On the convergence rate issues of general Markov search for global minimum

Abstract

This paper focuses on the convergence rate problem of general Markov search for global minimum. Many of existing methods are designed for overcoming a very hard problem which is how to efficiently localize and approximate the global minimum of the multimodal function f while all information which can be used are the f -values evaluated for generated points. Because such methods use poor information on f , the following problem may occur: the closer to the optimum, the harder to generate a "better" (in sense of the cost function) state. This paper explores this issue on theoretical basis. To do so the concept of lazy convergence for a globally convergent method is introduced: a globally convergent method is called lazy if the probability of generating a better state from one step to another goes to zero with time. Such issue is the cause of very undesired convergence properties. This paper shows when an optimization method has to be lazy and the presented general results cover, in particular, the class of simulated annealing algorithms and monotone random search. Furthermore, some attention is put on accelerated random search and evolution strategies.

On Asymptotic Convergence Rate of Evolutionary Algorithms

“…The above illustrative example leads to natural generalization. Pure Random Search is typical example of slow convergence and its bad convergence properties follow from more general idea called the lazy convergence, a concept introduced in [24].…”

“…Example 6: Lazy convergence. Now we will adopt the definition from [24] to Markov chains and define the lazy convergence as the property of probability kernel P . We will say that X t converges lazily to x 0 iff X t P → x 0 and :…”

“…Such Markov kernels are natural when the space A is homogeneous, for instance where A = R n or when A is a n-dimensional torus. The detailed analysis of the above examples (and the theoretical details of lazy convergence) may be found in [24]. Theorem 3 below gives a lower bound on ACR(X, h) which does not depend on h and it shows that the lazy convergence is a sufficient condition to have ACR(X, h) = 1 regardless of the function h.…”

On Asymptotic Convergence Rate of Evolutionary Algorithms

“…2 Now we will focus on Markov chains. We will start from the following definition which transfers the definition from [28] to Markov chains with probability kernel P .…”

“…The intuition behind the above is: the closer to the global solution, the harder to generate a better candidate than the current one because an optimization method uses poor information on the function f . Various lazy examples may be found in [28] ( including PRS and Algorithm 4 used for simulations below). Lazy methods do not converge exponentially fast for any nontrivial problem function h, see below.…”