ConstrainedL p approximation
01 Dec 1985-Constructive Approximation (Springer Science and Business Media LLC)-Vol. 1, Iss: 1, pp 93-102
TL;DR: This paper solves a class of constrained optimization problems that lead to algorithms for the construction of convex interpolants to convex data.
Abstract: In this paper, we solve a class of constrained optimization problems that lead to algorithms for the construction of convex interpolants to convex data.
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TL;DR: The hybrid steepest descent (HST) method as mentioned in this paper is a simple algorithmic solution to the variational inequality problem defined over the nonempty intersection of multiple fixed point sets of nonexpansive mappings in a real Hilbert space.
Abstract: This paper presents a simple algorithmic solution to the variational inequality problem defined over the nonempty intersection of multiple fixed point sets of nonexpansive mappings in a real Hilbert space. The algorithmic solution is named the hybrid steepest descent method, because it is constructed by blending important ideas in the steepest descent method and in the fixed point theory, and generates a sequence converging strongly to the solution of the problem. The remarkable applicability of this method to the convexly constrained generalized pseudoinverse problem as well as to the convex feasibility problem is demonstrated by constructing nonexpansive mappings whose fixed point sets are the feasible sets of the problems.
628 citations
TL;DR: The quadratic convergence of the proposed Newton method for the nearest correlation matrix problem is proved, which confirms the fast convergence and the high efficiency of the method.
Abstract: The nearest correlation matrix problem is to find a correlation matrix which is closest to a given symmetric matrix in the Frobenius norm. The well-studied dual approach is to reformulate this problem as an unconstrained continuously differentiable convex optimization problem. Gradient methods and quasi-Newton methods such as BFGS have been used directly to obtain globally convergent methods. Since the objective function in the dual approach is not twice continuously differentiable, these methods converge at best linearly. In this paper, we investigate a Newton-type method for the nearest correlation matrix problem. Based on recent developments on strongly semismooth matrix valued functions, we prove the quadratic convergence of the proposed Newton method. Numerical experiments confirm the fast convergence and the high efficiency of the method.
288 citations
TL;DR: In this article, the minimization of a convex integral functional over the positive cone of an $L_p $ space, subject to a finite number of linear equality constraints, is considered.
Abstract: This paper considers the minimization of a convex integral functional over the positive cone of an $L_p $ space, subject to a finite number of linear equality constraints. Such problems arise in spectral estimation, where the bjective function is often entropy-like, and in constrained approximation. The Lagrangian dual problem is finite-dimensional and unconstrained. Under a quasi-interior constraint qualification, the primal and dual values are equal, with dual attainment. Examples show the primal value may not be attained. Conditions are given that ensure that the primal optimal solution can be calculated directly from a dual optimum. These conditions are satisfied in many examples.
230 citations
TL;DR: In this paper, the authors considered minimization problems in Hilbert space with quadratic objective function and closed convex constraint set C. The regularizing properties of some gradient projection methods, i.e., convergence for exact data, order of convergence under additional assumptions on the solution and stability for perturbed data, are investigated.
Abstract: Minimization problems in Hilbert space with quadratic objective function and closed convex constraint set C are considered In case the minimum is not unique we are looking for the solution of minimal norm If a problem is ill-posed, ie if the solution does not depend continuously on the data, and if the data are subject to errors then it has to be solved by means of regularization methods The regularizing properties of some gradient projection methods—ie convergence for exact data, order of convergence under additional assumptions on the solution and stability for perturbed data—are the main issues of this paper
176 citations
TL;DR: In this article, a generic wheel-rail contact detection formulation is presented in order to determine online the contact points, even for the most general three-dimensional motion of the wheelset, which allows the study of lead and lag flange contact scenarios.
Abstract: The guidance of railway vehicles is determined by a complex interaction between the wheels and rails, which requires a detailed characterization of the contact mechanism in order to permit a correct analysis of the dynamic behavior. The kinematics of guidance of the wheelsets is based on the wheels and rails geometries. The movement of the wheelsets along the rails is characterized by a complex contact with relative motions on the longitudinal and lateral directions and relative rotations of the wheels with respect to the rails. A generic wheel–rail contact detection formulation is presented here in order to determine online the contact points, even for the most general three-dimensional motion of the wheelset. This formulation also allows the study of lead and lag flange contact scenarios, both fundamental for the analysis of potential derailments or for the study of the dynamic behavior in the presence of switches. The methodology is used in conjunction with a general geometric description of the track,...
153 citations
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TL;DR: In this paper, the authors explain how Favard solved the problem of minimizing √ f (k) √ ∞ under the constraint that ft i ) = f 0 (t i ), i = 1,…, n + k, for some given f 0 and given ( t i ) 1 n+ k, particularly since Favard's paper on the subject is rather sketchy in places.
62 citations
TL;DR: The existence of generalized perfect splines satisfying certain interpolation and/or moment conditions is established in this article, and precise criteria for the uniqueness of such interpolatory perfect spline are indicated.
Abstract: The existence of generalized perfect splines satisfying certain interpolation and/or moment conditions are established. In particular, the existence of ordinary perfect splines obeying boundary and interpolation conditions is demonstrated; precise criteria for the uniqueness of such interpolatory perfect splines are indicated. These are shown to solve a host of variational problems in certain Sobolev spaces.
45 citations
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TL;DR: In this article, the structure and characterization of such splines are studied and a special attention is paid to interpolation defined by linear functionals with support at more than one point.
13 citations