On relaxed constant rank regularity condition in mathematical programming

Minchenko, L. I.; Stakhovski, Sergey

doi:10.1080/02331930902971377

Cited by 82 publications

(108 citation statements)

References 12 publications

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“…Combining the properties verified above with [9, Theorem 1], we obtain local quadratic convergence of the LP-Newton method described by (31) under the assumptions of our Theorem 1, combined with the strict complementarity condition.…”

Section: Newton-type Methodsmentioning

confidence: 83%

“…It should be noted that writing the optimality conditions for (31), it can be observed that it is equivalent to an iteration of the Levenberg-Marquardt method for (29), with a special choice of the regularization parameter.…”

Section: Newton-type Methodsmentioning

confidence: 99%

“…Then for any starting point u 0 close enough toū = (x 1 ,x 2 ,μ 1 ,μ 2 ) the iterative scheme (31) for Φ given by (20) defines a sequence {u k } which converges quadratically to some u * ∈Ū , whereŪ is the solution set of system (2).…”

Section: Theoremmentioning

confidence: 99%

“…It is easy to see that MFCQ implies CPLD and, hence, RCPLD. Also, CPLD (and thus RCPLD) is implied by the well-known constant rank CQ (CRCQ) [22], and even by the relaxed constant rank CQ (RCRCQ) [31]. Note that RCPLD guarantees [1] that under twice differentiability of a and b near u, the following error bound holds:…”

Section: Introductionmentioning

confidence: 99%

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On error bounds and Newton-type methods for generalized Nash equilibrium problems

Izmailov

Solodov

2013

Comput Optim Appl

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Error bounds (estimates for the distance to the solution set of a given problem) are key to analyzing convergence rates of computational methods for solving the problem in question, or sometimes even to justifying convergence itself. That said, for the generalized Nash equilibrium problems (GNEP), the theory of error bounds had not been developed in depth comparable to the fields of optimization and variational problems. In this paper, we provide a systematic approach which should be useful for verifying error bounds for both specific instances of GNEPs and for classes of GNEPs. These error bounds for GNEPs are based on more general results for constraints that involve complementarity relations and cover those (few) GNEP error bounds that existed previously, and go beyond. In addition, they readily imply a Lipschitzian stability result for solutions of GNEPs, a subject where again very little had been known. As a specific application of error bounds, we discuss Newtonian methods for solving GNEPs. While we do not propose any significantly new methods in this respect, some new insights into applicability to GNEPs of various approaches and into their convergence properties are presented.

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Section: Newton-type Methodsmentioning

confidence: 83%

Section: Newton-type Methodsmentioning

confidence: 99%

Section: Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On error bounds and Newton-type methods for generalized Nash equilibrium problems

Izmailov

Solodov

2013

Comput Optim Appl

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“…It is well known that LICQ implies both MFCQ and RCRCQ. However, MFCQ neither is implied by nor implies RCRCQ [27]. Finally, we say that the local error bound condition for system (1.7) is satisfied atz if there exist ε > 0 and ω > 0 such that…”

mentioning

confidence: 99%

Convergence conditions for Newton-type methods applied to complementarity systems with nonisolated solutions

et al. 2015

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Abstract. We consider a class of Newton-type methods for constrained systems of equations that involve complementarity conditions. In particular, at issue are the constrained Levenberg-Marquardt method and the recently introduced Linear-Programming-Newton method, designed for the difficult case when solutions need not be isolated, and the equation mapping need not be differentiable at the solutions. We show that the only structural assumption needed for rapid local convergence of those algorithms is the piecewise error bound, i.e., a local error bound holding for the branches of the solution set resulting from partitions of the bi-active complementarity indices. The latter error bound is implied by various piecewise constraint qualifications, including some relatively weak ones. As further applications of our results, we consider Karush-Kuhn-Tucker systems arising from optimization or variational problems, and from generalized Nash equilibrium problems. In the first case, we show convergence under the assumption that the dual part of the solution is a noncritical Lagrange multiplier, and in the second case convergence follows under a certain relaxed constant rank condition. In both cases, this improves on results previously available for the corresponding problems.

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Constraint Qualifications

Solodov

2011

Wiley Encyclopedia of Operations Research and Management Science

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We discuss assumptions on the constraint functions that allow constructive description of the geometry of a given set around a given point in terms of the constraints derivatives. Consequences for characterizing solutions of variational and optimization problems are discussed. In the optimization case, these include primal and primal‐dual first‐ and second‐order necessary optimality conditions.

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On relaxed constant rank regularity condition in mathematical programming

Cited by 82 publications

References 12 publications

On error bounds and Newton-type methods for generalized Nash equilibrium problems

On error bounds and Newton-type methods for generalized Nash equilibrium problems

Convergence conditions for Newton-type methods applied to complementarity systems with nonisolated solutions

Constraint Qualifications

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