Introduction to Operational Research

Operational research applies mathematical models and algorithms to decision making under constraints.

The module covers 4 main areas:

Linear programming
Modelling and solving optimization problems with linear objectives and constraints.
Game theory
Decision making between competing agents with conflicting objectives.
Integer programming
Optimization problems where decision variables are restricted to integers.
Network optimization
Optimization problems modelled as flows or paths on a graph.

Hyperplane

Generalization of a line (from $\mathbb{R}^2$ ) to $\mathbb{R}^n$ .

\{x \in \mathbb{R}^n : a^Tx = c\}

Here $a \in \mathbb{R}^n$ and $c \in \mathbb{R}$ .

A hyperplane $a^Tx = c$ divides $\mathbb{R}^n$ into two half-spaces, the sets of points on either side of it.

Open half-space
$a^Tx < c$ or $a^Tx > c$ . Excludes the boundary hyperplane.
Closed half-space
$a^Tx \le c$ or $a^Tx \ge c$ . Includes the boundary hyperplane.

Every linear constraint in $n$ decision variables is a closed half-space. The feasible region is the intersection of all constraint half-spaces.

In $\mathbb{R}^2$ , a half-space is called a half-plane.