General Approach to Optimization Problems

Optimization problems are all about finding the best possible value for a certain quantity, usually described by a mathematical function. The aim is to figure out which input values give us the highest or lowest outcome, depending on the goal. These problems have the potential to show up everywhere, whether you’re trying to design a soup can to hold a fixed volume while using the least metal, maximize the brightness of a streetlight on a building by choosing the optimal height above the sidewalk, determine the angle to tilt a solar panel so it collects the most sunlight at a given latitude and time of year, or the price to charge for a muffin at a bakery to maximize profits.

Unconstrained Optimization

In unconstrained optimization, there are no extra limitations on the variable. You are free to pick any value within the valid range of the function. The goal is to find the absolute (global) maximum or minimum of the objective function, which is the function you are trying to optimize.

The first step is to decide which values the variable can actually take. This is called determining the domain. For example, if the problem deals with a physical length, negative values don’t make sense, so the domain would only include positive numbers.

Once you know the domain, you take the derivative of the objective function to find the critical points. Critical points occur where the derivative is zero or undefined. These are possible locations for a maximum or minimum.

After finding the critical points, the next step is to figure out what type of points they are. Not every critical point is a maximum or minimum. Some are just flat spots where the slope is zero but the function does not reach a peak or valley. To classify these points, you can use the first or second derivative test.

The first derivative test involves checking whether the function increases before and decreases after the point, which indicates a maximum, or decreases before and increases after, which indicates a minimum. If the direction does not change, it is not an extremum.

The second derivative test checks the concavity of the function. If the second derivative is positive at a critical point, the function is concave up and you have a minimum. If it is negative, the function is concave down and you have a maximum.

To find the absolute maximum or minimum, compare the values of the function at the critical points and at the endpoints of the domain, if the domain is closed. Sometimes the largest or smallest value is not at a critical point but at one of these endpoints.

Constrained Optimization

In most real-world situations, there are restrictions that limit which values are possible. Constrained optimization problems involve finding the best value under such limitations. For example, you might want to build the largest possible rectangular garden with only 100 meters of fencing. The amount of fencing is the constraint, since you cannot use more than you have. In this case, the area of the rectangle is what you want to maximize, and the perimeter equation connects the length and width. By solving the constraint equation for one variable, you can substitute it into the area formula, which reduces the problem to a single variable. This lets you take the derivative, find critical points, and determine the maximum area.

To solve a constrained problem, start by clearly identifying both the function you want to optimize and the constraint. For example, if you want to maximize the area of a rectangle with a fixed perimeter, the area is the objective function and the perimeter is the constraint.

Next, use the constraint to eliminate one variable from the objective function. This simplifies the problem to one variable, which makes it easier to analyze.

After that, check which values are actually allowed based on the context. Make sure you are not including negative values or values that don’t make sense for the problem.

Now, take the derivative of the new function and look for critical points. Use the first or second derivative test to decide whether these points are maxima, minima, or neither.

Finally, check both the critical points and the endpoints of the domain, if there are any. This will help you find the absolute highest or lowest value that fits within all the restrictions.