Solving an inequality for minimum value
equation, inequality, language-agnostic, linear-programming, python
Solution
This looks like a linear programming problem. The Simplex algorithm normally gives good results. It basically walks the boundaries of the subspace delimited by the inequalities, looking for the optimum.
Think of it visually: each inequality denotes a half-space, a plane in n-dimensional space that you have to be on the right side of. Your utility function is what you're trying to optimize. If the space is closed, the optimum is going to be at one of the apexes of the closed space; if it's open, it's possible that the optimum is infinite.
Problem
I'm working on a programming problem which boils down to a set of an equation and inequality: ``` x[0]*a[0] + x[1]*a[1] + ... x[n]*a[n] >= D x[0]*b[0] + x[1]*b[1] + ... x[n]*b[n] = C ``` I want to solve for the values of `X` that will give the absolute minimum of `C`, given the input `D` and lists and `A` and `B` consisting of `a[0 - n]` and `b[0 - n ]`. I'm doing the problem at the moment in Python, but the problem in general is language-agnostic. CLARIFICATION UPDATE: the coefficients `x[0 - n]` are restricted to the set of non-negative integers.