start ILP post

content/post/20220402-csips/index.md

@@ -1,5 +1,5 @@
 +++
-title = "Implementation of a Simple Integer Program Solver"
+title = "Implementation of a Basic Integer Linear Programming Solver"
 date = 2022-04-02T00:00:00
 lastmod = 2022-04-02T00:00:00
 draft = true
@@ -61,8 +61,6 @@ categories = []
     });
 </script>
 
-## Problem Formulation
-
 Inline math: \\(\varphi = \dfrac{1+\sqrt5}{2}= 1.6180339887…\\)
 
 Block math:
@@ -71,8 +69,45 @@ $$
  \varphi = 1+\frac{1} {1+\frac{1} {1+\frac{1} {1+\cdots} } }
 $$
 
+## Problem Formulation
+
+A [linear programming problem](https://en.wikipedia.org/wiki/Linear_programming) is an optimization problem with the following form:
+
+find \\(\mathbf{x}\\) to maximize
+$$
+\mathbf{c}^T\mathbf{x}
+$$
+
+subject to 
+
+$$\begin{aligned}
+A\mathbf{x} &\leq \mathbf{b} \cr
+0 &\leq \mathbf{x} \cr
+\end{aligned}$$
+
+For [integer linear programming (ILP)](https://en.wikipedia.org/wiki/Integer_programming), there is one additional constrait:
+
+$$
+\mathbf{x} \in \mathbb{Z}
+$$
+
+In plain english, \\(\mathbf{x}\\) is a vector, where each entry represents a variable we want to find the optimal value for.
+\\(\mathbf{c}\\) is a vector with one weight for each entry of \\(\mathbf{x}\\).
+We're trying to maximize the dot product  \\(\mathbf{c}^T\mathbf{x}\\); we're just summing up each  each entry of entry of \\(\mathbf{x}\\) weighted by the corresponding entry of \\(\mathbf{c}\\).
+For example, if we seek to maximize the sum of the first two entries of \\(\mathbf{x}\\), the first two entries of \\(\mathbf{c}\\) will be \\(1\\) and the rest \\(0\\).
+
+**What about Minimizing instead of Maximizing?** 
+Just maximize \\(-\mathbf{c}^T\mathbf{x}\\) instead of maximizing \\(\mathbf{c}^T\mathbf{x}\\).
+
+**What if one of my variables needs to be negative?** 
+At first glance, it appears \\(0 \leq \mathbf{x}\\) will be a problem.
+However, we'll just construct \\(A\\) and \\(\mathbf{b}\\) so that all constraints are relative to \\(-\mathbf{x}_i\\) instead of \\(\mathbf{x}_i\\) for variable \\(i\\).
+
 ## Linear Program Relaxation
 
+The way this is usually done is actually by initially ignoring the integer constraint ( \\(\mathbf{x} \in \mathbb{Z}\\) ), then going back and "fixing" your non-integer solution.
+When you ignore the integer constraint, you're doing what's called ["linear programming relaxation" (LP relaxation)](https://en.wikipedia.org/wiki/Linear_programming_relaxation).
+
 ## Branch and Bound
 
 ## User-friendly Interface