Global Optimization: A Quadratic Programming Perspective

Abstract

Global optimization is a highly active research field in the intersection of continuous and combinatorial optimization (a basic web search delivers overa million hits for this phrase and for its British cousin, Global Optimisation).A variety of methods have been devised to deal with this problem class, which – borrowing biological taxonomy terminology in a very superficial way – may be divided roughly into the two domains of exact/rigorous methods and heuristics, the difference between them probably being that you can prove less theorems in the latter domain. Breaking the domain of exact methods into two phyla of deterministic methods and stochastic methods, we may have the following further taxonomy of the deterministic phylum:

$$ \begin{array}{lll} {{\rm exhaustive methods}\left\{ {\begin{array}{lll} {{\rm passive/direct, streamlined enumeration}} \\ {{\rm homotopy, trajectory methods}} \\ \end{array}} \right.} \\ {{\rm methods using global structure}\left\{ {\begin{array}{lll} {{\rm smoothing, filling, parameter continuation}} \\ {{\rm hierarchical funnel, difference - of - convex}} \\ \end{array}} \right.} \\ {{\rm iterative improvement methods}\left\{ {\begin{array}{lll} {{\rm escape, tunneling, deflation, aux}{\rm .functions}} \\ {{\rm successive approximation, minorants}} \\ \end{array}} \right.} \\ \end{array} $$

implicit enumeration methods: branch & bound