A primal‐dual method for the Meyer model of cartoon and texture decomposition

In this paper, we study the original Meyer model of cartoon and texture decomposition in image processing. The model, which is a minimization problem, contains an l₁‐based TV‐norm and an l_∞‐based G‐norm. The main idea of this paper is to use the dual formulation to represent both TV‐norm and G‐norm. The resulting minimization problem of the Meyer model can be given as a minimax problem. A first‐order primal‐dual algorithm can be developed to compute the saddle point of the minimax problem. The convergence of the proposed algorithm is theoretically shown. Numerical results are presented to show that the original Meyer model can decompose better cartoon and texture components than the other testing methods.     

Transversality condition and optimization of higher order ordinary differential inclusions

In the present paper, a Bolza problem of optimal control theory with a fixed time interval given by convex and nonconvex second-order differential inclusions (P_H) is studied. Our main goal is to derive sufficient optimality conditions for Cauchy problem of sth-order differential inclusions. The sufficient conditions including distinctive transversality condition are proved incorporating the Euler–Lagrange and Hamiltonian type inclusions. The basic concepts involved in obtaining optimality conditions are the locally adjoint mappings. Furthermore, the application of these results is demonstrated by solving the problems with third-order differential inclusions.     

Zeroth order H ∞ norm approximation of multivariable systems

In this paper the H _∞ norm approximation of a given stable, proper, rational transfer function by a constant matrix is considered (Zeroth order H _∞ norm approximation problem). The solution method is based on the observation that the H _∞ norm approximation problem can be put into an allpass imbedding problem.     

Tikhonov regularization of nonlinear III-posed problems in hilbert scales

In this paper we consider nonlinear ill-posed problems F(x) = y ₀, where x and y ₀ are elements of Hilbert spaces X and Y, respectively. We solve these problems by Tikhonov regularization in a Hilbert scale. This means that the regularizing norm is stronger than the norm in X. Smoothness conditions are given that guarantee convergence rates with respect to the data noise in the original norm in X. We also propose a variant of Tikhonov regularization that yields these rates without needing the knowledge of the smoothness conditions. In this variant F is allowed to be known only approximately and X can be approximated by a finite-dimensional subspace. Finally, we illustrate the required conditions for a simple parameter estimation problem for regularization in Sobolev spaces.     

Necessary and sufficient optimality conditions for a class of nonsmooth minimization problems

The purpose of this paper is to derive, in a unified way, second order necessary and sufficient optimality criteria, for four types of nonsmooth minimization problems: thediscrete minimax problem, thediscrete l ₁-approximation, the minimization of theexact penalty function and the minimization of theclassical exterior penalty function. Our results correct and supplement conditions obtained by various authors in recent papers.     

Art networks with geometrical distances

In this paper, ART networks (Fuzzy ART and Fuzzy ARTMAP) with geometrical norms are presented. The category choice of these networks is based on the L_p norm. Geometrical properties of these architectures are presented. Comparisons between this category choice and the category choice of the ART networks are illustrated. And simulation results on the databases taken from the UCI repository are performed. It will be shown that using the L_p norm is geometrically more attractive. It will operate directly on the input patterns without the need for doing any preprocessing. It should be noted that the ART architecture requires two preprocessing steps: normalization and complement coding. Simulation results on different databases show the good generalization performance of the Fuzzy ARTMAP with L_p norm compared to the performance of a typical Fuzzy ARTMAP.     

Optimal treatment planning in radiotherapy based on Boltzmann transport calculations

A Boltzmann transport model for dose calculation in radiation therapy is considered. We formulate an optimal control problem for the desired dose. We prove existence and uniqueness of a minimizer. Based on this model we derive optimality conditions. The P_N discretization in angle of the full model is considered. We show that the P_N approximation of the optimality system is in fact the optimality system of the P_N approximation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On conditions for optimality of a class of nondifferentiable functions

The purpose of this paper is to derive first-order necessary conditions for optimality of a class of nondifferentiable functions. The first-order necessary conditions for optimality for the minimax function and thel ₁-function can be considered as special cases of the present method. Furthermore, the optimality conditions obtained are used to obtain threshold values for the controlling parameters of a class of exact penalty functions.     

Optimal Solutions in the Multi-location Inventory System with Transshipments

We consider a single-period multi-location inventory system where inventory choices at each location are centrally coordinated. Transshipments are allowed as recourse actions in order to reduce the cost of shortage or surplus inventory after demands are realized. This problem has not been solved to optimality before for more than two locations with general cost parameters. In this paper we present a simple and intuitive model that enables us to characterize optimal inventory and transshipment policies for three and four locations as well. The insight gained from these analytical results leads us to examine the optimality conditions of a greedy transshipment policy. We show that this policy will be optimal for two and three locations. For the n location model we characterize the necessary and sufficient conditions on the cost structure for which the greedy transshipment policy will be optimal.      

A Generalization of the Mean Size Formula of Wavelet Packets in Lp

The purpose of this paper is to investigate the mean size formula of wavelet packets in Lp for 0 〈 p ≤ ∞. We generalize a mean size formula of wavelet packets given in terms of the p-norm joint spectral radius and we also give some asymptotic formulas for the Lp-norm or quasi-norm on the subdivision trees. All results will be given in the general setting,     

Second order optimality conditions for theL 1-minimization problem

In this paper we establish necessary and sufficient second order optimality conditions for theL ₁-problem. The approach is based on optimality criteria in terms of a curved second directional derivative, discussed in [3]. Our conditions generalize conditions for theL ₁-problem given in [6]. An example demonstrates the usefulness of our criteria.This research was supported by NSF Grant No. ECS-8214081 and the Fund for Promotion of Research at the Technion, andDeutsche Forschungsgemeinschaft.     

Generalized Higher-Order Optimality Conditions for Set-Valued Optimization under Henig Efficiency

In this paper, generalized mth-order contingent epiderivative and generalized mth-order epiderivative of set-valued maps are introduced, respectively. By virtue of the generalized mth-order epiderivatives, generalized necessary and sufficient optimality conditions are obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a fixed set. Generalized Kuhn–Tucker type necessary and sufficient optimality conditions are also obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a set-valued map.     

Study of a special nonlinear problem arising in convex semi-infinite programming

We consider convex problems of semi-infinite programming (SIP) using an approach based on the implicit optimality criterion. This criterion allows one to replace optimality conditions for a feasible solution x ⁰ of the convex SIP problem by such conditions for x ⁰ in some nonlinear programming (NLP) problem denoted by NLP(I(x ⁰)). This nonlinear problem, constructed on the base of special characteristics of the original SIP problem, so-called immobile indices and their immobility orders, has a special structure and a diversity of important properties. We study these properties and use them to obtain efficient explicit optimality conditions for the problem NLP(I(x ⁰)). Application of these conditions, together with the implicit optimality criterion, gives new efficient optimality conditions for convex SIP problems. Special attention is paid to SIP problems whose constraints do not satisfy the Slater condition and to problems with analytic constraint functions for which we obtain optimality conditions in the form of a criterion. Comparison with some known optimality conditions for convex SIP is provided.     

Approximation and optimization of Darboux type differential inclusions with set-valued boundary conditions

The present paper is devoted to an optimal control problem given by hyperbolic discrete (P _D ) and differential inclusions (P _C ) of generalized Darboux type and ordinary discrete inclusions. The results are extended to non-convex problems. An approach concerning necessary and sufficient conditions for optimality is proposed. In order to formulate sufficient conditions of optimality for problem (P _C ) the approximation method is used. Formulation of these conditions is based on locally adjoint mappings. Moreover for construction of adjoint partial differential inclusions the equivalence theorems of locally adjoint mappings are proved. One example with homogeneous boundary conditions is considered.     

Stability criterion for a family of nonlocal difference schemes

We consider finite-difference schemes for the heat equation with nonlocal boundary conditions that contain a real parameter γ. A stability criterion for finite-difference schemes with respect to the initial data was earlier obtained for |γ| ≤ 1. In the present paper, we consider the case in which γ ∈ (−cosh π,−1) and the original differential problem is stable, while the stability conditions for the finite-difference schemes substantially depend on γ. We obtain estimates for the energy norm of the solution of the finite-difference problem via the same norm of the initial data and prove the equivalence of the energy norm and the grid L ₂-norm.     

Extension of the Weiszfeld procedure to a single facility minisum location model with mixed <Emphasis Type="Italic">ℓ</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> norms

This paper presents a general mixed-norm minisum problem for locating a single facility in continuous space. It is assumed that several transportation modes exist between the new facility and a given set of fixed points (the customers), each mode being represented by a different ? _p norm. A simple extension of Weiszfeld’s well known iterative procedure is proposed to solve the model. Convergence properties and optimality criteria are derived, and computational results are given.     

Sufficient conditions for a control to be a strong minimum

The control literature either presents sufficient conditions for global optimality (for example, the Hamilton-Jacobi-Bellman theorem) or, if concerned with local optimality, restricts attention to comparison controls which are local in theL -sense. In this paper, use is made of an exact expression for the change in cost due to a change in control, a natural extension of a result due to Weierstrass, to obtain sufficient conditions for a control to be a strong minimum (in the sense that comparison controls are merely required to be close in theL ₁-sense).     

On conditions for optimality of the nonlinearl 1 problem

The purpose of this paper is to present necessary and sufficient conditions for optimality in the nonlinearl ₁ problem. Furthermore, the relationship of thel ₁ problem and the Pietrzykowski's approach to solve the nonlinear programming problem is discussed in detail.     

Multigrid methods for the symmetric interior penalty method on graded meshes

The symmetric interior penalty (SIP) method on graded meshes and its fast solution by multigrid methods are studied in this paper. We obtain quasi‐optimal error estimates in both the energy norm and the L₂ norm for the SIP method, and prove uniform convergence of the W‐cycle multigrid algorithm for the resulting discrete problem. The performance of these methods is illustrated by numerical results. Copyright © 2009 John Wiley & Sons, Ltd.