Minimum-norm second-order correction for identified active constraints

Let $c_A:U\to {\mathbb{R}}^r$ collect the equality constraints and a specified set of active inequality constraints, in that order. Given $x,x+p\in U$, assume $A=D{c}_A(x)$ has full row rank and the SQP step satisfies $c_A(x)+Ap=0$. For $r>0$, compute $w=-A^T(A{A}^T{)}^{-1}{c}_A(x+p)$ and return $p+w$ as the corrected displacement. If the rank test fails, the inverse or evaluation fails, return correction-failure. If $r=0$, return $p$ with $w=0$. This is a correction for the specified active constraints; it does not certify feasibility of inactive constraints.

The full-row-rank inverse formula is supplied by If $A$ has full row rank, then $A^{+}=A^{\ast }(A{A}^{\ast }{)}^{-1}$, and the minimum-norm characterization by For every right-hand side $b$, $A^{+}b$ is the unique least-squares solution of minimum Euclidean norm, with right-hand side $-c_A(x+p)$. The zero-row branch returns the zero correction. Residual-order estimates under smoothness and uniform rank hypotheses are separate properties.