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 Abstract

  We consider an application of the ABS procedure to the linear systems arising from the primal-dual interior point methods where Newton method is used to compute path to the solution. When approaching the solution the linear system, which has the form of normal equations of the second kind, becomes more and more ill conditioned. We show how the use of the Huang algorithm in the ABS class can reduce the ill conditioning. Preliminary numerical experiments show that the proposed approach can provide a residual in the computed solution up to sixteen orders lower.

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 Abstract 

  This paper gives a survey of the theory and practice of nonlinear ABS methods including various types of generalizations and computer testing. We also show three applications to special problems, two of which are new.

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We consider a family of damped quasi-Newton methods for solving unconstrained optimization problems. This family resembles that of Broyden with line searches, except that the change in gradients is replaced by a certain hybrid vector before updating the current Hessian approximation. This damped technique modifies the Hessian approximations so that they are maintained sufficiently positive definite. Hence, the objective function is reduced sufficiently on each iteration. The recent result that the damped technique maintains the global and superlinear convergence properties of a restricted class of quasi-Newton methods for convex functions is tested on a set of standard unconstrained optimization problems. The behavior of the methods is studied on the basis of the numerical results required to solve these test problems. It is shown that the damped technique improves the performance of quasi-Newton methods substantially in some robust cases (as the BFGS method) and significantly in certain inefficient cases (as the DFP method)

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Here, a quasi-Newton algorithm for constrained multiobjective optimization is proposed. Under suitable assumptions, global convergence of the algorithm is established.