# Mathematical Optimization

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## What is mathematical optimization?

Mathematical optimization, also known as mathematical programming, is the process of finding the best solution to a problem from a set of possible solutions, often subject to constraints or conditions. The problem is typically formulated as an optimization problem consisting of an objective function to be maximized or minimized and a set of constraints that limit the feasible solutions.

The objective function is a mathematical expression that measures the performance or quality of the solution, and the constraints are mathematical expressions that specify the conditions the solution must satisfy. The feasible solutions are those that satisfy all the constraints, and the optimal solution is the one that maximizes or minimizes the objective function while satisfying the constraints.

### Example

Suppose we have a budget of $5000 to purchase two types of products: product A and product B. Each unit of product A costs$25, and each unit of product B is $35. We want to maximize the total number of units purchased subject to the following constraints: • The total purchase cost must not exceed the budget of$5000.
• The number of units of product A must be at least 50.
• The number of units of product B must be at least 60.

Let $x$ be the number of units of product A purchased and $y$ be the number of units of product B purchased. Then, the objective function to be maximized is:

$f(x,y) = x + y$

The constraints can be written as follows:

\begin{aligned} 25x + 35y & <= 5000 \\ x &>= 50 \\ y &>= 60 \end{aligned}

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