Unformatted text preview: blem will ‘constrain’ the
variables in the right way, and we have for ourselves a solution to the problem we’re really
interested in (the constrained problem).
It is somewhat interesting to note that Lagrange’s approach is one of method – it is almost
entirely based on off‐the‐shelf mathematical concepts.
We’ll motivate this section by considering a simple constrained optimization problem with one
constraint only. The method ‘works’ in a very similar way for more complicated problems, as
long as the number of constraints is less than the number of critical points we are trying to find.
Example: Let ∶
→ . We want to choose
minimize, depending on the context of the problem) subject to the constraint
The general (unconstrained) problem would not be too difficult, if we had the proper
assumptions on the function ∙ . But with this new angle, i.e., the restriction that
0, we have a somewhat different problem in force.
Side note #3: if is a linear function, as would be a budget constraint, the problem then can be written as a fairly
easy problem of reduction the choice problem to one variable, say, . That is, first solve for
in terms of using
0, and then find the critical point ∗ and finally, with ∗ we obtain ∗ . However, this
approach doesn’t really help us much if a) is not linear and/or, b) there are more than 2 variables. Method of Lagrange, Cont.
We form what is referred to as a Lagrangian:
, , ≡ ,
is an ancillary one, that is, a variable we have introduced to convert our
constrained optimization problem into an unconstrained one. It is of very little general
interest to us or to the problem at hand. We refer to it as a Lagrange Multiplier.
The critical points , ∗ , ∗ of this problem satisfy: ∗, ∗ 2.
∗ ∗ ∗, ∗ ∗, ∗ 1. 3. ∗ ∗ ∗, ∗ , ∗ 0
0 0 These three first order conditions give us 3 equations in 3 unknowns, ∗ , ∗ , and ∗ . Notice
that what we are finding are critical points ∗ , ∗ that, as we can see...
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