Data Randomness Makes Optimization Problems Easier to Solve?

Optimization algorithms have been recently applied to solver problems where data possess certain randomness, partly because data themselves contain randomness in a big-data environment or data are randomly sampled from their populations. It has been shown that data randomness typically makes algorithms run faster in the so-called “average behavior analysis”. In this short note, we give an example to show that a general non-convex quadratically constrained quadratic optimization problem, when data are randomly generated and the variable dimension is relatively higher than the number of constraints, can be globally solved with high probability via convex optimization algorithms. The proof is based on the fact that the semidefinite relaxation of the problem with random data would likely be exact in such cases. This implies that certain randomness in the gradient vectors and/or Hessian matrices may help to solve non-convex optimization problems.