“To solve” vs “For solving”: what is the difference?

Question

What are the differences between:

An algorithm to solve optimization problems

and

An algorithm for solving optimization problems

I think only the second one is correct but I can’t explain why the first one is wrong.

Answer 1

They are both grammatical, but have different structures, and in some contexts could have different meanings.

Algorithm takes a "for" phrase (usually containing an "-ing" clause) as its complement:

An [algorithm for solving optimization problems]]

Its complement is closely bound to it, and typically forms part of the meaning - an algorithm is underspecified if you don't say what it is for.

Algorithm cannot take a "to"-infinitive clause as a complement, so in your other example, the "to"-infinitive clause must be an adjunct, less tightly bound to the headword: it is not part of the meaning of the headword, but additional information about the instance being referred to:

[An algorithm] [to solve optimization problems]

Other examples of adjuncts are:

An algorithm in my book

An algorithm invented by Knuth.

The meaning of a "to"-infinitive clause as an adjunct is the purpose for which something is done.

In some contexts these will in practice have the same meaning:

This is an algorithm to solve optimization problems.

cannot reasonably mean anything different from the other sentence (with the complement).

But consider

I want an algorithm for solving optimization problems.

What do I want? An algorithm, and specifically one that solves ...

I want an algorithm to solve optimization problems.

what do I want? An algorithm. What do I want it for? To solve optimization problems. What kind of algorithm? I don't care what it's for, as long as it solves my problems.

By the way, there is no rule that explains why algorithm takes a for complement and not a to complement: it's just a fact about present day English.