Constrained Minimization: Equality Constraints
To have an optimization problem, the number of variables must exceed the number of constraints. Thus, the subject of constrained optimization is introduced by minimizing a function of two variables where the variables are related through one algebraic constraint. First, the function is minimized using the direct approach, Here, differentials of the performance index and the constraint are taken and the dependent differential is eliminated to form the first and second differentials in terms of the independent differential. Next, the Lagrange multiplier approach is introduced. Here, the constraint is multiplied by an undetermined constant (Lagrange multiplier) and is added to the performance index. The multiplier is used later to eliminate the dependent differential. The results are compared with those of the direct approach to confirm the validity of the multiplier method. This part of the chapter is concluded by a discussion of higher-order differentials. Next, the problem of n constrained variables is analyzed using an indicial notation; the same problem is investigated in Chapter 5 using matrix notation.
KeywordsLagrange Multiplier Performance Index Equality Constraint Minimal Point Optimal Point
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