The marginal value formula in input optimization

The derivative of the optimal value function is obtained for perturbations over certain regions of stability for bi-convex optimization models. The formula has two terms: the familiar one, related to the derivative of the Lagrangian with respect to the parameters, and a new one, related to the derivative of the optimal solution function. The latter term yields an ‘orthogonality condition’; The derivative of the optimal solution is orthogonal to the polar set of cones of directions of constancy of ‘badly behaved’ constraints. The formula is used in various theoretical and practical situations: First conditions are derived for an optimal input on stable paths. Then an error analysis is developed for two general classes of numerical methods of input optimization. Usefulness of the formula is demonstrated by solving a real-life problem involving the operation of a textile mill. It is shown how the formula can be used in nonlinear programming to calculate structural optima. Also, it is applied to the least-squares problem to derive new error estimates, and finally its role in a duality theory is emphasized. The basic philosophy of this paper is that natural processes are essentially stable (continuous). Therefore a mathematical description of stability should be included in the definition and study of optimality and in the formulations of numerical methods.