A primal-dual trust region algorithm for nonlinear optimization

Gertz, E. Michael; Gill, Philip E.

doi:10.1007/s10107-003-0486-3

Cited by 4 publications

(3 citation statements)

References 61 publications

(80 reference statements)

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“…The approximate Newton equations for problem (NIP) are derived by Gill and Zhang [1]. As is the case for problem (NIPs) the principal work at each iteration is the solution of a reduced (n + m) × (n + m) KKT system analogous to (14). Each KKT matrix was factored using the Matlab built-in command LDL, which uses the routine MA57 [28].…”

Section: Proofmentioning

confidence: 99%

“…In this context, the penalty-barrier function may be regarded as a merit function for forcing convergence of the sequence of Newton iterates of the path-following method. For examples of this approach, see Byrd et al [11], Wächter and Biegler [12], Forsgren and Gill [13], and Gertz and Gill [14].…”

Section: Introductionmentioning

confidence: 99%

“…The matrix (14) has inertia (n, m, 0). If this condition does not hold for H = H (x, y), a common choice of H is the matrix H (x, y) + δ I n for some positive scalar δ (see Sect.…”

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confidence: 99%

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A projected-search interior-point method for nonlinearly constrained optimization

Gill,

Zhang

2024

Comput Optim Appl

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This paper concerns the formulation and analysis of a new interior-point method for constrained optimization that combines a shifted primal-dual interior-point method with a projected-search method for bound-constrained optimization. The method involves the computation of an approximate Newton direction for a primal-dual penalty-barrier function that incorporates shifts on both the primal and dual variables. Shifts on the dual variables allow the method to be safely “warm started” from a good approximate solution and avoids the possibility of very large solutions of the associated path-following equations. The approximate Newton direction is used in conjunction with a new projected-search line-search algorithm that employs a flexible non-monotone quasi-Armijo line search for the minimization of each penalty-barrier function. Numerical results are presented for a large set of constrained optimization problems. For comparison purposes, results are also given for two primal-dual interior-point methods that do not use projection. The first is a method that shifts both the primal and dual variables. The second is a method that involves shifts on the primal variables only. The results show that the use of both primal and dual shifts in conjunction with projection gives a method that is more robust and requires significantly fewer iterations. In particular, the number of times that the search direction must be computed is substantially reduced. Results from a set of quadratic programming test problems indicate that the method is particularly well-suited to solving the quadratic programming subproblem in a sequential quadratic programming method for nonlinear optimization.

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Section: Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A projected-search interior-point method for nonlinearly constrained optimization

Gill,

Zhang

2024

Comput Optim Appl

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Numerical methods for large-scale nonlinear optimization

Gould¹,

Orban²,

Toint³

2005

Acta Numerica

115

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Recent developments in numerical methods for solving large differentiable nonlinear optimization problems are reviewed. State-of-the-art algorithms for solving unconstrained, bound-constrained, linearly-constrained and nonlinearly-constrained problems are discussed. As well as important conceptual advances and theoretical aspects, emphasis is also placed on more practical issues, such as software availability.

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A trust region method using subgradient for minimizing a nondifferentiable function

Gardašević-Filipović

2009

YUGOSLAV J OPERATION

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The minimization of a particular nondifferentiable function is considered. The first and second order necessary conditions are given. A trust region method for minimization of this form of the objective function is presented. The algorithm uses the subgradient instead of the gradient. It is proved that the sequence of points generated by the algorithm has an accumulation point which satisfies the first and second order necessary conditions

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A primal-dual trust region algorithm for nonlinear optimization

Cited by 4 publications

References 61 publications

A projected-search interior-point method for nonlinearly constrained optimization

A projected-search interior-point method for nonlinearly constrained optimization

Numerical methods for large-scale nonlinear optimization

A trust region method using subgradient for minimizing a nondifferentiable function

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