Towards Optimal Techniques for Solving Global Optimization Problems: Symmetry-Based Approach

Abstract

Most techniques for solving global optimization problems have parameters that need to be adjusted to the problem or to the class of problems: for example, in gradient methods, we can select different step sizes. When we have a single parameter (or few parameters) to choose, it is possible to empirically try many values and come up with an (almost) optimal value. Thus, in such situations, we can come up with optimal version of the corresponding technique.In other approaches, e.g., in methods like convex underestimators, instead of selecting the value of single number-valued parameter, we have select the auxiliary function. It is not practically possible to test all possible functions, so it is not easy to come up with an optimal version of the corresponding technique.In this paper, we show that in many such situations, natural symmetry requirements enable us either to analytically solve the problem of finding the optimal auxiliary function, or at least reduce this problem to the easier-to-solve problem of finding a few parameters.

A Generalization of the Classical αBB Convex Underestimation via Diagonal and Nondiagonal Quadratic Terms

“…Akrotirianakis and Floudas [11] discussed their computational experience for the γ BB underestimators on box-constrained problems. Floudas and Kreinovich [60,61] proved that the classical and the exponential αBB underestimators are the only two functional forms possessing both shift-invariance and sign-invariance, and are therefore the two natural choices for any αBB-like convexification methodology.…”

On the functional form of convex underestimators for twice continuously differentiable functions

“…shifts but w.r.t scalings as well? Theoretical results on the scale invariance are presented in [9].…”

Swarm Intelligence: Theoretical Proof That Empirical Techniques are Optimal

“…Empirically, among these techniques, the best are αBB method [4,5,17,27] and its modifications recently proposed in [6,7]. It turns out [18] that this empirical optimality can also be explained via shift-and scale-invariance.…”