Stability in the presence of degeneracy and error estimation

Hager, William W.; Gowda, M. Seetharama

doi:10.1007/s101070050051

Cited by 66 publications

(65 citation statements)

References 12 publications

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“…Hence, according to (12), ξ k = −x k (note that by (A1), the existence of (H (λ)) −1 implies nonsingularity of H k for all k ∈ K large enough). But from (5) we then obtain that…”

Section: The Case Of Complete Degeneracymentioning

confidence: 99%

“…Applying (I n − ) and to both sides of (12), and taking into account (18), (24), (25) and the inclusion a ∈ im B T 1 , we obtain…”

Section: The General Casementioning

confidence: 99%

“…For example, it is known that for degenerate problems of the form of (1) multipliers associated with a local solution exist generically, provided the number of variables is large enough with respect to the number of constraints (see Lemma 1 and subsequent discussion in [4,5]). The case of violation of classical constraint qualifications has been a subject of considerable interest in the past decade, both in the general case (e.g., [1,4,8,11,12,15,16,18,32]) and in the special case of equilibrium or complementarity constraints (e.g., [2,3,9,17,22,[26][27][28]). …”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On attraction of Newton-type iterates to multipliers violating second-order sufficiency conditions

Izmailov

Solodov

2007

Math. Program.

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Assuming that the primal part of the sequence generated by a Newtontype (e.g., SQP) method applied to an equality-constrained problem converges to a solution where the constraints are degenerate, we investigate whether the dual part of the sequence is attracted by those Lagrange multipliers which satisfy second-order sufficient condition (SOSC) for optimality, or by those multipliers which violate it. This question is relevant at least for two reasons: one is speed of convergence of standard methods; the other is applicability of some recently proposed approaches for handling degenerate constraints. We show that for the class of damped Newton methods, convergence of the dual sequence to multipliers satisfying SOSC is unlikely to occur. We support our findings by numerical experiments. We also suggest a simple auxiliary procedure for computing multiplier estimates, which does not have this

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“…Hence, according to (12), ξ k = −x k (note that by (A1), the existence of (H (λ)) −1 implies nonsingularity of H k for all k ∈ K large enough). But from (5) we then obtain that…”

Section: The Case Of Complete Degeneracymentioning

confidence: 99%

“…Applying (I n − ) and to both sides of (12), and taking into account (18), (24), (25) and the inclusion a ∈ im B T 1 , we obtain…”

Section: The General Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On attraction of Newton-type iterates to multipliers violating second-order sufficiency conditions

Izmailov

Solodov

2007

Math. Program.

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show abstract

“…The case of violation of classical constraint qualifications has been a subject of considerable interest in the past decade, both in the general case (e.g., [2,6,11,13,14,20,21,24,25,39]) and in the special case of equilibrium or complementarity constraints (e.g., [3,4,12,22,29,[33][34][35]). …”

Section: F (X) G I (X) (X) = L + |I (X)|mentioning

confidence: 99%

Examples of dual behaviour of Newton-type methods on optimization problems with degenerate constraints

Izmailov

Solodov

2007

Comput Optim Appl

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We discuss possible scenarios of behaviour of the dual part of sequences generated by primal-dual Newton-type methods when applied to optimization problems with nonunique multipliers associated to a solution. Those scenarios are: (a) failure of convergence of the dual sequence; (b) convergence to a so-called critical multiplier (which, in particular, violates some second-order sufficient conditions for optimality), the latter appearing to be a typical scenario when critical multipliers exist; (c) convergence to a noncritical multiplier. The case of mathematical programs with complementarity constraints is also discussed. We illustrate those scenarios with examples, and discuss consequences for the speed of convergence. We also put together a collection of examples of optimization problems with constraints violating some standard constraint qualifications, intended for preliminary testing of existing algorithms on degenerate problems, or for developing special new algorithms designed to deal with constraints degeneracy.

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“…Recall that, according to [23,Lemma 2] and [14, Theorem 2] (see also [31,Section 1.3.3]), the SOSC (4) implies the following error bound:…”

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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Stability in the presence of degeneracy and error estimation

Cited by 66 publications

References 12 publications

On attraction of Newton-type iterates to multipliers violating second-order sufficiency conditions

On attraction of Newton-type iterates to multipliers violating second-order sufficiency conditions

Examples of dual behaviour of Newton-type methods on optimization problems with degenerate constraints

Combining stabilized SQP with the augmented Lagrangian algorithm

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