A simple question, but (I'm quite sure) not a superficial one: is the basic distinction between algorithms and much of the rest of math that algorithms try to tackle problems for which we lack global information, or alternatively, lack a complete, instantaneous understanding of the structure of the problem?

A concrete example: consider any simple quadratic polynomial function. If I wanted to know its zeroes, I could just solve for x. Boom: instant analytic solution. Or I could use the Netwon-Raphson method, algorithmically getting closer to the final solution, step-by-step, in accordance with a simple rule.

Obviously there are more complex problems where I can do something like the second approach (use some algorithm), but nothing like the first approach (no clean, instant analytical solution). So is the implication of this that the basic point of algorithms are the problems for which we lack complete, global information, which in turn prevents us from quickly producing the clean analytical solution?

This question is motivated by a recent encounter with the subject of non-convex optimization. A peer explained to me that non-convex optimization is so difficult (and requires difficult algorithms) because basically, you never know if a local optimum is also a global optimum. This remark confused me, because if you had global information, aka the functional form of whatever you were trying to optimize, you could just get an analytical solution. The only way I can make sense of this remark is if we just don't have the functional form -> we don't have global information -> we have to try to screw around with algorithms.

notgive "boom, instant solution", because you need an algorithm to compute the square root. And if you say: "I can leave the square root as is", then you're comparing apples and oranges... Sorry, but very naive question IMHO. $\endgroup$ – Thierry Zell Sep 27 '12 at 15:13