Date: Apr 30, 2013 1:57 PM
Author: gus gassmann
Subject: Re: Is it possible to maximize/minimize several variables in LP?

On 30/04/2013 1:25 PM, Xu Wang wrote:
> Hello,
>
> I want to maximize (or minimize) several variables in a LP problem as following.
>
> maximize: x1 + x2 + x3
> subject to: constrains contain xi, i = 1, ..., n
>
> When the object function is maximized, are the values of x1, x2 and x3 maxmized?


Depends on what you mean by "maxmized". Let's look at a simple example:

max x1 + x2 + x3

s.t. x1 + 2 x2 + 3 x3 <= 10
x1 >= 0,
x2 >= 0,
x3 >= 0

Any linear programming solver (and there are many, including several
open-source solvers) will report as optimal solution x1 = 10, x2 = 0, x3
= 0. However, this clearly does not "maximize" x2, as x1=0, x2=5, x3=0
satisfies the constraints, and so x2 _could_ be as high as 5.

When you have multiple objectives (i.e., you truly want to maximize ---
in some sense --- x1 AND x2 AND x3), an ordinary linear programming
solver is not going to help you much. Look up "Multi-objective
programming" or "Pareto optimum" on the internet. That should get you
started, at least.

> Thank you in advance.
>
> Regards,
> Xu
>