Stationary conditions for mathematical programs with vanishing constraints using weak constraint qualifications

In this paper, we considered the mathematical programs with vanishing constraints or MPVC. We proved that an MPVC-tailored penalty function, introduced in [5], is still exact under a very weak and new constraint qualification. Most importantly, this constraint qualification is shown to be strictly stronger than MPVC-Abadie constraint qualification.

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Constraint Qualifications and Stationary Conditions for Mathematical Programming with Non-differentiable Vanishing Constraints

Journal of Optimization Theory and Applications ◽

10.1007/s10957-018-1373-7 ◽

2018 ◽

Vol 179 (3) ◽

pp. 800-819 ◽

Cited By ~ 2

Author(s):

Sajjad Kazemi ◽

Nader Kanzi

Keyword(s):

Mathematical Programming ◽

Constraint Qualifications ◽

Vanishing Constraints ◽

Stationary Conditions

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Mathematical programs with vanishing constraints: constraint qualifications, their applications, and a new regularization method

Optimization ◽

10.1080/02331934.2018.1542531 ◽

2018 ◽

Vol 68 (2-3) ◽

pp. 509-538 ◽

Cited By ~ 1

Author(s):

J.-P. Dussault ◽

M. Haddou ◽

T. Migot

Keyword(s):

Regularization Method ◽

Constraint Qualifications ◽

Mathematical Programs ◽

Vanishing Constraints

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Enhanced Fritz John stationarity, new constraint qualifications and local error bound for mathematical programs with vanishing constraints

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2018.11.063 ◽

2019 ◽

Vol 472 (1) ◽

pp. 1042-1077

Author(s):

Abeka Khare ◽

Triloki Nath

Keyword(s):

Error Bound ◽

Constraint Qualifications ◽

Local Error ◽

Local Error Bound ◽

Mathematical Programs ◽

Vanishing Constraints

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Exact penalty results for mathematical programs with vanishing constraints

Nonlinear Analysis ◽

10.1016/j.na.2009.10.047 ◽

2010 ◽

Vol 72 (5) ◽

pp. 2514-2526 ◽

Cited By ~ 30

Author(s):

Tim Hoheisel ◽

Christian Kanzow ◽

Jiří V. Outrata

Keyword(s):

Exact Penalty ◽

Mathematical Programs ◽

Vanishing Constraints

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Relations between Abs-Normal NLPs and MPCCs. Part 2: Weak Constraint Qualifications

Journal of Nonsmooth Analysis and Optimization ◽

10.46298/jnsao-2021-6673 ◽

2021 ◽

Vol Volume 2 (Original research articles>) ◽

Author(s):

Lisa C. Hegerhorst-Schultchen ◽

Christian Kirches ◽

Marc C. Steinbach

Keyword(s):

Normal Form ◽

Optimization Problems ◽

Constraint Qualifications ◽

Inequality Constraints ◽

First Order ◽

Nonlinear Programs ◽

Mathematical Programs ◽

Ongoing Effort ◽

Equality And Inequality Constraints ◽

Smooth Optimization

This work continues an ongoing effort to compare non-smooth optimization problems in abs-normal form to Mathematical Programs with Complementarity Constraints (MPCCs). We study general Nonlinear Programs with equality and inequality constraints in abs-normal form, so-called Abs-Normal NLPs, and their relation to equivalent MPCC reformulations. We introduce the concepts of Abadie's and Guignard's kink qualification and prove relations to MPCC-ACQ and MPCC-GCQ for the counterpart MPCC formulations. Due to non-uniqueness of a specific slack reformulation suggested in [10], the relations are non-trivial. It turns out that constraint qualifications of Abadie type are preserved. We also prove the weaker result that equivalence of Guginard's (and Abadie's) constraint qualifications for all branch problems hold, while the question of GCQ preservation remains open. Finally, we introduce M-stationarity and B-stationarity concepts for abs-normal NLPs and prove first order optimality conditions corresponding to MPCC counterpart formulations.

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