On the characterization of 𝑞-superlinear convergence of quasi-Newton methods for constrained optimization

Stoer, J.; Tapia, R. A.

doi:10.1090/s0025-5718-1987-0906190-8

Cited by 1 publication

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“…It is well known that the Dennis and Moré condition [7] is necessary and sufficient for superlinear convergence of a quasi-Newton method for solving nonlinear equations or unconstrained optimization problems. Boggs, Tolle, and Wang [3] extended this result to the quasi-Newton method for solving equality constrained optimization problems (see also [33]). We will extend this result to our algorithm.…”

Section: Superlinear Convergencementioning

confidence: 92%

A Feasible Sequential Linear Equation Method for Inequality Constrained Optimization

Yang¹,

Li²,

Qi³

2003

SIAM J. Optim.

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In this paper, by means of the concept of the working set, which is an estimate of the active set, we propose a feasible sequential linear equation algorithm for solving inequality constrained optimization problems. At each iteration of the proposed algorithm, we first solve one system of linear equations with a coefficient matrix of size m × m (where m is the number of constraints) to compute the working set; we then solve a subproblem which consists of four reduced systems of linear equations with a common coefficient matrix. Unlike existing QP-free algorithms, the subproblem is concerned with only the constraints corresponding to the working set. The constraints not in the working set are neglected. Consequently, the dimension of each subproblem is not of full dimension. Without assuming the isolatedness of the stationary points, we prove that every accumulation point of the sequence generated by the proposed algorithm is a KKT point of the problem. Moreover, after finitely many iterations, the working set becomes independent of the iterates and is essentially the same as the active set of the KKT point. In other words, after finitely many steps, only those constraints which are active at the solution will be involved in the subproblem. Under some additional conditions, we show that the convergence rate is two-step superlinear or even Q-superlinear. We also report some preliminary numerical experiments to show that the proposed algorithm is practicable and effective for the test problems.

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Section: Superlinear Convergencementioning

confidence: 92%

A Feasible Sequential Linear Equation Method for Inequality Constrained Optimization

Yang¹,

Li²,

Qi³

2003

SIAM J. Optim.

View full text Add to dashboard Cite

show abstract

On the characterization of 𝑞-superlinear convergence of quasi-Newton methods for constrained optimization

Abstract: Abstract. In this paper we present a short, straightforward and self-contained derivation of the Boggs-Tolle-Wang characterization of those quasi-Newton methods for equality-constrained optimization which produce iterates which are g-superlinearly convergent.

Cited by 1 publication

References 8 publications

A Feasible Sequential Linear Equation Method for Inequality Constrained Optimization

A Feasible Sequential Linear Equation Method for Inequality Constrained Optimization

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