In complexity, why do we find upper bounds, not lower bounds?

In algorithms we use to find Big-O (upper bound), Big-omega (lower bound) and Big-Theta but why we are always interested in finding upper bounds instead of lower bounds?

Lower bounds are of great interest and an active topic of research. However, we tend to be interested in lower bounds for problems and upper bounds for algorithms.

An upper bound for an algorithm is a performance guarantee: the algorithm will not use more than such-and-such an amount of memory or processing steps when run on an input of a particular size. On the other hand, lower bounds for algorithms aren't especially interesting, since most algorithms can be modified to be fast on certain instances and knowing that an algorithm might terminate very quickly isn't often a useful guarantee.

Conversely, any algorithm is an upper bound for the problem it solves. Lower bounds for problems, on the other hand, tell you whether it's worthwhile looking for a faster algorithm than the one you already have. Examples of lower bounds you'll probably be familiar with are completeness results (especially NP-completeness) and the fact that comparison-based sorting takes time $\Omega(n\log n)$.

StackExchange Computer Science Q#44325, answer score: 11