An interior point method for nonlinear programming with infeasibility detection capabilities

Nocedal, Jorge; Oztoprak, Figen; Waltz, Richard A.

doi:10.1080/10556788.2013.858156

Cited by 50 publications

(41 citation statements)

References 24 publications

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“…One possibility is to introduce slack variables to inequality constraints and apply augmented Lagrangian method in the case of simple bounds as in [5]. On the other hand, we note that infeasibility detection has been used in the framework of interior point methods [24,25]. However, these works did not report complete global and local convergence analyses of their methods, despite good numerical results.…”

Section: Discussionmentioning

confidence: 97%

An Augmented Lagrangian Method for Equality Constrained Optimization with Rapid Infeasibility Detection Capabilities

Armand

Tran

2018

J Optim Theory Appl

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We present a primal-dual augmented Lagrangian method for solving an equality constrained minimization problem, which is able to rapidly detect infeasibility. The method is based on a modification of the algorithm proposed in [1]. A new parameter is introduced to scale the objective function and, in case of infeasibility, to force the convergence of the iterates to an infeasible stationary point. It is shown, under mild assumptions, that whenever the algorithm converges to an infeasible stationary point, the rate of convergence is quadratic. This is a new convergence result for the class of augmented Lagrangian methods. The global convergence of the algorithm is also analysed. It is also proved that, when the algorithm converges to a stationary point, the properties of the original algorithm [1] are preserved. The numerical experiments show that our new approach is as good as the original one when the algorithm converges to a local minimum, but much more efficient in case of infeasibility.

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Section: Discussionmentioning

confidence: 97%

An Augmented Lagrangian Method for Equality Constrained Optimization with Rapid Infeasibility Detection Capabilities

Armand

Tran

2018

J Optim Theory Appl

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“…The interior-point approach has been shown to be robust and efficient in solving linear and nonlinear programs (for example, see [2,3,9,10,13], [14]- [22] and [24,26,27,30,31,32,34,35]. Among all interior-point methods, the primal-dual interior-point methods have drawn considerable attention.…”

Section: Introductionmentioning

confidence: 99%

“…These methods can roughly and mainly be summarized into three kinds by the order of using the penalty technique. The first kind of methods firstly reformulate the original program to a problem with only equality constraints by interior barrier technique and then prompt the global convergence of these methods by different penalty functions, such as [9,10,13], [18]- [21], [24,26,27,30]. The second kind of methods first use the penalty strategy to obtain a new formulation of the original program with only inequality constraints and then use the interiorpoint methods to solve the formulation, such as [22].…”

Section: Introductionmentioning

confidence: 99%

A primal-dual interior-point method capable of rapidly detecting infeasibility for nonlinear programs

Dai¹,

Li²,

Sun³

2020

Journal of Industrial &Amp; Management Optimization

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With the help of a logarithmic barrier augmented Lagrangian function, we can obtain closedform solutions of slack variables of logarithmic-barrier problems of nonlinear programs. As a result, a two-parameter primal-dual nonlinear system is proposed, which corresponds to the Karush-Kuhn-Tucker point and the infeasible stationary point of nonlinear programs, respectively, as one of two parameters vanishes. Based on this distinctive system, we present a primal-dual interior-point method capable of rapidly detecting infeasibility of nonlinear programs. The method generates interior-point iterates without truncation of the step. It is proved that our method converges to a Karush-Kuhn-Tucker point of the original problem as the barrier parameter tends to zero. Otherwise, the scaling parameter tends to zero, and the method converges to either an infeasible stationary point or a singular stationary point of the original problem. Moreover, our method has the capability to rapidly detect the infeasibility of the problem. Under suitable conditions, not only the method can be superlinearly or quadratically convergent to the Karush-Kuhn-Tucker point as the original problem is feasible, but also it can be superlinearly or quadratically convergent to the infeasible stationary point when a problem is infeasible. Preliminary numerical results show that the method is efficient in solving some simple but hard problems and some standard test problems from the CUTE collection, where the superlinear convergence is demonstrated when we solve two infeasible problems and one well-posed feasible counterexample presented in the literature.Key words: nonlinear programming, constrained optimization, infeasibility, interior-point method, global and local convergence AMS subject classifications. 90C26, 90C30, 90C51 * This work was firstly submitted to SIOPT (#099809) on December 2, 2014, the current version was a submitted version (MOR-

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“…Of all the methods of this kind, the step-decomposition approaches, which splits the total step into a normal step and a tangential step, and which integrates ideas of interior point methods and Byrd-Omojokun's trust region idea [3,19], have been proven practical. Two nice features of this strategy is the consistent subproblems and the capacity of infeasibility detection [18].…”

Section: Introductionmentioning

confidence: 99%

“…where Lemma 3.1, (12), (18) and (25) are used. Since α k ≥ 0.5α h k , where α h k given by (39), the previous inequalities give…”

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confidence: 99%

An interior point method for nonlinear optimization with a quasi-tangential subproblem

Qiu

Chen

2018

Journal of Computational and Applied Mathematics

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In this paper, we proposed an interior point method for constrained optimization, which is characterized by the using of quasi-tangential subproblem. This algorithm follows the main ideas of primal dual interior point methods and Byrd-Omojokun's step decomposition strategy. The quasi-tangential subproblem is obtained by penalizing the null space constraint in the tangential subproblem. The resulted quasi-tangential step is not strictly lying in the null space of the gradients of constraints. We also use a line search trust-funnel-like strategy, instead of penalty function or filter technology, to globalize the method. Global convergence results were obtained under standard assumptions.

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An interior point method for nonlinear programming with infeasibility detection capabilities

Cited by 50 publications

References 24 publications

An Augmented Lagrangian Method for Equality Constrained Optimization with Rapid Infeasibility Detection Capabilities

An Augmented Lagrangian Method for Equality Constrained Optimization with Rapid Infeasibility Detection Capabilities

A primal-dual interior-point method capable of rapidly detecting infeasibility for nonlinear programs

An interior point method for nonlinear optimization with a quasi-tangential subproblem

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