Transcription of Lagrangian Methods for Constrained Optimization
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Lagrangian Methods for Constrained Optimization
Appendix ALagrangian Methods forConstrained regional and functional constraintsThroughout this book we have considered Optimization problems that were subject to con-straints. These include the problem of allocating a finite amounts of bandwidth to maximizetotal user benefit (page 17), the social welfare maximization problem (page 129) and thetime of day pricing problem (page 213). We make frequent use of the Lagrangian method tosolve these problems. This appendix provides a tutorial on the method. Take, for example,NETWORK: maximizex 0nr r=1wrlogxr,subject toAx C,posed on page 271. This is an example of the generic Constrained Optimization problem:P: maximizex Xf(x),subject tog(x)= to be maximized subject to constraints that are of two types.
is a regional constraint. For example, it might be x ≥ 0. The constraint g(x)=b is a functional constraint. Sometimes the functional constraint is an inequality constraint, like g(x) ≤ b. But if it is, we can always add a slack variable, z, and re-write it as the equality constraint g(x)+z = b, re-defining the regional constraint as x ∈ ...
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