An algorithm for the minimization of convex,nondifferentiable functionals in Hilbert space

In this paper, an algorithm for minimizing a convex functionalF(x), defined by the maximum of convex smooth functionalsF _i (x), is described. The sequence of valuesF(x _n ) is monotonically decreasing, the algorithm is convergent, and it attains quadratic convergence near the solution.This work was supported by the National Research Council of Italy.     

Lyapunov functionals for certain linear delay differential equations in a Hilbert space

A theory of stability via Lyapunov functionals is developed for a general class of autonomous delay differential equation whose values lie in a Hilbert space.     

First-order methods for certain quasi-variational inequalities in a Hilbert space

Sufficient conditions are obtained for quasi-variational inequalities of a special type with nonlinear operators in a Hilbert space to be uniquely solvable. A first-order continuous method and its discrete variant are constructed for inequalities of this kind. The strong convergence of these methods is proved.     

Rate of convergence bound for approximating sequences in minimization problems for functionals

A general approximation scheme for minimization of functionals in a Banach space is considered. Inequalities are proved which supply bounds on the rate of convergence of the approximate solutions to the exact solution. These bounds are applied to an optimal control problem for an abstract operator equation in a Hilbert space with control in the right-hand side.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 117–121, 1986.     

Regularization methods for certain quasi-variational inequalities with inexactly given data in a Hilbert space

For quasi-variational inequalities of a special type with inexactly given data in a Hilbert space, continuous and discrete operator regularization methods are used to construct approximations that strongly converge to the solution to the original inequality.     

Strong convergence theorem for nonexpansive semigroups in Hilbert space

The purpose of this paper is to prove the strong convergence of a method combining the descent method and the hybrid method in mathematical programming for finding a point in the common fixed point set of a semigroup of nonexpansive mappings in Hilbert space.     

On the variational convergence of non-coercive quadratic integral functionals and semicontinuity problems

In this paper we give some results about convergence of non coercive quadratic integral functionals by examining the behaviour of coefficients. We apply our results to semicontinuity problems and we illustrate them by some examples.AMS Subject Classification: 40A10, 49J45.     

Asymptotic convergence of nonlinear contraction semigroups in Hilbert space

Let S be a contraction semigroup on a closed convex subset C of a Hilbert space. If the generator of S satisfies a strengthened monotonicity condition then the weak lim_{t → ∞}S(t)x exists for all x in C. As one consequence, the method of steepest descent converges weakly for convex functions in Hilbert space; and it converges strongly for even convex functions.     

Some examples of cycling in variable metric methods for constrained minimization

Although variable metric methods for constrained minimization generally give good numerical results, many of their convergence properties are still open. In this note two examples are presented to show that variable metric methods may cycle between two points instead of converging to the required solution.     

On the relation between quadratic termination and convergence properties of minimization algorithms

Summary It is shown that the theory developed in part I of this paper [22] can be applied to some well-known minimization algorithms with the quadratic termination property to prove theirn-step quadratic convergence. In particular, some conjugate gradient methods, the rank-1-methods of Pearson and McCormick (see Pearson [18]) and the large class of rank-2-methods described by Oren and Luenberger [16, 17] are investigated.This work was supported in part at Stanford University, Stanford, California, under Energy Research and Development Administration, Contract E(04-3) 326 PA No. 30, and National Science Foundation Grant DCR 71-01996 A04 and in part by the Deutsche Forschungsgemeinschaft     

Strong convergence of projection-like methods in Hilbert spaces

Many mathematical and applied problems can be reduced to finding a common point of a system of convex sets. The aim of this paper is twofold: first, to present a unified framework for the study of all the projection-like methods, both parallel and serial (chaotic, mostremote set, cyclic order, barycentric, extrapolated, etc.); second, to establish strong convergence results for quite general sets of constraints (generalized Slater, generalized uniformly convex, made of affine varieties, complementary, etc.). This is done by introducing the concept of regular family. We proceed as follows: first, we present definitions, assumptions, theorems, and conclusions; thereafter, we prove them.     

On the convergence of alternating minimization methods in variational PGD

The approximation of solutions to partial differential equations by tensorial separated representations is one of the most efficient numerical treatment of high dimensional problems. The key step of such methods is the computation of an optimal low-rank tensor to enrich the obtained iterative tensorial approximation. In variational problems, this step can be carried out by alternating minimization (AM) technics, but the convergence of such methods presents a real challenge. In the present work, the convergence of rank-one AM algorithms for a class of variational linear elliptic equations is studied. More precisely, we show that rank-one AM-sequences are in general bounded in the ambient Hilbert tensor space and are compact if a uniform non-orthogonality condition between iterates and the reaction term is fulfilled. In particular, if a rank-one AM-sequence is weakly convergent then it converges strongly and the common limit is a solution of the rank-one optimization problem.     

Strong convergence result for solving monotone variational inequalities in Hilbert space

Numerical Algorithms - In this paper, we study strong convergence of the algorithm for solving classical variational inequalities problem with Lipschitz-continuous and monotone mapping in real...