 Solution

The sequential quadratic programming step minimizes a quadratic local objective model while enforcing the first-order constraint expansion $c(x_k+s)\approx c_k+A_{k}s=0$. When the iterate and multipliers are close to a regular solution and $B_k$ approximates the Lagrangian Hessian, the resulting correction approximates a Newton step for the nonlinear KKT conditions.

Full row rank of $A_k$ ensures that $A_{k}s=-c_k$ is solvable for every $c_k$ and that the multiplier of the quadratic subproblem is unique. A positive-definite $B_k$ makes the objective strictly convex and coercive on the affine feasible set, giving a unique global subproblem minimizer. In fact positivity on $\ker A_k$, rather than on every vector, is enough for uniqueness and boundedness on that set; full positive definiteness is a convenient stronger assumption.