Shadow price of the constraint

In nonlinear programming (NLP) problems, either the objective function, the constraints, or both the objective and the constraints are nonlinear. Unlike LP, NLP solution does not always lie at the vertex of the feasible region. NLP optimum lies where the Jacobean of the function obtained by combining constraints with the objective function (using Lagrange multiphers as follows) is zero. The solution is local minimum if the Jacobian J is zero and the Hessian H is positive definite, and it is a local maximum if J is zero and H is negative definite. 

The first order Kuhn-Tucker conditions necessary for optimality can be written as follows  

It should be remembered that the direction of inequality is very important here. The non-negativity requirement (for a minimization problem) above ensures that the constraint direction is not violated and the solution is in the feasible region. The sufficiency condition depends on Hessian as stated earlier. 

Example 5.2 The following reaction is taking place in a batch reactor. A R S, where R is the desired product and A is the reactant, with an initial concentration, CAo = 10 moles/volume. The rate equations for this reaction are provided below. Solve the problem to obtain maximiun concentration of R. 

Converting the NLP into a minimization problem and formulating the augmented Lagrangian function results in the following imconstrained NLP. 

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