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From: 海涛 朱 on 23 Jul 2010 14:23
Given some variables a1, a2, a3... an, and some constraints based on
these variables, like:
a1 + a2 >0,
a3 - a5 >0,
a4 - a3 + a2 <0,
.....

how can I find a solution for variables, that maximum the positive
value of variables? I even don't know which area this topic should
belong to. Please help. Thanks in advance!

steve
From: Bastian Erdnuess on 23 Jul 2010 15:02
海涛 朱 wrote:

> Given some variables a1, a2, a3... an, and some constraints based on
> these variables, like:
> a1 + a2 >0,
> a3 - a5 >0,
> a4 - a3 + a2 <0,
> ....

That looks very linear. (Why "nonlinear" in the subject?)

> how can I find a solution for variables, that maximum the positive
> value of variables? I even don't know which area this topic should
> belong to. Please help. Thanks in advance!

Google "simplex algorithm" or more general "linear programming".

Bastian
From: 海涛 朱 on 23 Jul 2010 15:15
Let me make it more clear. How to get a solution for variables, that
maximun the positive number of variables. That means, make as much of
the variables positive. I think that's not linear problem
From: Greg Neill on 23 Jul 2010 15:35
?? ? wrote:
> Let me make it more clear. How to get a solution for variables, that
> maximun the positive number of variables. That means, make as much of
> the variables positive. I think that's not linear problem

It's a linear programming problem.

It is equivalent to plotting the areas or volumes
represented by each of the constraints on a set of
multidimensional cartesian axes, then concentrating
on the volume that represents the solution set (the
set of points that could possibly satisfy all the
constraints). In that volume you want to look for
the point that best satisfies your goal. Often this
point will lie at a vertex of the resulting volume's
shape.


From: Ray Vickson on 23 Jul 2010 16:58
On Jul 23, 11:23 am, 海涛 朱 <zhuht...(a)gmail.com> wrote:
> Given some variables a1, a2, a3... an, and some constraints based on
> these +variables, like:
> a1 + a2 >0,
> a3 - a5 >0,
> a4 - a3 + a2 <0,
> ....
>
> how can I find a solution for variables, that maximum the positive
> value of variables?

Do you mean you want to maximize the positive parts of a1, a2, ...?
Or, do you mean that you want a1, a2, a3, ... to be positive and you
want to maximize some function f(a1,a2,a3,...)? Anyway: be careful:
optimization problems with POSITIVE variables and STRICT inequalities
(like a1 > 0 and a1 + a2 > 0) may, and often do, fail to have
solutions. You should use non-negative variables and non-strict
inequalities (like a1 >= 0 and a1 + a2 >= 0). If you want to maximize
or minimize a _linear_ function of the form f(a1,a2,...) = c1*a1 +
c2*a2 + ..., subject to the types of (linear) constraints that you
wrote, you have a standard "linear programming problem"; see, eg.,
http://www.usna.edu/Users/weapsys/avramov/Compressed%20sensing%20tutorial/LP.pdf
or
http://www.sce.carleton.ca/faculty/chinneck/po/Chapter2.pdf .

If your function f(a1,a2,...) is nonlinear in the variables a_i, you
have a (linearly-constrained) "nonlinear programming problem".

R.G. Vickson

> I even don't know which area this topic should
> belong to. Please help. Thanks in advance!
>
> steve

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