large-scale instances. i i