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On the Weak Stationarity Conditions for Mathematical Programs with Cardinality Constraints: A Unified Approach

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Abstract

In this paper, we study a class of optimization problems, called Mathematical Programs with Cardinality Constraints (MPCaC). This kind of problem is generally difficult to deal with, because it involves a constraint that is not continuous neither convex, but provides sparse solutions. Thereby we reformulate MPCaC in a suitable way, by modeling it as mixed-integer problem and then addressing its continuous counterpart, which will be referred to as relaxed problem. We investigate the relaxed problem by analyzing the classical constraints in two cases: linear and nonlinear. In the linear case, we propose a general approach and present a discussion of the Guignard and Abadie constraint qualifications, proving in this case that every minimizer of the relaxed problem satisfies the Karush–Kuhn–Tucker (KKT) conditions. On the other hand, in the nonlinear case, we show that some standard constraint qualifications may be violated. Therefore, we cannot assert about KKT points. Motivated to find a minimizer for the MPCaC problem, we define new and weaker stationarity conditions, by proposing a unified approach.

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Acknowledgements

We are thankful to the remarks and suggestions of the reviewers, which helped us to improve the presentation of this work.

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  1. Federal University of Paraná, Curitiba, Brazil

    Evelin H. M. Krulikovski, Ademir A. Ribeiro & Mael Sachine

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  1. Evelin H. M. Krulikovski
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  2. Ademir A. Ribeiro
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  3. Mael Sachine
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Correspondence to Ademir A. Ribeiro.

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This work was partially supported by CNPq (Grant 309437/2016-4)

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Krulikovski, E.H.M., Ribeiro, A.A. & Sachine, M. On the Weak Stationarity Conditions for Mathematical Programs with Cardinality Constraints: A Unified Approach. Appl Math Optim 84, 3451–3473 (2021). https://doi.org/10.1007/s00245-021-09752-0

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