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The purpose of this paper is to study the strong convergence of an implicit iteration process with errors to a common fixed point for a finite family of asymptotically pseudocontractive mappings and nonexpansive mappings in normed linear spaces. The results in this paper improve and extend the corresponding results of Xu and Ori, Zhou and Chang, Sun, Yang and Yu in some aspects. [ABSTRACT FROM AUTHOR]

The matrix rank minimization problem has applications in many fields, such as system identification, optimal control, low-dimensional embedding, etc. As this problem is NP-hard in general, its convex relaxation, the nuclear norm minimization problem, is often solved instead. Recently, Ma, Goldfarb and Chen proposed a fixed-point continuation algorithm for solving the nuclear norm minimization problem ( Math. Program., doi:, ). By incorporating an approximate singular value decomposition technique in this algorithm, the solution to the matrix rank minimization problem is usually obtained. In this paper, we study the convergence/recoverability properties of the fixed-point continuation algorithm and its variants for matrix rank minimization. Heuristics for determining the rank of the matrix when its true rank is not known are also proposed. Some of these algorithms are closely related to greedy algorithms in compressed sensing. Numerical results for these algorithms for solving affinely constrained matrix rank minimization problems are reported. [ABSTRACT FROM AUTHOR]

A VU-decomposition method for solving a second-order cone problem is presented in this paper. It is first transformed into a nonlinear programming problem. Then, the structure of the Clarke subdifferential corresponding to the penalty function and some results of its VU-decomposition are given. Under a certain condition, a twice continuously differentiable trajectory is computed to produce a second-order expansion of the objective function. A conceptual algorithm for solving this problem with a superlinear convergence rate is given. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in some convergent formulations for the unsymmetric collocation method or the so-called Kansa’s method. We review some newly developed theories on solvability and convergence. The rates of convergence of these variations of Kansa’s method are examined and verified in arbitrary–precision computations. Numerical examples confirm with the theories that the modified Kansa’s method converges faster than the interpolant to the solution; that is, exponential convergence for the multiquadric and Gaussian radial basis functions (RBFs). Some numerical algorithms are proposed for efficiency and accuracy in practical applications of Kansa’s method. In double–precision, even for very large RBF shape parameters, we show that the modified Kansa’s method, through a subspace selection using a greedy algorithm, can produce acceptable approximate solutions. A benchmark algorithm is used to verify the optimality of the selection process. [ABSTRACT FROM AUTHOR]

In this paper, we consider algorithms for constructing n-terms approximations with nonnegative coefficients. The convergence theorem is proved for a “positive” analog of the Pure Greedy Algorithm. We establish a condition on the sequence of weakness coefficients which is sufficient for the convergence of the Positive Weak Greedy Algorithm. This condition is also necessary for the class of monotone sequences. [ABSTRACT FROM AUTHOR]

In the present paper, we investigate the asymptotic behavior of the solution of a variational inequality with one-sided constraints on ɛ-periodically located subsets G ε belonging to the boundary ∂Ω of the domain Ω ⊂ ℝ3. We construct a limit (homogenized) problem and prove the strong (in H 1(Ω)) convergence of the solutions of the original inequality to the solution the limit nonlinear boundary-value problem as ɛ → 0 in the so-called critical case. [ABSTRACT FROM AUTHOR]

In this paper, we consider special classes of strongly convex functionals in Sobolev spaces. It is proved that functionals from such classes have the so-called H-property: weak convergence of sequences of arguments and convergence of such sequences with respect to a given functional imply strong convergence. [ABSTRACT FROM AUTHOR]