An Approximate Lagrange Multiplier Rule

In this paper, we show that for a large class of optimization problems, the Lagrange multiplier rule can be derived from the so-called approximate multiplier rule. In establishing the link between the approximate and the exact multiplier rule we first derive an approximate multiplier rule for a very general class of optimization problems using the approximate sum rule and the chain rule. We also provide a simple proof to the approximate chain rule based on a fundamental result in parametric optimization. In the end we derive a mixed approximate multiplier rule for an equality and inequality constrained optimization problem and outline an approach to use the mixed approximate multiplier rule in studying the computational aspect associated with such a problem.