An application of matrix computations to classical second-order optimality conditions

For finite dimensional optimization problems with equality and inequality constraints, a weak constant rank condition (WCR) was introduced by Andreani–Martinez–Schuverdt (AMS) (Optimization 5–6:529–542, 2007) to study classical necessary second-order optimality conditions. However, this condition is not easy to check. Using a polynomial and matrix computation tools, we can substitute it by a weak constant rank condition (WCRQ) for an approximated problem and we present a method for checking points that satisfy WCRQ. We extend the result of (Andreani et al. in Optimization 5–6:529–542, 2007), we show that WCR can be replaced by WCRQ and we prove that these two conditions are independent.