I have an optimization problem in the form

max \((a-\bar{a})(b-\bar{b})\)

subject to \(a+b=1\)

Here \(\bar{a}\) and \(\bar{b}\) are known values and both of them are >0.

Let \(a_{opt}\) and \(b_{opt}\) are the solutions or optimal values

then

\(a_{opt}-\bar{a}>0\) and \(b_{opt}-\bar{b}>0\)

How to find this solution for this optimization problem?

asked 09 Feb '16, 04:13

dimitrios's gravatar image

dimitrios
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accept rate: 0%

edited 09 Feb '16, 04:17

1

How big could be those \(a\) and \(b\)?

(09 Feb '16, 05:48) Slavko

Substitute \(b=1-a\) into your objective function, yielding a quadratic polynomial in \(a\). To find \(a_\text{opt}\) and \(b_\text{opt}\), use either single-variable calculus or a well-known fact about the location of a parabola's vertex with respect to its intercepts.

(09 Feb '16, 10:53) Rob Pratt

@Rob Pratt, Thank you very much for the hints, i.e., the solutions!

(09 Feb '16, 20:34) dimitrios

As Rob suggests, substitute to get a function of one variable (a). Then first derivative to find turning point gives optimal a (second derivative will show it's a maximum). Substitute back to get b_opt. You should find

a = (1 + a¯ - b¯)/2,
b = (1 - a¯ + b¯)/2

Also, your last set of constraints on the optimal values constrains the sum of a¯ + b¯ < 1

link

answered 09 Feb '16, 11:55

Grallator's gravatar image

Grallator
212
accept rate: 0%

@Grallator, thanks for the answer!

(09 Feb '16, 20:35) dimitrios
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Asked: 09 Feb '16, 04:13

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Last updated: 09 Feb '16, 20:35

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