A generalization of the steepest descent and other methods for solving a large scale symmetric positive definitive system Ax = b is presented. Given a positive integer m, the new iteration is given by x k+1 = x k − λ(x ν(k))(Ax k − b), where λ(x ν(k)) is the steepest descent step at a previous iteration ν(k) ∈ {k, k − 1,. .. , max{0, k − m}}. The global convergence to the solution of the problem is established under a more general framework, and numerical experiments are performed that suggest that some strategies for the choice of ν(k) give rise to efficient methods for obtaining approximate solutions of the system.

No abstract

In this paper we discuss a specialization of the augmented Lagrangian-type algorithm of Conn, Gould, and Toint to the solution of strictly convex quadratic programming problems with simple bounds and equality constraints. The new feature of the presented algorithm is the adaptive precision control of the solution of auxiliary problems in the inner loop of the basic algorithm which yields a rate of convergence that does not have any term that accounts for inexact solution of auxiliary problems. Moreover, boundedness of the penalty parameter is achieved for the precision control used. Numerical experiments illustrate the efficiency of the presented algorithm and encourage its usage.

Nonlinear programming, Inexact restoration, Line search, Penalty function, Complementarity constraints,

This papeT investigates a kind of resource allocation problem which maximizes a strictly concave objective function with double layers of constraints oll the total amount ef resources, Resources are distributed on a two-dimensional space, say, a geographical space with time flow, and are doubly constrained in the sense that the total amount is limited on the whole space and the subtotal amount is constrained at each time too. We derive necessary and suMcient conditions for an eptimal $olution and propose two methods of solving it. Both rnethods manipulate Lagrange multipliers amd make a sequence of feasible solntions that ultimately satisfy necessary and suficient conditions for optimality. It is shown by numerical complltation that the proposed methods are faster than other well-known methods.