Search

The conjugate gradient (CG) method is an efficient iterative method for solving large-scale strongly convex quadratic programming (QP). In this paper, we propose some generalized CG (GCG) methods for solving the [l.sub.1] -regularized (possibly not strongly) convex QP that terminate at an optimal solution in a finite number of iterations. At each iteration, our methods first identify a face of an orthant and then either perform an exact line search along the direction of the negative projected minimum-norm subgradient of the objective function or execute a CG subroutine that conducts a sequence of CG iterations until a CG iterate crosses the boundary of this face or an approximate minimizer of over this face or a subface is found. We determine which type of step should be taken by comparing the magnitude of some components of the minimum-norm subgradient of the objective function to that of its rest components. Our analysis on finite convergence of these methods makes use of an error bound result and some key properties of the aforementioned exact line search and the CG subroutine. We also show that the proposed methods are capable of finding an approximate solution of the problem by allowing some inexactness on the execution of the CG subroutine. The overall arithmetic operation cost of our GCG methods for finding an [epsilon]-optimal solution depends on [epsilon] in O(log(l/[epsilon])), which is superior to the accelerated proximal gradient method (Beck and Teboulle [Beck A, Teboulle M (2009) A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(l):183-202], Nesterov [Nesterov Yu (2013) Gradient methods for minimizing composite functions. Math. Program. 140(1):125-161]) that depends on [member of] in O(l/[square root of [epsilon]])- In addition, our GCG methods can be extended straightforwardly to solve box-constrained convex QP with finite convergence. Numerical results demonstrate that our methods are very favorable for solving ill-conditioned problems. Funding: The first author's work was supported in part by Natural Sciences and Engineering Research Council of Canada. The second author's work was supported in part by Hong Kong Research Council [Grant PolyU153000/15p]. Keywords: conjugate gradient method * convex quadratic programming * [l.sub.1]-regularization * sparse optimization * finite convergence, 1. Introduction The conjugate gradient (CG) method is an efficient numerical method for solving strongly convex quadratic programming (QP) in the form of [mathematical expression not reproducible] (1) or equivalently, [...]

Given a Markov decision process (MDP) with n states and a total number m of actions, we study the number of iterations needed by policy iteration (PI) algorithms to converge [...]

We consider stochastic optimal control models with Borel spaces and universally measurable policies. For such models the standard policy iteration is known to have difficult measurability issues and cannot be [...]

An investigation was conducted to identify and characterize the relaxation spectrum of polymer melts from transient capillary rheological data. System identification techniques using an iterative prediction-error minimization (PEM) method were applied to isolate the viscoelastic melt response from the apparatus dynamics. Parameters for linear time invariant (LTI) models of varying complexity were estimated using capillary rheology data across a range of temperatures and shear rates using the principle of time-temperature superposition. Melt capillary rheology data for polystyrene was found to exhibit viscoleastic behavior. Subsequently, three viscoelastic constitutive models were then implemented and their model coefficients directly fit using optimization techniques. The implemented methods provided useful relaxation behavior from melt capillary rheology data while also explaining much of the residual error in the purely viscous response as traditionally fit to the Cross-WLF model., INTRODUCTION Fluids are a class of matter that flow under an applied stress. The rate of deformation is dependent on the fluid's viscosity, such that less viscous fluids exhibit greater [...]

In this paper, we model a steady thin film flow of an Oldroyd 8-constant fluid on a vertically moving belt. The governing nonlinear differential equation is first integrated exactly and then solved by applying the variational iteration method and the Adomian decomposition method. The numerical results obtained by these methods are then compared through graphs and tables and no visible difference is observed. This study highlights the significant features of the proposed methods and their ability for solving nonlinear problems arising in non-Newtonian fluid mechanics. Moreover, a reasonable estimation for the belt speed to lift the non-Newtonian fluid is also recorded. This estimation can be used for experimental verification. PACS Nos.: 47.50.-d, 66.20.-d, 47.40.-x, 47.85.-g. Nous modelisons ici l'ecoulement d'un fluide d'Oldroyd a 8 constantes sur une courroie se deplacant verticalement. L'equation differentielle directrice non lineaire est d'abord integree exactement et ensuite solutionnee numeriquement a l'aide de la methode d'iteration variationnelle et de la methode de decomposition d'Adomian. Les valeurs numeriques obtenues de ces methodes sont alors comparees via tables et graphiques et nous n'observons aucune difference. Cette etude souligne les caracteristiques significatives de ces methodes et leur capacite a resoudre des problemes non lineaires rencontres en mecanique des fluides non newtoniens. Nous notons aussi l'estimation raisonnable de la vitesse de courroie necessaire pour elever le fluide. Cette estimation peut etre utilisee pour verification experimentale. [Traduit par la Redaction], 1. Introduction During the past few decades, significant research efforts have been devoted to investigating thin film flows of non-Newtonian fluids in a variety of situations. Specifically, such flows are [...]

We consider the classical finite-state discounted Markovian decision problem, and we introduce a new policy iteration-like algorithm fnr finding the optimal state costs or Q-factors. The main difference is in [...]