Computing shadow prices with multiple Lagrange multipliers

Slopes of shadow prices and Lagrange multipliers

Manyeconomicmodels andoptimizationproblemsgenerate (endogenous) shadow prices—alias dual variables or Lagrange multipliers. Frequently the “slopes” of resulting price curves—that is, multiplier derivatives—are of great interest. These objects relate to the Jacobian of the optimality conditions. That particular matrix often has block structure. So, we derive explicit formulas for the inverse of ...

Domain Decomposition with Lagrange Multipliers

Computing True Shadow Prices in Linear Programming

It is well known that in linear programming, the optimal values of the dual variables can be interpreted as shadow prices (marginal values) of the right-hand-side coefficients. However, this is true only under nondegeneracy assumptions. Since real problems are often degenerate, the output from conventional LP software regarding such marginal information can be misleading. This paper surveys and...

Substructuring by Lagrange Multipliers

Lagrange Multipliers and Optimality

Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write first-order optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions than equations, have demanded deeper understanding of the concept and how it fits into a larger t...