Two New Weak Constraint Qualifications and Applications

We present two new constraint qualifications (CQ) that are weaker than the recently introduced Relaxed Constant Positive Linear Dependence (RCPLD) constraint qualification. RCPLD is based on the assumption that many subsets of the gradients of the active constraints preserve positive linear dependence locally. A major open question was to identify the exact set of gradients whose properties had to be preserved locally and that would still work as a CQ. This is done in the first new constraint qualification, that we call Constant Rank of the Subspace Component (CRSC) CQ. This new CQ also preserves many of the good properties of RCPLD, like local stability and the validity of an error bound. We also introduce an even weaker CQ, called Constant Positive Generator (CPG), that can replace RCPLD in the analysis of the global convergence of algorithms. We close this work extending convergence results of algorithms belonging to all the main classes of nonlinear optimization methods: SQP, augmented Lagrangians, interior point algorithms, and inexact restoration. ∗This work was supported by PRONEX-Optimization (PRONEX-CNPq/FAPERJ E26/171.510/2006-APQ1), Fapesp (Grants 2006/53768-0, 2009/09414-7, and 2010/19720-5), and CNPq (Grants 300900/2009-0, 303030/2007-0, 305740/2010-5, and 474138/2008-9). †Department of Applied Mathematics, Institute of Mathematics, Statistics and Scientific Computing, University of Campinas, Campinas, SP, Brazil. Email: [email protected]. Phone: +55-19-3521 5960. ‡Institute of Science and Technology, Federal University of São Paulo, São José dos Campos, SP, Brazil. Email: [email protected]. Phone: +55-12-3309 9585. §CONICET, Department of Mathematics, FCE, University of La Plata, CP 172, 1900 La Plata Bs. As., Argentina. Email: [email protected]. Phone: +54-221-422 9850. ¶Institute of Mathematics and Statistics, University of São Paulo, São Paulo, SP, Brazil. Email: [email protected]. Phone: +55-11-3091 5178.