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How Stringent Is the Linear Independence Assumption for Mathematical Programs with Complementarity Constraints?

University of Cambridge, Judge Institute of Management and Department of Engineering, Cambridge CB2 1AG, England
Lufthansa Cargo AG, 60546 Frankfurt, Germany
s.scholtes{at}jims.cam.ac.uk
michael.stoehr{at}dlh.de

The linear independence constraint qualifications (LICQ) plays an important role in the analysis of mathematical programs with complementarity constraints (MPCCs) and is a vital ingredient to convergence analyses of SQP-type or smoothing methods, cf., e.g., Fukushima and Pang (1999), Luo et al. (1996), Scholtes and Stöhr (1999), Scholtes (2001), Stöhr (2000). We will argue in this paper that LICQ is not a particularly stringent assumption for MPCCs. Our arguments are based on an extension of Jongen's (1977) genericity analysis to MPCCs. His definitions of nondegenerate critical points and regular programs extend naturally to MPCCs and his genericity results generalize straightforwardly to MPCCs in standard form. An extension is not as straightforward for MPCCs with the particular structure induced by lower-level stationarity conditions for variational inequalities or optimization problems. We show that LICQ remains a generic property for this class of MPCCs.

Key Words: Constraint qualification; complementarity constraints; critical point
History: Received: April 15, 2000; revision received: January 28, 2001;revision received: May 16, 2001;

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