TITLE: Two fold optimization: is there an established approach for this kind of problem?
QUESTION [5 upvotes]: I have a list of thousands of linear expressions, with less than 100 total variables. Each expression is associated with a positive point value. I want to make as many of the expressions greater than zero as possible, maximizing the total points of the expressions I can make positive.
For example:
x1 + x2 > 0 (5 points)
x1 - 3*x2 > 0 (1 point)
-x1 + x2 > 0 (4 points) 
I can choose x1=1, x2=2 to maximize the score (9 points).
Is there an established algorithmic approach to solve this kind of problem?

REPLY [3 votes]: If the coefficients of your linear expressions are integers and you want the variables $x_i$ to be integers as well then the problem can be written as an integer program:
\begin{align*}
\text{Maximize }\sum_{i=1}^m&c_iy_i\\
\text{subject to }My_i &\leqslant M-1+\sum_{j=1}^na_{ij}x_j &&i\in\{1,\ldots,m\},\\
x_j &\in\mathbb{Z}&&j\in\{1,\ldots,n\},\\
y_i &\in\{0,1\} && i\in\{1,\ldots,m\}. 
\end{align*} 
Here $\displaystyle\sum_{j=1}^na_{ij}x_j$ is the $i$-th linear expression, $c_i$ is its point value, and $M$ is a sufficiently large number.
Throwing this formulation at a MIP solver might be worth a try.