XNOR as linear inequalities

Boolean expressions are found in many MIP models. AND and OR are the most common. When we have an expression like $x {\bf{\ and\ }} y$ or $x {\bf{\ or\ }} y$, where all variables are binary, we typically reformulate this as a set of inequalities. XOR is a bit more exotic, but I have never seen a usage for XNOR. Until now [1].

As this is about boolean logic, in the discussion below all variables $x$, $y$, $z$ are binary.   

AND  

Goal Programming

In [1] a goal programming [2] model was stated, although the poster did not use that term.

The model is fairly simple:

• We have 6 different raw materials that need to be blended into a final product.
• The raw materials have 4 different ingredients.
• We want to blend one unit of final product that has some target values for some formulas. We have three of these goals. 

The data looks like:

MAX-CUT

The MAX-CUT problem is quite famous [1]. It can be stated as follows:

Given an undirected graph $G=(V,E)$, split the nodes into two sets. Maximize the number of edges that have one node in each set.

Here is an example using a random sparse graph:

Maximizing Standard Deviation

In [1] a question was posed: how to maximize the standard deviation of a vector $\color{darkred}x_i$, subject to side constraints using an LP/MIP formulation?

The standard deviation is usually defined as: $$\mathbf{sd}(x) = \sqrt{\frac{1}{n-1}\sum_i \left(x_i-\mu\right)^2}$$ where $$\mu = \frac{\sum_i x_i}{n}$$ and $n$ is the number of elements in $x$.

A subset selection problem

Selecting $k$ out of $n$ can be a computational difficult task. Here we look at a special case and see how we can exploit the properties of the data. Suppose we have $n$ data points. We want to select $k$ points such that the average of these $k$ points is as close to a target value $\color{darkblue}t$ as possible [1]. I.e. we want $$\min_{\color{darkred}S} \left| \frac{\sum_{s \in \color{darkred}S}\color{darkblue}p_s}{\color{darkblue}k} - \color{darkblue}t \right|$$ where $|\color{darkred}S|=k$. The special property of the data is that it is composed of just a few different values. Let's say our data looks like: