A General Framework for Constructing Cooperative Global Optimization Algorithms

Abstract

Many well-structured problems in many areas can be defined as the optimization of a multivariate energy function E(x ₁,x ₂,…,x _n). In most cases, a problem of that type is computationally hard. The only way we know before to find the global optimum is by making an algorithm that runs in exponential time which is too expensive in practice. The other traditional methods, like Local Search, Simulated Annealing, Tabu Search, and the evolutionary algorithms, have no general conditions to identify the global optimum and stop searching. A new cooperative optimization algorithm is presented in this paper that is capable of finding the global optimum within a polynomial time, though not guaranteed. It has both sufficient conditions and necessary conditions to identify the global optimums and to trim the search space. It is guaranteed to converge linearly to a solution, which must be the global one if it is a consensus solution. The convergence is also insensitive to the disturbances to its initial or intermediate solutions. A lower bound on the energy function is also provided by the algorithm which is guaranteed to be improved after each iteration. Its power is demonstrated in solving nonlinear optimization problems raised from an early vision problem, shape from shading of a polyhedron. Most importantly in this paper, to make the algorithm complete, i.e. be always capable of finding the global optimum, we propose a general framework for constructing the cooperative global optimization algorithms more powerful than the primary one based on the lattice concept from Abstract Algebra.