Jacobians in Constrained Nonlinear Optimization

In constrained optimization problems the rates of change of the objective function with respect to the independent (nonbasic) variables have been interpreted as shadow prices, dual variables, and Lagrange multipliers. These derivatives are shown here to be ratios of Jacobians of the transformations between certain sets of variables. This result extends the Kuhn-Tucker conditions of nonlinear programming to optimum seeking problems in which information about the objective and constraint functions can be obtained only by direct measurement, the functions not being given in closed analytic form. Other derivatives helpful for maintaining feasibility during a search are also shown to be ratios of Jacobians. An example shows how these results might be incorporated into algorithms for seeking a constrained optimum.