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Now that you are convinced (perhaps) that the binary GA solves many optimization problems that stump traditional techniques, let s look a bit closer at the quantization limitation What if you are attempting to solve a problem where the values of the variables are continuous and you want to know them to the full machine precision In such a problem each variable requires many bits to represent it If the number of variables is large, the size of the chromosome is also large Of course, 1s and 0s are not the only way to represent a variable One could, in principle, use any representation conceivable for encoding the variables When the variables are naturally quantized, the binary GA ts nicely However, when the variables are continuous, it is more logical to represent them by oating-point numbers.

likelihood allow large spatial estimation problems to be carried out in a reasonable amount of time. Because Bayesian models also rely on the likelihood, many of the computational advances described here for maximum likelihood estimation can also be applied to Bayesian estimation. In this connection it should be noted that it took only 342 seconds to produce 2500 draws for the problem involving 10,514 census block observations. This time is based on using the Barry and Pace (1999) Monte Carlo estimator for the log determinant as well as univariate numerical integration of the conditional distribution for on every pass through the sampler. This timing result is based on using MATLAB version 6.1 on a 1200-megahertz Athalon processor. Use of c-language functions that interface with MATLAB reduced this time to 228 seconds. Another issue discussed was the trade-off between computational speed and accuracy. One can analyze the impact of: (1) using approximations, (2) changing tolerances, and (3) placing reasonable restrictions or bounds on parameter values during optimization. Flexible software algorithms should allow users to change these aspects of the program so that exploratory work might be carried out using faster but less accurate versions of the algorithm. Public-domain algorithms for use with MATLAB that take this approach are available at <http://www. spatial-econometrics.com>, along with c-language functions that can be used in conjunction with MATLAB or as stand-alone functions.

The voltage at the ith port location can be determined from the field inside the cavity. This voltage must be identical with Vi Thus we can write Vi = hEz xi yi = h

In addition, since the binary GA has its precision limited by the binary representation of variables, using oating point numbers instead easily allows representation to the machine precision This continuous GA also has the advantage of requiring less storage than the binary GA because a single oating-point number represents the variable instead of Nbits integers The continuous GA is inherently faster than the binary GA, because the chromosomes do not have to be decoded prior to the evaluation of the cost function The purpose of this chapter is to introduce the continuous GA Most sources call this version of the GA a real-valued GA We use the term continuous rather than real-valued to avoid confusion between real and complex numbers The development here closely parallels the last chapter We primarily dwell upon the differences in the two algorithms The continuous example introduced in 1 is our primary example problem.

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This allows the reader to compare the continuous GA performance with the more traditional optimization algorithms introduced in 1..