Directional shadow price in linearly constrained nonconvex optimization models

The shadow price is, along with its close cousin, the Lagrange multiplier, one of the most important and fundamental concepts in operations research. However, the classical shadow prices have some significant drawbacks in nonconvex optimization models. First, the shadow prices are designed for the single resource, and are not applicable to explain the marginal utility when multiple resources are simultaneously changed. Moreover, in nonconvex optimization models, the Lagrange multipliers only provide the upper and lower bound of the shadow prices, which implies a practical difficulty in computing shadow prices. More importantly, the shadow prices are not continuous with respect to the variation of parameters of the optimization models, unless very strong assumptions are imposed. These drawbacks motivate us to develop a generalization of the shadow price, i.e., the directional shadow price, by using the lower and upper directional derivatives of the optimal value function. We also show that the minimum norm Lagrange multiplier is a kind of directional shadow price, which provides a tool for the computation of the directional shadow price. Moreover, we show that if the optimal solution set is lower semicontinuous with respect to the parameter, then the minimum norm Lagrange multiplier is continuous with respect to the variation of the parameter. Compared with other existing researches, the continuity of the minimum norm Lagrange multiplier holds under the weakest, to the best of our knowledge, conditions. The numerical examples support our results.