Second-Order Necessary Conditions for Nonlinear Optimization Problems with Abstract Constraints: The Degenerate Case

Gfrerer, Helmut

doi:10.1137/050641387

Cited by 9 publications

(4 citation statements)

References 27 publications

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“…By definition of the appearing index sets, we directly see that (5.4) is not stronger than (5.2) while validity of the stationarity condition (5.5) implies validity of (5.3). We note that these necessary optimality conditions are different from those ones from Avakov (1989); Avakov et al (2007); Gfrerer (2007Gfrerer ( , 2014b which hold without further assumptions but comprise a (potentially vanishing) leading multiplier associated with the objective function.…”

Section: Standard Nonlinear Programmingmentioning

confidence: 72%

“…The associated mixedorder necessary optimality conditions and qualification conditions are worked out in Section 4.2. Let us note that necessary optimality conditions for degenerate optimization problems which comprise a mixture of first-and second-order information at the same time can be found in Avakov (1989); Avakov et al (2007); Gfrerer (2007Gfrerer ( , 2014b. All these results comprise a multiplier associated with the objective function which is allowed to be zero.…”

Section: Introductionmentioning

confidence: 99%

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On the directional asymptotic approach in optimization theory Part A: approximate, M-, and mixed-order stationarity

Benko¹,

Mehlitz²

2022

Preprint

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We show that, for a fixed order γ ≥ 1, each local minimizer of a rather general nonsmooth optimization problem in Euclidean spaces is either Mstationary in the classical sense (corresponding to stationarity of order 1), satisfies stationarity conditions in terms of a coderivative construction of order γ, or is approximately stationary with respect to a critical direction as well as γ in a certain sense. By ruling out the latter case with a constraint qualification not stronger than directional metric subregularity, we end up with new necessary optimality conditions comprising a mixture of limiting variational tools of order 1 and γ. These abstract findings are carved out for the broad class of geometric constraints. As a byproduct, we obtain new constraint qualifications ensuring M-stationarity of local minimizers. The paper closes by illustrating these results in the context of standard nonlinear, complementarity-constrained, and nonlinear semidefinite programming.

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Section: Standard Nonlinear Programmingmentioning

confidence: 72%

Section: Introductionmentioning

confidence: 99%

On the directional asymptotic approach in optimization theory Part A: approximate, M-, and mixed-order stationarity

Benko¹,

Mehlitz²

2022

Preprint

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“…In this section, we review some classical and some recent results on second-order optimility conditions. Several second-order optimility conditions have been proposed in the literature, both from a theoretical and practical point of view, see [15,52,49,33,32,17,20,51,22,19,12,14,26,34,18] and references therein.…”

Section: Second-order Optimality Conditionsmentioning

confidence: 99%

On a Conjecture in Second-Order Optimality Conditions

Behling

Haeser

Ramos

et al. 2018

J Optim Theory Appl

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In this paper we deal with optimality conditions that can be verified by a nonlinear optimization algorithm, where only a single Lagrange multiplier is avaliable. In particular, we deal with a conjecture formulated in [R. Andreani, J.M. Martínez, M.L. Schuverdt, "On second-order optimality conditions for nonlinear programming", Optimization, 56:529-542, 2007], which states that whenever a local minimizer of a nonlinear optimization problem fulfills the Mangasarian-Fromovitz Constraint Qualification and the rank of the set of gradients of active constraints increases at most by one in a neighborhood of the minimizer, a second-order optimality condition that depends on one single Lagrange multiplier is satisfied. This conjecture generalizes previous results under a constant rank assumption or under a rank deficiency of at most one. In this paper we prove the conjecture under the additional assumption that the Jacobian matrix has a smooth singular value decomposition, which is weaker than previously considered assumptions. We also review previous literature related to the conjecture.

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“…In more recent years several authors have been concerned with second-order (necessary and/or sufficient) optimality conditions, both from a theoretic and from an algorithmic point of view, producing various interesting papers. See, e. g., Andreani, Behling, Haeser and Silva (2017), Andreani, Birgin, Martinez and Schuverdt (2010), Andreani, Echagüe and Schuverdt (2010), Andreani, Martinez and Schuverdt (2007), Arutyunov (1998), Arutyunov and Pereira (2006), Baccari (2004), Baccari and Trad (2005), Bomze (2015), Bonnans and Shapiro (2000), Brinkhuis (2009), Casas and Troltzsch (2002), Cominetti (1990), Di (1996), Gfrerer (2007), Huy and Tuyen (2016), Kawasaki (1988), Minchenko and Leschov (2016), Penot (1998Penot ( , 2000, Shen, Xue and An (2015).…”

Section: Introductionmentioning

confidence: 99%

Notes on Constraint Qualifications for Second-Order Optimality Conditions

Giorgi¹

2019

JMR

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In the first part of this paper we point out some basic properties of the critical cones used in second-order optimality conditions and give a simple proof of a strong second-order necessary optimality condition by assuming a “modified” first-order Abadie constraint qualification. In the second part we give some insights on second-order constraint qualifications related to second-order local approximations of the feasible set.

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Second-Order Necessary Conditions for Nonlinear Optimization Problems with Abstract Constraints: The Degenerate Case

Cited by 9 publications

References 27 publications

On the directional asymptotic approach in optimization theory Part A: approximate, M-, and mixed-order stationarity

On the directional asymptotic approach in optimization theory Part A: approximate, M-, and mixed-order stationarity

On a Conjecture in Second-Order Optimality Conditions

Notes on Constraint Qualifications for Second-Order Optimality Conditions

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