Superlinear and quadratic convergence of some primal–dual interior point methods for constrained optimization

Superlinear and quadratic convergence of some primal–dual interior point methods for constrained optimization

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Article ID: iaor1998435
Country: Netherlands
Volume: 75
Issue: 3
Start Page Number: 377
End Page Number: 397
Publication Date: Dec 1996
Journal: Mathematical Programming
Authors: ,
Keywords: interior point methods
Abstract:

This paper proves local convergence rates of primal–dual interior point methods for general nonlinearly constrained optimization problems. Conditions to be satisfied at a solution are those given by the usual Jacobian uniqueness conditions. Proofs about convergence rates are given for three kinds of step size rules. They are: (i) the step size rules adopted by Zhang et al. in their convergence analysis of a primal–dual interior point method for linear programs, in which they used single step size for primal and dual variables; (ii) the step size rule used in the software package OB1, which uses different step sizes for primal and dual variables; and (iii) the step size rule used by Yamashita for his globally convergent primal–dual interior point method for general constrained optimization problems, which also uses different step sizes for primal and dual variables. Conditions to the barrier parameter and parameters in step size rules are given for each case. For these step size rules, local and quadratic convergence of the Newton method and local and superlinear convergence of the quasi-Newton method are proved.

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