Mathematical Programming

- Overview

Mathematical programming refers to mathematical models used to solve problems such as decision problems. These terms are meant to contrast with computer programming, which solves such problems by executing algorithms designed specifically for a particular problem. 

Mathematical programming refers to a declarative approach. This means that we consider the representation of a problem through a mathematical model separate from the solution of the problem. 

The idea is that the solution can be done using a mathematical model that captures the problem through a general method (such as branching methods). For example, consider a decision problem that can be represented as a graph, with variables representing the presence or absence of certain vertices and edges in the solution. 

- Categories of Mathematical Programming Problems

Mathematical programming refers to mathematical models used to solve problems such as decision problems. It involves separating the representation of a problem through a mathematical model from the solution of the problem. 

The solution can be done using the mathematical model used to describe the problem through general methods such as branching methods. For example, consider a decision problem that can be represented by a graph, with variables representing whether certain vertices and edges are present in the solution. 

Mathematical programming problems can be divided into several categories based on the nature of the objective function and constraints. These categories include unconstrained problems, linear programming (LP) problems, integer linear programming (ILP) problems, mixed integer linear programming (MILP or MIP) problems, quadratic programming (QP) problems, nonlinear programming (NLP) problems, convex programming or convex optimization problems, and discrete or combinatorial optimization problems. 

Different categories of mathematical programming problems require different solution techniques and may have different computational complexities. Linear programming and convex optimization problems can be solved efficiently and reliably.

- Applications of Mathematical Programming

Mathematical programming is used in a wide range of fields, including:

• Production Scheduling: Determining the optimal production quantities over time to meet demand while minimizing costs.
• Transportation: Finding the best routes and schedules for delivering goods.
• Military Logistics: Optimizing supply chains and resource allocation.
• Economic Growth: Modeling economic systems and predicting future growth.

- The Process of Mathematical Programming

Mathematical programming involves:

• Formulating the problem as a mathematical model.
• Defining the objective function (what needs to be maximized or minimized).
• Identifying constraints (limitations or restrictions).
• Solving the model using mathematical algorithms (often with the help of computers).