Shadow price multiplier

Lagrange multipliers are often referred to as shadow prices, adjoint variables, or dual variables, depending on the context. Suppose the variables are at an optimum point for the problem. Perturb the variables such that only constraint hj changes. We can write... 

The multiplier tells how the optimal value of the objective func tion changes for this small change in the value of a constraint while holding all the other constraints at zero. It is for this reason that they are often called shadow prices. 

At convergence the Lagrange multipliers have some physical meaning similar to the "shadow prices" discussed in Section 1.2. The interested reader may consult (refs. 28-29) where nonideality, treatment of condensed phases, numerical difficulties and other problems are also discussed. Handbooks like (ref. 38) contain the necessary standard Gibbs free energy data for a great number of substances. 

The Lagrange multipliers in a constrained nonlinear optimization problem have a similar interpretation to the dual variables or shadow prices in linear programming. To provide such an interpretation, we will consider problem (3.3) with only equality constraints that is,... 

The Lagrangian multipliers may be interpreted as prices which reveal the economic value associated with a differential change in the corresponding state variables, and are referred to as shadow prices (5). 

As the goal is to minimize the objective function, releasing the constraint into the feasible region must not decrease the objective function. Using the shadow price argument above, it is evident that the multiplier must be nonnegative (Ref. 177). 

The following Eqs. (10) and (11) give the changes of the variables in the basis and profit for changes Afo,. Referring to Eq. (11), the units on the Lagrange multiplier are dollars per unit mass or volume, hence the name shadow prices. For the optimal solution to... 

The sensitivity of the linear programming solution is expressed in terms of shadow prices (dual price/simplex multipliers) and opportunity (reduced) cost. A shadow... 

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