  In view of solving nonsmooth and nonconvex problems involving complex
constraints (like standard NLP problems), we study general
maximization-minimization procedures produced by families of strongly convex
sub-problems. Using techniques from semi-algebraic geometry and variational
analysis -in particular Lojasiewicz inequality- we establish the convergence of
sequences generated by this type of schemes to critical points. The broad
applicability of this process is illustrated in the context of NLP. In that
case critical points coincide with KKT points. When the data are semi-algebraic
or real analytic our method applies (for instance) to the study of various SQP
methods: the moving balls method, Sl1QP, ESQP. Under standard qualification
conditions, this provides -to the best of our knowledge- the first general
convergence results for general nonlinear programming problems. We emphasize
the fact that, unlike most works on this subject, no second-order assumption
and/or convexity assumptions whatsoever are made. Rate of convergence are shown
to be of the same form as those commonly encountered with first order methods.
