optimize for $p_{i}$, $i$ = 1,2,3.
$(\frac{p_{3}}{1-p_3}(\mu_3 - \sum_{i=1}^{3}p_{i}\mu_i)^2)$
given that $\sum_{i=1}^{3}p_{i}\mu_{i}-\mu_3\leq 0$ and $\sum_{i=1}^{3}p_{i} = 1$. $0\leq p_i\leq 1$ for $i = 1,2,3.$
KKT may help
Non Linear optimization of non central parameter
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nonlinear-optimization
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0the question is: what is the question? do you want an algorithm? or do you need to find an exact solution? hint for the second case: start with the first order system. additional hint: by your equality constraints this can be considerd a 2 variable problem with inequality constraints only. – 2017-01-15
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0exact solution will be great help.. – 2017-01-15
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0have you tried anything? maybe with the first hint i gave you? where did you get stuck? (this is not a have-your-homework-done-by-someone-else forum) – 2017-01-15
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0I tried using KKT slackness but not working... – 2017-01-15