Constraint qualifications in terms of convexificators for nonsmooth programming problems with mixed constraints

Abstract

The main aim of the paper is to introduce certain constraint qualifications for a nonsmooth programming problem in terms of semi-regular convexificators and investigate their relations with other existing notions of constraint qualifications. The programming problem under consideration has mixed constraints, that is, it involves both inequality and equality constraints. All these notions are in terms of upper semi-regular convexificators of inequality constraints and pseudo-differentials of equality constraints. Based on a sufficient condition for error bound property, the implication relation between quasinormality and error bound property in terms of convexificators is investigated in this paper. Three conditions are introduced, namely constant positive linear dependence condition (CPLD), constant rank constraint qualification (CRCQ) and Mangasarian–Fromovitz constraint qualification (MFCQ) in terms of convexificators. These conditions are in fact shown to be constraint qualifications as Karush–Kuhn–Tucker optimality conditions hold when CPLD holds and both MFCQ and CRCQ imply CPLD. Further, it is observed that CPLD and quasinormality conditions are independent for nonsmooth problems in terms of convexificators.

Keywords:

• Constraint qualifications
• optimality conditions
• nonsmooth analysis
• convexificators

AMS CLASSIFICATIONS:

• 90C30
• 49J52
• 90C46

Acknowledgments

The authors are grateful to the anonymous referee for the valuable suggestions and comments, which led to improvement of some results in the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research work of the first author is supported by the Council of Scientific and Industrial Research (CSIR), National R&D Organisation [file number 09/045(1560)/2018-EMR-I], India and the second author is supported by the research grant under the Faculty Research Programme of the IoE scheme, University of Delhi [grant number IoE/2021/12/FRP].