A new asynchronous parallel global optimization method based on simulated annealing and differential evolution

Abstract

This paper presents a new asynchronous parallel global optimization method and its application to the automated device sizing in analog integrated circuit (IC) design. The method is based on the simulated annealing algorithm (SA), but incorporates features from differential evolution (DE) to improve the sampling efficiency and avoid the problems involved with the cooling schedule selection. A simple local search procedure is also incorporated to improve the fine tuning capabilities of the method. To reduce the optimization time, the method is designed as an asynchronous master-slave parallel system that allows simultaneous evaluation of several trial solutions. Comparison with simple SA and DE on a set of well-known analytical test functions confirms the method’s efficiency. The parallel efficiency of the method is also verified by optimizing the functions with 1, 2, 4, and 8 processors. The proposed approach is also applied to several real world cases of device sizing in analog IC design. The optimization results indicate that the method is capable of finding near optimal circuits. The parallel efficiency of the method is confirmed with optimization runs on 1, 2, 4, and 8 processors.

Introduction

Global optimization problems arise in almost every field of scientific research, engineering, chemistry, economy, etc. Many real world problems can be formulated as global optimization problems of the following form:

$$\begin{array}{c}\hfill x^{\ast }=\arg \underset{x\in S}{\min }f(x)\hfill \\ \hfill f:S\to R\hfill \\ \hfill S=\{x\text{,}x\in R^{N_v}\text{,}{l}_i\le x_i\le u_i\text{,}i=1\text{,}\dots \text{,}{N}_v\}\hfill \end{array}$$

where $f(x)$ is the so-called cost function (CF), x is an $N_v$-dimensional vector of optimization variables, and $l_i$ and $u_i$ are the lower and the upper bound for the ith variable, respectively. $x^{\ast }$ denotes the global minimum of the CF. In this paper we only consider problems with simple box constraints.

In practical applications the actual shape of the CF is usually unknown. Often the CF values are the result of expensive and time consuming measurements or simulations. In such cases problem (1) cannot be solved analytically. Many different classes of optimization methods have been developed to solve the problem numerically. Gradient methods are the fastest, but they require the derivatives of the CF and work only on differentiable functions. They are also extremely local by nature and sensitive to noise. This reduces their suitability for many practical applications. The alternative to the gradient methods are the direct search methods. They do not require gradients of the CF and can handle noisy and multimodal functions. Optimization methods can also be classified as local or global. The former are designed to find a minimum as fast as possible, even though it may not be the true global minimum. The latter are usually slower but can find the true global minimum with high probability. There are also many different hybrid methods that try to exploit the fast convergence of the local methods and the global search capabilities of the global methods [1], [2], [3], [4], [5], [6].

The basis of the method presented in this paper is a hybrid method (DESA) [7] that combines simulated annealing (SA) and differential evolution (DE). The random sampling and the Metropolis criterion from SA [8] are combined with the population of points and the sampling mechanism from DE [9] to balance global and local search. We extend DESA so that it can be run in parallel on a cluster of computers. The new method is referred to as Parallel Simulated Annealing Differential Evolution (PSADE).

The paper is organized as follows. In Sections 2 Simulated annealing, 3 Differential evolution the simple SA and the basic DE algorithms are summarized. Section 4 presents a brief classification of parallel optimization approaches. In Section 5 PSADE is presented in more detail. Section 6 compares PSADE with the original DE and simple SA on a set of 23 well-known mathematical test functions. In Section 7 PSADE is applied to the problem of device sizing in analog integrated circuit (IC) design. Section 8 gives the concluding remarks.

We use $U(1\text{,}N)$ to denote a uniformly distributed random integer from $\{1\text{,}2\text{,}\dots \text{,}N\}$, and $U(0\text{,}1)$ to denote a uniformly distributed random number from the (0,1) interval. Superscripts denote different vectors, while subscripts denote vector components. Parentheses are used to denote iteration numbers. $x_n^i(k)$ for example denotes the n th component of the ith vector in k th iteration.