In this paper, we propose two interior proximal algorithms inspired by the logarithmic-quadratic proximal method. The first method we propose is for general linearly constrained quasiconvex minimization problems. For this method, we prove global convergence when the regularization parameters go to zero. The latter assumption can be dropped when the function is assumed to be pseudoconvex. We also obtain convergence results for quasimonotone variational inequalities, which are more general than monotone ones.
We study two outer approximation schemes, applied to the variational inequality problem in reflexive Banach spaces. First we pro- * Research of this author was supported by CAPES Grant BEX 0664-02/2. † Partially supported by PICDT/UFPI-CAPES ‡ Partially supported by CNPq Grant 301200/93-9(RN) and by PRONEXOptimization.1 pose a generic outer approximation scheme, and its convergence analysis unifies a wide class of outer approximation methods applied to the constrained optimization problem. As is standard in this setting, boundedness and optimality of weak limit points are proved to hold under two alternative conditions: (i) boundedness of the feasible set, or (ii) coerciveness of the operator. In order to develop a convergence analysis where (i) and (ii) do not hold, we consider a second scheme in which the approximated subproblems use a coercive approximation of the original operator. Under conditions alternative to both (i) and (ii), we obtain standard convergence results. Furthermore, when the space is uniformly convex, we establish full strong convergence of the second scheme to a solution.
Abstract.We propose an infeasible interior proximal method for solving variational inequality problems with maximal monotone operators and linear constraints. The interior proximal method proposed by Auslender, Teboulle and Ben-Tiba [3] is a proximal method using a distance-like barrier function and it has a global convergence property under mild assumptions. However, this method is applicable only to problems whose feasible region has nonempty interior. The algorithm we propose is applicable to problems whose feasible region may have empty interior. Moreover, a new kind of inexact scheme is used. We present a full convergence analysis for our algorithm.Mathematical subject classification: 90C51, 65K10, 47J20, 49J40, 49J52, 49J53.
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