A new line search inexact restoration approach for nonlinear programming

Fischer, Andreas; Friedlander, Ana

doi:10.1007/s10589-009-9267-0

Cited by 45 publications

(59 citation statements)

References 19 publications

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“…We will use the approach of Fischer and Friedlander [14], which consists of a simplified variation of the method introduced by Martínez [17], with improved global convergence properties.…”

Section: Inexact Restoration Methodsmentioning

confidence: 99%

“…In this paper we develop an Inexact Restoration algorithm [12,13,14,15,16,17,18] for solving (3-4) which, potentially, satisfies the requirements above.…”

mentioning

confidence: 99%

“…This is the basic "local" form of the method, which essentially corresponds to the IR description of Birgin and Martínez [13] that provides local quadratic convergence. Here we consider the modification introduced by Fischer and Friedlander [14] by means of which the method converges to KKT stationary points. This paper is organized as follows:…”

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confidence: 99%

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Inexact restoration method for minimization problems arising in electronic structure calculations

et al. 2010

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An inexact restoration (IR) approach is presented to solve a matricial optimization problem arising in electronic structure calculations. The solution of the problem is the closed-shell density matrix and the constraints are represented by a Grassmann manifold. One of the mathematical and computational challenges in this area is to develop methods for solving the problem not using eigenvalue calculations and having the possibility of preserving sparsity of iterates and gradients. The inexact restoration approach enjoys local quadratic convergence and global convergence to stationary points and does not use spectral matrix decompositions, so that, in principle, large-scale implementations may preserve sparsity. Numerical experiments show that IR algorithms are competitive with current algorithms for solving closed-shell Hartree-Fock equations and similar mathematical problems, thus being a promising alternative for problems where eigenvalue calculations are a limiting factor.

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“…We will use the approach of Fischer and Friedlander [14], which consists of a simplified variation of the method introduced by Martínez [17], with improved global convergence properties.…”

Section: Inexact Restoration Methodsmentioning

confidence: 99%

“…In this paper we develop an Inexact Restoration algorithm [12,13,14,15,16,17,18] for solving (3-4) which, potentially, satisfies the requirements above.…”

mentioning

confidence: 99%

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Inexact restoration method for minimization problems arising in electronic structure calculations

et al. 2010

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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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“…Another direction of thinking is whether it can be useful to somehow avoid dealing with multipliers. Some of the restoration techniques (see Martínez and Pilotta 2000;Fischer and Friedlander 2010) do not use multipliers (with the effect of local slow convergence). Results in Birgin and Martínez (2005) and show that superlinear convergence is possible (by means of multipliers).…”

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confidence: 99%

Comments on: Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it

Izmailov

Solodov

2015

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The paper by Izmailov and Solodov provides an excellent overview of recent research on a phenomenon of slow convergence of optimization algorithms. The phenomenon was unknown about 10 years ago. For me, one of the most striking outcomes of this research is that the phenomenon may cause serious problems to software if applied to certain optimization problems. Although the occurrence of the phenomenon also depends on the point where an algorithm starts from, the sole existence of a critical multiplier frequently forces an algorithm to show this slow convergence. By now, this behavior is not fully understood and remedies are sought. The invited paper is well suited for researchers to become interested in the topic.In the remainder of the discussion, I would like to first make a short excursion to a research problem that started to attract people in the second half of the nineties. Then, I will raise some questions related to both the excursion and the phenomenon of slow convergence.A root of the invited paper's topic goes back to about 20 years. At that time, researchers thought about algorithmically dealing with optimization problems whose set of Lagrange multipliers associated to a stationary point contains more the one element (degenerate case). More in detail, let us consider the optimization problemwhere f : R n → R, g : R n → R m , and R n → R l are sufficiently smooth functions. For the sake of simplicity, Izmailov and Solodov deal with equality constraints only. Here, inequality constraints are involved as well. A pointx ∈ R n is called stationaryThis comment refers to the invited paper available at

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A new line search inexact restoration approach for nonlinear programming

Abstract: Nonlinear programming, Inexact restoration, Line search, Penalty function, Complementarity constraints,

Cited by 45 publications

References 19 publications

Inexact restoration method for minimization problems arising in electronic structure calculations

Inexact restoration method for minimization problems arising in electronic structure calculations

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

Comments on: Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it

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