In this note we give a new, simple proof of the standard first and second order necessary conditions, under the Mangasarian–Fromovitz constraint qualification (MFCQ), for non-linear programming problems. We work under a mild constraint qualification, which is implied by MFCQ. This makes it possible to reduce the proof to the relatively easy case of inequality constraints only under MFCQ. This reduction makes use of relaxation of inequality constraints and it makes use of a penalty function. The new proof is based on the duality theorem for linear programming; the proofs in the literature are based on results of mathematical analysis. This paper completes the work in a recent note of Birbil et al. where a linear programming proof of the first order necessary conditions has been given, using relaxation of equality constraints.

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Brinkhuis, J. (2009). A linear programming proof of the second order conditions of nonlinear programming. European Journal of Operational Research, 192(3), 1001–1007. doi:10.1016/j.ejor.2007.10.062