# Solving Optimization Problems with Calculus: Expanding from Simple to Complex Examples

Optimization problems are a significant part of calculus, as they involve finding the maximum or minimum value of a function within given constraints. In this blog post, we’ll briefly explore how to solve a simple optimization problem using calculus and then discuss how the same methods can be applied to more complex problems with multiple variables. It will help to have this understanding as we convert a FanDuel lineup into an optimization problem to be solved with our code.

## The Basics of Solving an Optimization Problem

Consider a straightforward example: you have a limited amount of fencing material to create a rectangular enclosure for your garden. Your goal is to maximize the area of the enclosure while staying within the constraint of the available fencing material. Let’s say you have 100 meters of fencing available.

First, we need to set up the constraint equation. Since the perimeter of a rectangle is given by $P = 2l + 2w$ (where $l$ is the length and $w$ is the width), our constraint equation is $100 = 2l + 2w$. We can simplify this to $50 = l + w$ or $w = 50 - l$.

Next, we need to express the area we want to maximize in terms of a single variable. The area of a rectangle is given by $A = lw$. Using our constraint equation, we can rewrite this as $A = l(50 - l) = 50l - l^2$.

Now, we’ll use calculus to find the maximum area. We first find the derivative of the area function, $A'(l) = 50 - 2l$. To find the critical points, we set the derivative equal to zero and solve for $l$: $0 = 50 - 2l \Rightarrow l = 25$. Plugging this back into our constraint equation, we find that $w = 25$ as well. Thus, the maximum area occurs when the enclosure is a square with side lengths of 25 meters.

## Expanding to More Complex Problems

The method we used for solving the simple optimization problem can be applied to more complex problems involving multiple variables. For instance, imagine you need to optimize a function of nine variables instead of just two (like in a FanDuel Full Roster contest).

The process remains essentially the same:

1. Establish the constraint equation(s) involving the nine variables.
2. Express the function you want to maximize or minimize in terms of a single variable or a reduced set of variables using the constraint equation(s).
3. Use calculus techniques, such as partial derivatives, to find the critical points of the function.
4. Analyze the critical points to determine the optimal solution.

While the calculations become more complicated as the number of variables increases, the fundamental approach remains consistent. By understanding the basics of solving optimization problems in calculus, you can tackle more complex situations and make the most of your mathematical skills.

Fortunately, we will be able to use code for steps 2-4, but I think it’s helpful to understand what it will be doing behind the scenes. We also need to understand how to do step 1, and set up our equations for optimization.

## What’s Next?

In the next post, we will start talking about what the optimization equations will need to be for our optimizer. This will help us plan out our code before we start writing it. We will also touch on how we can adjust these equations based on specific DFS strategies we want to test.

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