AnswerBun.com

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.

One Answer

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

Add your own answers!

Related Questions

CPLEX MIP warm start seems slow down the program?

1  Asked on August 19, 2021 by mengfan-ma

   

Two M/M/1 queues working independently

0  Asked on August 19, 2021 by cookiemonster

 

Scheduling with setup cost

0  Asked on August 19, 2021 by dirk-nachbar

 

MINLP Solution same as Global Optimum?

2  Asked on August 19, 2021 by clement

   

Meta papers on operations research

2  Asked on August 19, 2021 by luke599999

 

How can I use warm start in C#

1  Asked on August 19, 2021 by fhm-ider

   

Feasible sets represented as point clouds

1  Asked on August 19, 2021 by harry-cohen

   

Is this the same as Agent Based DES or something different?

0  Asked on August 19, 2021 by brendan-hill

 

Is there any OR way to solve this problem?

3  Asked on August 19, 2021 by samiczy

 

Ask a Question

Get help from others!

© 2022 AnswerBun.com. All rights reserved. Sites we Love: PCI Database, MenuIva, UKBizDB, Menu Kuliner, Sharing RPP