# Can we get closed form solution for such a problem?

Operations Research Asked on August 19, 2021

begin{align}min&quadsum_{i=1}^Nfrac{A_i}{x_i}\text{s.t.}&quadsum x_i le X\&quad x_i ge 0end{align}

wherein $$A_i>0, (iin{1,dots,N})$$ is constant, $$x_i, (iin{1,dots,N})$$ is a continuous optimization variable.

If some $$A_i$$ is negative the problem is unbounded: we can make the objective arbitrarily small by making $$x_i$$ arbitrarily close to 0.

Assuming $$A_i geq 0$$, the optimality is obtained when all $$frac{A_i}{x_i^2}$$ are equal (KKT optimality conditions), or equivalently all $$frac{x_i^2}{A_i}$$ are equal, and $$sum x_i = X$$ (otherwise you can increase some $$x_i$$ and reduce the objective).

Stating $$x_i = sqrt{A_i} y$$, you can deduce $$y = frac{X}{sum sqrt{A_i}}$$ from the equality.

Therefore $$x_i = frac{sqrt{A_i}}{sum sqrt{A_j}}X$$

Answered by Gabriel Gouvine on August 19, 2021

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