What is the difference between optimal and optimum? [closed]

Question

My question is exactly as the title says. Someone used "optimum" in an email to me and in my response I used "optimal," so I began wondering what the difference between them is.

Answer 1

I'd like to add the following lines as an addendum to other answers.

In some areas of mathematics, an optimal value is essentially a local optimum; i.e. it's an optimum value in its neighborhood. For example:

In parallel, the same applies to maximal and minimal values. In other words, a maximal value is essentially a local maximum; a minimal value is essentially a local minimum. For example:

Answer 2

The adjective forms of optimal and optimum are synonymous:

optimal, adjective: most desirable or satisfactory : OPTIMUM [optimal concentration of a drug]

optimum, adjective: 1. most favorable or most conducive to a given end especially under fixed conditions [question is one of combining these various techniques to optimum advantage]; 2. greatest or best possible under a restriction expressed or implied [an optimum return on capital]

However, optimum also has a noun form:

optimum, noun: 1. the amount or degree of something that is most favorable to some end; especially : the most favorable condition (as of temperature, light, moisture, food) for the growth and reproduction of an organism; 2 a : greatest degree (as of growth, activity, or effectiveness) attained under implied or specified conditions [this pest reaches its optimum further south]; b : a period of warmer and drier climate than that of the present [he post-Wisconsin optimum of the northern hemisphere is considered to have occurred between 6000 B.C. and 3000 B.C]

[All from Merriam-Webster Unabridged]

Answer 3

My usual reference on such matters, 'The Cambridge Guide to English Usage', sees no difference in meaning, but concludes that optimal is the more formal of the two. Optimal seems to be less common in American English than in British English.