There's three particular optimization problems I see quite often(in various applications). They seem to me to be specific cases of a more general problem, for which there perhaps exists a technique with which they can be solved.

1) Suppose $x_{1}...x_{n}$ are positive reals such that $x_{1}+...+x_{n} = K$. Then maximize the product $x_{1}*x_{2}*...*x_{n}$.

2) Suppose $x_{1}...x_{n}$ are positive reals such that $x_{1}*...*x_{n} = K$. Then minimize the sum $x_{1}+x_{2}+...+x_{n}$.

3) Suppose $p_{1}...p_{n}$ are positive reals, each less than or equal to 1, i.e. $ 0 < p_{i} \leq 1 \ $. Then minimize $ p_{1}*log(p_{1})+...+p_{n}*log(p_{n}) $ (this number is going to be negative, so make it as negative as possible.)

How could I solve these three problems? I'm versed in calculus, and also analysis, but can't seem to think of an appropriate technique.

I was reading about convex functions, what bearing do they have on the problem? What about techniques from elementary calculus?


1 Answer 1


This type of optimisation problems can be solved using the method of Lagrange multipliers , a method for finding the local maxima and minima of a function subject to equality constraints. Given a function to maximise (minimise) $f(x,y)$ and a constraint $h(x,y)=0$ you write down an auxiliary function $\mathcal{L}(x,y,\lambda_1)=f(x,y)-h(x,y)\lambda_1$. Then, derive $\mathcal{L}(x,y,\lambda_1)$ wrt to $x,y,\lambda_1$, set the derivatives equal zero and solve the system in $x,y,\lambda_1$ to find the optimal values $(x^*,y^*)$ satisfying the constraint and the minimisation/maximization problem. You can generalise it to the case of more variables and/or constraints.


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