A relaxed constant positive linear dependence constraint qualification and applications

Andreani, Roberto; Haeser, Gabriel; Schuverdt, María Laura; Silva, Paulo J. S.

doi:10.1007/s10107-011-0456-0

Cited by 134 publications

(152 citation statements)

References 17 publications

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“…If (x 1 ,x 2 ) ∈ R n 1 × R n 2 is an equilibrium, and the constraints of the first problem in (1) with x 2 =x 2 satisfy an appropriate constraint qualification (CQ) [37] atx 1 , while the constraints of the second problem in (1) with x 1 =x 1 satisfy a CQ atx 2 , then there exists (μ 1 ,μ 2 ) ∈ R m × R m such that (x 1 ,x 2 ,μ 1 ,μ 2 ) solves (2). Conversely, if f 1 (·, x 2 ) and the components of g(·, x 2 ) are convex for each x 2 ∈ R n 2 , while f 2 (x 1 , ·) and the components of g(x 1 , ·) are convex for each x 1 ∈ R n 1 , then for every solution (x 1 ,x 2 ,μ 1 ,μ 2 ) of (2) it holds that (x 1 ,x 2 ) is an equilibrium. In particular, under the appropriate assumptions, by estimating the distance to the solution set of (2) one also estimates the distance to the solution set of GNEP (1).…”

Section: Introductionmentioning

confidence: 99%

“…Conversely, if f 1 (·, x 2 ) and the components of g(·, x 2 ) are convex for each x 2 ∈ R n 2 , while f 2 (x 1 , ·) and the components of g(x 1 , ·) are convex for each x 1 ∈ R n 1 , then for every solution (x 1 ,x 2 ,μ 1 ,μ 2 ) of (2) it holds that (x 1 ,x 2 ) is an equilibrium. In particular, under the appropriate assumptions, by estimating the distance to the solution set of (2) one also estimates the distance to the solution set of GNEP (1).…”

Section: Introductionmentioning

confidence: 99%

“…In Sect. 3, we first put in evidence that the weakest general CQ that is currently known to imply an error bound for "generic" (without special structure) systems-the relaxed constant positive linear dependence condition [1]-cannot be expected to hold for the KKT systems associated to GNEP. Thus, a special analysis is needed.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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: Introductionmentioning

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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“…Taking this point of view, it appears very natural to combine sSQP with the usual augmented Lagrangian algorithm. One reason is that Aug-L methods are very robust and have good convergence properties [2,3,7], including when applied to degenerate problems [13,33]. Moreover, it is known that sSQP and Aug-L methods are related: in a sense, the former can be considered as linearization of the iterative subproblems of the latter.…”

Section: Introductionmentioning

confidence: 99%

“…We refer to [3] and [13,25] for state-of-the-art on global and local convergence properties of Aug-L methods, respectively. For many other issues, see [5].…”

Section: Introductionmentioning

confidence: 99%

Combining stabilized SQP with the augmented Lagrangian algorithm

2015

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For an optimization problem with general equality and inequality constraints, we propose an algorithm which uses subproblems of the stabilized SQP (sSQP) type for approximately solving subproblems of the augmented Lagrangian method. The motivation is to take advantage of the well-known robust behavior of the augmented Lagrangian algorithm, including on problems with degenerate constraints, and at the same time try to reduce the overall algorithm locally to sSQP (which gives fast local convergence rate under weak assumptions). Specifically, the algorithm first verifies whether the primal-dual sSQP step (with unit stepsize) makes good progress towards decreasing the violation of optimality conditions for the original problem, and if so, makes this step. Otherwise, the primal part of the sSQP direction is used for linesearch that decreases the augmented Lagrangian, keeping the multiplier estimate fixed for the time being. The overall algorithm has reasonable global convergence guarantees, and inherits strong local convergence rate properties of sSQP under the same weak assumptions. Numerical results on degenerate problems and comparisons with some alternatives are reported.

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On Local Error Bound in Nonlinear Programs

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Sirotko

2021

Optimization and Applications

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A relaxed constant positive linear dependence constraint qualification and applications

Cited by 134 publications

References 17 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

Combining stabilized SQP with the augmented Lagrangian algorithm

On Local Error Bound in Nonlinear Programs

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