I have a problem where I have a bunch of inequalities in the form:
$a_{1,1}b_{1,1} + a_{1,2}b_{1,2} + ... + a_{1,n}b_{1,n} > a_{2,1}b_{2,1} + a_{2,2}b_{2,2} + ... + a_{2,n}b_{2,n}$
$a_{2,1}b_{2,1} + a_{2,2}b_{2,2} + ... + a_{2,n}b_{2,n} > a_{3,1}b_{3,1} + a_{3,2}b_{3,2} + ... + a_{3,n}b_{3,n}$
$a_{1,1}b_{1,1} + a_{1,2}b_{1,2} + ... + a_{1,n}b_{1,n} < a_{3,1}b_{3,1} + a_{3,2}b_{3,2} + ... + a_{3,n}b_{3,n}$
etc.
I've randomly distributed the <
and >
signs in the inequalities for the example; they can be any of ≤
, ≥
, <
, or >
. All the b
variables are dummy variables (either 0 or 1).
I want to find values of all the a
variables, assuming possible given the data. This seems like a linear programming problem, but I'm struggling to convert it to one.
One further problem is that I may come to a point where there is not solution by classic linear programming (ex. the example above). In this case, I would want to find the point that is "closest" to being a valid point (think OLS error reduction). I know how to do this for linear programming (another reason I jumped to lp), but am unsure on how to do it for the problem statement above. As a side note, I plan to do this mainly computationally.
What method would I use to solve the above?
b
variables are there for expository ease $\endgroup$