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→Equality constraints: substitution method |
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===Equality constraints===
====Substitution method====
For very simple problems, say a function of two variables subject to a single constraint, it is most practical to apply the method of substitution.<ref>{{cite book |first=Mike |last=Prosser |title=Basic Mathematics for Economists |___location=New York |publisher=Routledge |year=1993 |isbn=0-415-08424-5 |chapter=Constrained Optimization by Substitution |pages=338–346 }}</ref> The idea is to substitute the constraint into the objective function to create a [[Function composition|composite function]] that incorporates the effect of the constraint. For example, assume the objective is to maximize <math>f(x,y) = x \cdot y</math> subject to <math>x + y = 10</math>. The constraint implies <math>y = 10 - x</math>, which can be substituted into the objective function to create <math>g(x) = x (10 - x) = 10x - x^{2}</math>. The first-order necessary condition gives <math>\frac{\partial g}{\partial x} = 10 - 2x = 0</math>, which can be solved for <math>x=5</math> and, consequently, <math>y = 10 - 5 = 5</math>.
====Lagrange multiplier====
If the constrained problem has only equality constraints, the method of [[Lagrange multipliers]] can be used to convert it into an unconstrained problem whose number of variables is the original number of variables plus the original number of equality constraints. Alternatively, if the constraints are all equality constraints and are all linear, they can be solved for some of the variables in terms of the others, and the former can be substituted out of the objective function, leaving an unconstrained problem in a smaller number of variables.
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