On the convergence of the exponential multiplier method for convex programming

Cover title.Includes bibliographical references (p. 18-20).Research supported by the Army Research Office. DAAL03-86-K-0171 Research supported by the National Science Foundation. DDM-8903385by Paul Tseng and Dimitri P. Bertsekas

Existence of Local Saddle Points for a New Augmented Lagrangian Function

We give a new class of augmented Lagrangian functions for nonlinear programming problem with both equality and inequality constraints. The close relationship between local saddle points of this new augmented Lagrangian and local optimal solutions is discussed. In particular, we show that a local saddle point is a local optimal solution and the converse is also true under rather mild conditions

On the finite termination of an entropy function based smoothing Newton method for vertical linear complementarity problems

By using a smooth entropy function to approximate the non-smooth max-type function, a vertical linear complementarity problem (VLCP) can be treated as a family of parameterized smooth equations. A Newton-type method with a testing procedure is proposed to solve such a system. We show that the proposed algorithm finds an exact solution of VLCP in a finite number of iterations, under some conditions milder than those assumed in literature. Some computational results are included to illustrate the potential of this approach.Newton method;Finite termination;Entropy function;Smoothing approximation;Vertical linear complementarity problems

On the finite termination of an entropy function based smoothing Newton method for vertical linear complementarity problems

By using a smooth entropy function to approximate the non-smooth max-type function, a vertical linear complementarity problem (VLCP) can be treated as a family of parameterized smooth equations. A Newton-type method with a testing procedure is proposed to solve such a system. We show that the proposed algorithm finds an exact solution of VLCP in a finite number of iterations, under some conditions milder than those assumed in literature. Some computational results are included to illustrate the potential of this approach

Efficient Variational Bayesian Approximation Method Based on Subspace optimization

International audienceVariational Bayesian approximations have been widely used in fully Bayesian inference for approx- imating an intractable posterior distribution by a separable one. Nevertheless, the classical variational Bayesian approximation (VBA) method suffers from slow convergence to the approximate solution when tackling large-dimensional problems. To address this problem, we propose in this paper an improved VBA method. Actually, variational Bayesian issue can be seen as a convex functional optimization problem. The proposed method is based on the adaptation of subspace optimization methods in Hilbert spaces to the function space involved, in order to solve this optimization problem in an iterative way. The aim is to determine an optimal direction at each iteration in order to get a more efficient method. We highlight the efficiency of our new VBA method and its application to image processing by considering an ill-posed linear inverse problem using a total variation prior. Comparisons with state of the art variational Bayesian methods through a numerical example show the notable improved computation time

On the Finite Termination of An Entropy Function Based Smoothing Newton Method for Vertical Linear Complementarity Problems

By using a smooth entropy function to approximate the non-smooth max-type function, a vertical linear complementarity problem (VLCP) can be treated as a family of parameterized smooth equations. A Newton-type method with a testing procedure is proposed to solve such a system. We show that the proposed algorithm finds an exact solution of VLCP in a finite number of iterations, under some conditions milder than those assumed in literature. Some computational results are included to illustrate the potential of this approach