There is a wide consensus that the shadow prices of certain resources in an economic system are equal to Lagrange multipliers. However, this is misleading with respect to multiple Lagrange multipliers. In this paper, we propose a new type of Lagrange multiplier, the weighted minimum norm Lagrange multiplier, which is a type of shadow price. An attractive aspect of this type of Lagrange multiplier is that it conveys the sensitivity information when resources are required to be proportionally input. To compute the weighted minimum norm Lagrange multiplier, we propose two algorithms. One is the penalty function method with numeric stability, and the other is the accelerated gradient method with fewer arithmetic operations and a convergence rate of $ O(\frac{1}{k^2}) $. Furthermore, we propose a two-phase procedure to compute a particular subset of shadow prices that belongs to the set of bounded Lagrange multipliers. This subset is particularly attractive since all its elements are computable shadow prices. We report the numerical results for randomly generated problems.

中文翻译：

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用多个拉格朗日乘数计算影子价格

人们普遍认为，经济系统中某些资源的影子价格等于拉格朗日乘数。然而，这对于多个拉格朗日乘数是误导性的。在本文中，我们提出了一种新型的拉格朗日乘数，即加权最小范数拉格朗日乘数，它是一种影子价格。这种类型的拉格朗日乘数的一个吸引人的方面是它在需要按比例输入资源时传达敏感性信息。为了计算加权最小范数拉格朗日乘数，我们提出了两种算法。一种是数值稳定的惩罚函数法，另一种是算术运算较少、收敛速度为$O(\frac{1}{k^2})$的加速梯度法。此外，我们提出了一个两阶段程序来计算属于有界拉格朗日乘数集的特定影子价格子集。这个子集特别有吸引力，因为它的所有元素都是可计算的影子价格。我们报告随机生成问题的数值结果。