The Use of Multi-Objective Genetic Algorithms for Accelerator and Light Source Optimization

The nonlinear effects are very important in development of new accelerators and synchrotron light sources. Nowadays they are one of the main factors limiting the achievement of the required facility parameters. In many cases in development of new accelerators the analytical estimations give very rough results and in some cases they don’t apply at all. Therefore, the best way to research and design accelerators is to use numerical simulation. Nevertheless, very often during complex physical process simulation (taking into account many nonlinear effects) the use of classical optimization methods is difficult and does not give the desired results. The article deals with the application of multi-objective optimization using genetic algorithms for accelerators and light sources. These algorithms allow both simple linear and complex nonlinear accelerator structures to be optimized with the same effectiveness when obtaining the required facility parameters. INTRODUCTION There are many methods of optimization. All of them can be divided into three groups: determinate, random (stochastic) and combined. Most accelerator and light source optimization problems can be attributed to combinatorial problems with many different quality solutions. An exhaustive search of all solutions or only subset of solutions is the main feature of combinatorial algorithms. To find the best solution directed, random and combined an exhaustive search of all possible problem variables is used. Therefore, the search of proper solutions often becomes the art. After all, very often if you want to optimize nonlinear multi-objective problem (for example – beam emittance minimization and dynamic aperture maximization) with many variable parameters and restrictions you will face serious difficulties (most rapid and effective optimization methods can’t be used, there are many local minima solutions, solving time is directly related to the number of variable parameters, etc.). One of the effective way to solve combinatorial problems within a reasonable time is the use of genetic algorithms. Genetic algorithms are heuristic search algorithms used to solve optimization problems by random selection, combining and modification of desired parameters using process like the biological evolution. Genetic algorithms as any other optimization algorithms have their own advantages and disadvantages. Their most important advantages may be said to be:  Any information about the fitness function behavior is not required.  Discontinuities of the fitness function don’t have a significant effect on optimization.  Methods are relatively stable to fall into local minima. Their most important disadvantages may be said to be:  Methods are inefficient for optimizing fitness functions which have a long calculation time.  A large number of parameters often turns «work with genetic algorithm» to «play with genetic algorithms».  In the case of simple fitness functions, genetic algorithms are slower than specialized optimization algorithms. Nowadays, genetic algorithms are powerful computing tool to solve different multidimensional multi-objective optimization problems. The use of genetic algorithms for accelerator and light source optimization allows to simplify and speed up the search of proper solutions. The common block diagram for optimization process using genetic algorithms is shown in Fig. 1. Figure 1: The block diagram of the optimization process using a genetic algorithm. DESCRIPTION OF OPTIMIZATION METHOD In general, all optimization problems can be divided into two groups. The first group contains only one fitness function optimization problems, the second one – at the same time two or more fitness function optimization problems. To solve the problems of each group it is advantageous to use a little different algorithms. One fitness function optimization problem is the simplest situation with easy-to-analyse results. These kinds of problems can be efficiently solved with the help of the differential evolution method [1] is well suited. ____________________________________________ *The reported study was funded by RFBR according to the research project No. 16-32-00336 о _а. yafomin@gmail.com WEPSB010 Proceedings of RuPAC2016, St. Petersburg, Russia ISBN 978-3-95450-181-6 376 C op yr ig ht © 20 17 C C -B Y3. 0 an d by th e re sp ec tiv e au th or s Particle dynamics, new methods of acceleration and cooling Differential evolution is used for multidimensional realvalued functions but does not use the gradient of the problem being optimized, which means differential evolution does not require for the optimization problem to be differentiable as is required by classic optimization methods such as gradient descent and quasi-newton methods. Differential evolution can therefore also be used on optimization problems that are noisy, not continuous, changing over time, etc. To solve two or more fitness functions optimization problems it is well suited another methods – the so-called multi-objective genetic algorithms. The aim of our multiobjective optimization problem is to find all possible tradeoffs among multiple objective functions that are usually conflicting. Since it is difficult to choose a single solution for a multi-objective optimization problem without iterative interaction with the decision maker, one general approach is to show the set of Pareto optimal solutions to the decision maker. Then one of the Pareto optimal solutions can be chosen depending on the preference. Pareto frontier and Pareto optimal solutions for example emittance and dynamic aperture optimization is shown in Fig 2. Blue dots – Pareto optimal (nondominated) solutions, red dots – other (dominated or not Pareto optimal) solutions. Figure 2: Pareto frontier. Blue dots – Pareto optimal solutions, red dots – other solutions. A multi-objective optimization is an evolution of conventional numerical or combinatorial optimization, therefore many existing methods could be applied to this general case which makes defining of fitness function the main issue needs to be resolved. In recent decades, a number of approaches was developed to do it. For solving accelerator lattice optimization problems only Step 2 (selection) and Step 5 (elitist strategy) in optimization process (see Fig. 1) are of particular interest. For selection we chose the Nondominated Sorting Genetic Algorithm [2] as one of the most powerful and faster algorithm up to date. As an alternative, we also tried to use simpler and faster Random Weights Genetic Algorithm [3], but results were not as good as with the previous one. In scenario when optimization is being done with smooth and continuous fitness function the genetic algorithms prove to be effective and everything works well. However, this is not the case when optimization needs to be performed for circular accelerators since its fitness function is not smooth, nor continuous, for which reason using of the genetic algorithms becomes quite problematic. Mostly because of a large amount of not periodic or incorrect solutions which means that no optimization is really accomplished. This problem can be successfully addressed through the use of elitist strategy. During every new generation after fitness functions evaluation a certain number of best solutions are selected as elite individuals. When mutation is done N solutions are randomly removed from the current population and replaced by solutions from elite individuals. This elite preserve strategy has an effect in keeping the variety of each population.