Assume I have some non-convex function $f(x_1, x_2, ...)$ and I want to optimize it to find a global minimum. I feel like it is easy to show that this problem is in the class NP with the decision problem
Is there a set of points such that f < C?
Where C is some constant. However, I am not sure if these problems are in the class of NP-Complete, and if so, what would you say the size of the input is? Complexity of the function?
Thanks!
Yes, non-convex optimization is NP-hard. For a simple proof, consider the following reduction from Subset-Sum. The Subset-Sum problem asks whether there is a subset of the input integers $a_1, \dots, a_n$ which sums to zero. To reduce to non-convex programming, let $x_1, \dots, x_n$ be variables encoding the subset and consider the following non-convex program:
$$ \begin{align*} \text{minimize }\quad&(a\cdot x)^2 + \sum_{i=1}^n x_i^2(1 - x_i)^2\\ \text{subject to}\quad& \sum_{i=1}^n x_i \ge 1. \end{align*} $$
Note that the optimum of this program is zero iff the Subset-Sum instance has a subset which sums to zero.
Quadratic programming is an example of a non-convex optimization problem that is NP-hard. See Transforming SAT to Quadratic Programming in polynomial time for a proof.
