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Abstract: In this paper we present a new primal-dual path-following interior-point algorithm for semidefinite optimization. The algorithm is based on a new technique for finding the search direction and the strategy of the central path. At each iteration, we use only full Nesterov–Todd step. Moreover, we obtain the currently best known iteration bound for the algorithm with small-update method, namely, , which is as good as the linear analogue. [Copyright &y& Elsevier]

Abstract: In this paper, we show that minimization problems involving sublinear regularizing terms are ill-posed, in general, although numerical experiments in image processing give very good results. The energies studied here are inspired by image restoration and image decomposition. Rewriting the nonconvex sublinear regularizing terms as weighted total variations, we give a new approach to perform minimization via the well-known Chambolle''s algorithm. The approach developed here provides an alternative to the well-known half-quadratic minimization one. [Copyright &y& Elsevier]

Abstract: A family of algorithms for approximate solution of the bound-constrained minimization problem was introduced in [K.A. Ariyawansa, W.L. Tabor, A class of collinear scaling algorithms for bound-constrained optimization: Derivation and computational results, Technical Report 2003-1, Department of Mathematics, Washington State University, Pullman, WA, 2003, submitted for publication. Available at http://www.math.wsu.edu/math/TRS/2003-1.pdf]. These algorithms employ the standard barrier method, with the inner iteration based on trust region methods. Local models are conic functions rather than the usual quadratic functions, and are required to match first and second derivatives of the barrier function at the current iterate. The various members of the family are distinguished by the choice of a vector-valued parameter, which is the zero vector in the degenerate case that quadratic local models are used. This paper presents a convergence analysis of the family of algorithms presented in [K.A. Ariyawansa, W.L. Tabor, A class of collinear scaling algorithms for bound-constrained optimization: Derivation and computational results, Technical Report 2003-1, Department of Mathematics, Washington State University, Pullman, WA, 2003, submitted for publication. Available at http://www.math.wsu.edu/math/TRS/2003-1.pdf]. Specifically, convergence properties similar to those of barrier methods using quadratic local models are established. [Copyright &y& Elsevier]

Abstract: Proximal point algorithms (PPA) are attractive methods for monotone variational inequalities. The approximate versions of PPA are more applicable in practice. A modified approximate proximal point algorithm (APPA) presented by Solodov and Svaiter [Math. Programming, Ser. B 88 (2000) 371–389] relaxes the inexactness criterion significantly. This paper presents an extended version of Solodov–Svaiter''s APPA. Building the direction from current iterate to the new iterate obtained by Solodov–Svaiter''s APPA, the proposed method improves the profit at each iteration by choosing the optimal step length along this direction. In addition, the inexactness restriction is relaxed further. Numerical example indicates the improvement of the proposed method. [Copyright &y& Elsevier]

In this paper we consider a learning rule whose underlying space, possibly infinite dimensional, is equipped with an inner product. The rule proposed is a generalization of Oja''s maximum eigenfilter algorithm. We study its convergence properties and iterative behavior. We observe a whole variety of dynamical behaviors. We establish conditions on parameter values generating chaoticity as well as asymptotic convergence. [Copyright &y& Elsevier]