Summary: We analyze the mixed penalty methods introduced in the classic book of Fiacco and McCormick using two distinct penalty parameters \(r\), \(t\). The two penalty coefficients induce a two-parameter differentiable trajectory. We analyze the numerical behaviour of an extrapolation strategy that follows the path of the two-parameter trajectory. We show also how to remove the ill-conditioning by suitable transformations of the equations. In the resulting theory, we show that function values as well as distances to the optimum are both governed by the same behaviour as interior methods (two-step superlinearly convergent, with limiting exponent \({4\over 3}\)).