New results on constraint qualifications for nonlinear extremum problems and extensions
The authors compare different new constraint qualifications for nonlinear optimization problems with equality and inequality constraints according to \[ \min f(x)\quad\text{s.t.}\quad g(x)\leq 0,\;h(x)= 0, \] where all functions \(f: \mathbb{R}^n\to\mathbb{R}\), \(g: \mathbb{R}^n\to\mathbb{R}^p\), \(h:\mathbb{R}^n\to\mathbb{R}^q\) are assumed to be continuously differentiable. The discussed constraint qualifications are modifications of the well-known Mangasarian-Fromovitz constraint qualification: \(*\) the constant rank Mangasarian-Fromovitz constraint qualification (CRMFCQ), \(*\) the relaxed constant positive linear dependence condition (RCPLD), \(*\) the constant rank of the subspace component condition (CRSC), \(*\) the relaxed Mangasarian-Fromovitz constraint qualification (RMFCQ). It is shown that \[ \text{CRMFCQ}\Longrightarrow \text{RCPLD} \Longrightarrow\text{CRSC/RMFCQ} \] (especially CRSC and RMFCQ are equivalent), that CRMFCQ is well-posed (i.e. this condition persists also in some feasible neighborhood) and that all the mentioned conditions imply the existence of local error bounds for the feasible set (and hence they imply the Abady constraint qualification). The results are extended to optimization problems with equilibrium constraints. It is shown that also here the associated RCPLD implies the existence of local error bounds for the mixed complementarity system.
mathematical programs with equilibrium constraints
scientific article; zbMATH DE number 6379728
scientific article