Informative Lagrange Multipliers in the Nonlinear Parametric Programming Model

The shadow price expresses the marginal cost with respect to the variation of constraints, and it is extremely useful in the sensitivity analysis of nonlinear programming models. However, the shadow price may fail to exist in particular parametric programming models, and the informative Lagrange multipliers are proposed to supplement the theory of the shadow price. The traditional analysis of informative Lagrange multipliers is based on the right hand side perturbation model, in which the resource constraints are assumed to be arbitrarily violated, and the variation of resources are measured by the perturbations. In the line of traditional analysis, the minimum norm Lagrange multiplier is proved to be informative since it expresses the rate of cost improvement per unit constraints violation along the steepest descent direction. However, the internal cause of the resource variations is neglected, and the minimum norm Lagrange multiplier may fail to be informative when the perturbations are not only on the right hand side of the constraints. In this paper, we extend the classical constraint violation condition to the generalized constraint violation condition, which captures the characteristic of the problem structure of nonlinear parametric programming models. Based on the generalized constraint violation condition, we provide sufficient conditions for the minimum norm Lagrange multiplier to be informative. Furthermore, we propose a kind of penalty function method to derive the informative LM in fully parametric programming models, which means that the perturbations are not only on the right hand side of the constraints. Finally, we use examples to support our theoretic results.