Definition

The minima and maxima, plurals of minimum and maximum, are collectively known as extrema, the smallest and largest values in a given field. There can only be one global maximum in any domain, and any smaller peaks are local maxima. Similarly, there is a global minimum, and smaller valleys are local minima. 

As a concept, they are valuable for determining the peaks and valleys of life and work.

Why Use It

Scaling a peak in any discipline will likely result in climbing local maxima and require an astute observer to view the territory more broadly.

By identifying local maxima, one can discern a global maximum and strive for that. The local minima work in the same fashion and permit one to view the bottom — the global minimum. Discovering minima is helpful for understanding personal or market failures. They shed light on how much farther there is to fall as well.

One employs a “hill climbing” optimization technique to arrive at a global maximum or minimum. It is an algorithm that works to find a solution and then continues to find incrementally better solutions. Ultimately, the algorithm iterates until it discovers no other possible solutions, surfacing the global maximum in some cases. It is imperfect in that search space is pre-defined, so the algorithm may find local optima, a solution that cannot be improved, in that specific set of circumstances. One uses multiple search space techniques to understand the territory holistically and mitigate this limitation.

When to Use It

When climbing a hill in life, it’s challenging to comprehend if you’ve reached the highest peak. By reaching local maxima, one knows what it took to climb there, and from that vantage point can see the descent afterward.

The highs and lows of a given scenario can illuminate minima and maxima. The idea is to contemplate what has been accomplished, recognize where optimization is possible, and step back to see more of the territory.

After successfully climbing a metaphorical mountain, is there room for improvement? If not, you have reached the global maximum. Conversely, the global minimum may still be somewhere below local minima.

How to Use It

There can only be one global minimum and maximum in a domain, but there are many local minima and maxima in a range.

Mathematically speaking:

• A function can have numerous minima and maxima.
• The lowest value is the global minimum. All other values are local minima.
• The highest value is the global maximum. All other values are local maxima.
• Depending on the range, a subset of the entire domain may only be local minima or maxima.
• The global maximum or minimum are in the universal set and are discoverable.
• A maximum value is found where the function’s gradient is zero — the highest peak.
• Immediately before a maximum value is positive and afterward is negative.
• Determining the pattern leads to discovering the global minimum or maximum.

For example, a machine learning algorithm is programmed to seek maxima and minima to ascertain the highest possible value. Algorithms employ optimization strategies on a subset of the domain and can get stuck in local minima and maxima during training. An algorithm is run many times or has random restarts to deliver better results.

Apart from computer science, minima and maxima have broad applications in network analysis, operations, material design, and many other disciplines. It can function as a tool for personal growth and a barometer for success and failure. When learning a new skill, you have to start at the bottom, grow competency, and work your way to mastery. A sound way to life design is to consider where you are in the ascent, where else you can go, and how far you’ll need to descend to rise even higher.

How to Misuse It

Calculating minima and maxima is not a guideline for what action to take next. Instead, it illuminates the terrain of any domain and where an entity stands. The concept’s utility is in situating the person or organization to inform future decisions.

Next Step

Improving one’s ability to notice the level of success and failure in a given industry is advantageous. Understanding the global minimum and maximum can level-set expectations for the future and couch opportunities for growth from the current location.

The highs and lows of life are natural, and moving between peaks and valleys is how you get from point A to point B. Sometimes, one has to descend from the current local maxima and challenge themself to scale the next highest peak — the global maximum.

Look for the places in your personal and professional life where you may be scaling a peak or recently landed in a valley, take a closer look at where you are, and decide on the next best course of action that can lead you higher.

Where it Came From

Pierre de Fermat introduced the adequality technique in his 1636 treatise Methodus ad disquirendam maximam et minimam, which allows folks to calculate minima and maxima and other calculus problems.