Extremum conditions for smooth problems with equality-type constraints

Avakov, E. R.

doi:10.1016/0041-5553(85)90069-2

“…Another important concept of second-order regularity was developed in [30,31] for the case when there are no inequality constraints, under the name of 2-regularity. In somewhat different terms, we say that this condition holds atx ∈ D in a direction d ∈ R n if im h (x) + h (x)[d, ker h (x)] = R l .…”

Section: Error Bounds and Metric Regularitymentioning

confidence: 99%

Constraint Qualifications

Solodov

¹

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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“…Another important concept of second-order regularity was developed in [30,31] for the case when there are no inequality constraints, under the name of 2-regularity. In somewhat different terms, we say that this condition holds atx…”

Section: Error Bounds and Metric Regularitymentioning

confidence: 99%

Constraint Qualifications

2005

Nonlinear Programming

0

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“…If k * is a regular zero of the Hessian H A , then π(Q(k * )) = 0, so Q(k * ) belongs to LX, and then there exists ξ * ∈ X such that 1 2…”

Section: Therefore (20) Implies If We Writementioning

confidence: 99%

“…[1,2]). Avakov's sufficient condition for openness-called "2-regularity" by some authors, e.g., Ledzewicz and Schättler, cf.…”

Section: Therefore (20) Implies If We Writementioning

confidence: 99%

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High-Order Open Mapping Theorems

Sussmann

¹

Directions in Mathematical Systems Theory and Optimization

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The well-known finite-dimensional first-order open mapping theorem says that a continuous map with a finite-dimensional target is open at a point if its differential at that point exists and is surjective. An identical result, due to Graves, is true when the target is infinite-dimensional, if "differentiability" is replaced by "strict differentiability." We prove general theorems in which the linear approximations involved in the usual concept of differentiability is replaced by an approximation by a map which is homogeneous relative to a one-parameter group of dilations, and the error bound in the approximation involves a "homogeneous pseudonorm" or a "homogeneous pseudodistance," rather than the ordinary norm. We outline how these results can be used to derive sufficient conditions for openness involving higher-order derivatives, and carry this out in detail for the second-order case.

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Extremum conditions for smooth problems with equality-type constraints

Cited by 67 publications

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Constraint Qualifications

Constraint Qualifications

Constraint Qualifications

High-Order Open Mapping Theorems

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