How to solve a system of inequalities?

Question

I\'ve reduced my problem (table layout algorithm) to the following problem:

Imagine I have N variables X₁, X₂, ..., X_N. I also have some

Answer 1

Minimizing x1 + x2 + ... where the xi satisfy linear constraints is called Linear Programming. It's covered in some detail in Wikipedia

Answer 2

You may also post directly your linear model to NEOS platform (http://neos.mcs.anl.gov/neos/solvers/index.html) . What you simply have to do first is write your model in an algebraic language such as AMPL. Then NEOS will solve the model and returns the results by e-mail.

Answer 3

What you have there is a pretty basic Linear Programming problem. You want to maximize the equation X_1 + ... + X_n subject to

X_1 >= 2
X_2 + X_3 >= 13
etc.

There are numerous algorithms to solve this type of problem. The most well known is the Simplex algorithm which will solve your equation (with certain caveats) quite efficiently in the average case, although there exist LP problems for which the Simplex algorithm will require exponentially many steps to solve (in the problem size).

Various implementations of LP solvers exist. For example LP_Solve should satisfy most of your requirements

Answer 4

Linear programming