Local Convergence of an Inexact-Restoration Method and Numerical Experiments

Birgin, Ernesto G.; Martínez, José Mário

doi:10.1007/s10957-005-6537-6

Cited by 39 publications

(58 citation statements)

References 25 publications

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“…Another formulation [17] solves a minimization problem where the objective function is replaced by its Lagrangian function:…”

Section: Inexact Restoration Methodsmentioning

confidence: 99%

“…In the minimization phase we compute a trial point belonging to π k solving a trust-region subproblem such that the functional value at the trial point is less than the functional value at the restored point. A Lagrangian function can be also used at the minimization phase as it is presented in [16,17]. By means of a merit function, the new iterate is accepted or rejected.…”

Section: Introductionmentioning

confidence: 99%

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An Inexact Restoration Package for Bilevel Programming Problems

Pilotta¹,

Torres²

2012

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Bilevel programming problems are a class of optimization problems with hierarchical structure where one of the constraints is also an optimization problem. Inexact restoration methods were introduced for solving nonlinear programming problems a few years ago. They generate a sequence of, generally, infeasible iterates with intermediate iterations that consist of inexactly restored points. In this paper we present a software environment for solving bilevel programming problems using an inexact restoration technique without replacing the lower level problem by its KKT optimality conditions. With this strategy we maintain the minimization structure of the lower level problem and avoid spurious solutions. The environment is a user-friendly set of Fortran 90 modules which is easily and highly configurable. It is prepared to use two well-tested minimization solvers and different formulations in one of the minimization subproblems. We validate our implementation using a set of test problems from the literature, comparing different formulations and the use of the minimization solvers.

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“…Another formulation [17] solves a minimization problem where the objective function is replaced by its Lagrangian function:…”

Section: Inexact Restoration Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An Inexact Restoration Package for Bilevel Programming Problems

Pilotta¹,

Torres²

2012

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show abstract

“…Clearly, this procedure produces severe simplex deformations which impair its chances of practical convergence. However, the idea of using decreasing infeasibility tolerances is quite valuable and has been employed in several modern methods for constrained (not necessarily derivative-free) optimization [7,9,16,23,24].…”

Section: Introductionmentioning

confidence: 99%

Constrained derivative-free optimization on thin domains

Martínez

Sobral

2012

J Glob Optim

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Many derivative-free methods for constrained problems are not efficient for minimizing functions on "thin" domains. Other algorithms, like those based on Augmented Lagrangians, deal with thin constraints using penalty-like strategies. When the constraints are computationally inexpensive but highly nonlinear, these methods spend many potentially expensive objective function evaluations motivated by the difficulties of improving feasibility. An algorithm that handles efficiently this case is proposed in this paper. The main iteration is splitted into two steps: restoration and minimization. In the restoration step the aim is to decrease infeasibility without evaluating the objective function. In the minimization step the objective function f is minimized on a relaxed feasible set. A global minimization result will be proved and computational experiments showing the advantages of this approach will be presented.

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“…This method belongs to the class of two-phase trust-region methods, e.g., Byrd, Schnabel, and Shultz [3], Dennis, El-Alem, and Maciel [6], Gomes, Maciel, and Martínez [13], Gould and Toint [15], Lalee, Nocedal, and Plantenga [17], Omojokun [21], and Powell and Yuan [23]. Also, our method, since it deals with two steps, can be classified in the area of inexact restoration methods proposed by Martínez, e.g., [1,2,8,18,19,20].…”

Section: Introductionmentioning

confidence: 99%

A new double trust regions SQP method without a penalty function or a filter

Zhu¹,

Pu²

2012

Comput. Appl. Math.

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Abstract.A new trust-region SQP method for equality constrained optimization is considered.This method avoids using a penalty function or a filter, and yet can be globally convergent to first-order critical points under some reasonable assumptions. Each SQP step is composed of a normal step and a tangential step for which different trust regions are applied in the spirit of Gould and Toint [Math. Program., 122 (2010), pp. 155-196]. Numerical results demonstrate that this new approach is potentially useful.Mathematical subject classification: 65K05, 90C30, 90C55.

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Local Convergence of an Inexact-Restoration Method and Numerical Experiments

Abstract: Local convergence of an inexact-restoration method for nonlinear programming is proved. Numerical experiments are performed with the objective of evaluating the behavior of the purely local method against a globally convergent nonlinear-programming algorithm.

Cited by 39 publications

References 25 publications

An Inexact Restoration Package for Bilevel Programming Problems

An Inexact Restoration Package for Bilevel Programming Problems

Constrained derivative-free optimization on thin domains

A new double trust regions SQP method without a penalty function or a filter

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