Document Sample

Proceedings of The National Conference On Undergraduate Research (NCUR) 2006 The University of North Carolina at Asheville Asheville, North Carolina April 6 – 8, 2006 A Heuristic Rule for Varying Exploration and Convergence in Iterated Genetic Algorithms Bryan Culbertson, Mark Kokoska Department of Computer Science Lafayette College 1 Markle Hall Easton, PA 18042. USA Faculty Advisor: Chun Wai Liew Abstract Current techniques used by most genetic algorithms do not effectively manage real world optimization problems. Our approach, however, applies an iterative genetic algorithm that uses new patterns of convergence and exploration to quickly converge on a suboptimal but satisfactory (satisficing) answer, and then explore more possible solutions. We control this process of convergence and exploration by altering the number of individuals transferred from each generation. The number of individuals that we reseed varies inversely with the number of iterations that have occurred. In such a way, exploration increases as more iterations are performed. We examined the performance of our algorithm by optimizing two biological problems each with complex search spaces. The first such model involves optimizing eight parameters that describe the angles of various bones within a snake's jaw to find the maximum cross sectional area of the jaw's gap. The second model involves determining sixteen parameter values that minimize the difference between the calculated swimming motion of a sunfish generated by the model and the actual motion deduced from video of the modeled sunfish swimming in a flume. The difficulties in optimizing this model for accuracy include the size of the search space, the interdependence of variables, and the number of unusable points. These obstacles manifest themselves in an irregularly shaped landscape that standard optimization algorithms cannot easily navigate. Our algorithms patterns of exploration and convergence coupled with other mechanics for moving between one iteration and the next allow the iterated genetic algorithm to be more effective. Results indicate that our algorithm reaches a satisficing answer in 29% less time than previous algorithms and more consistently generates accurate solutions than previous algorithms. Our approach of converging before exploration could potentially be applied to any such sufficiently complex optimization problem. The method has the potential to be used to optimize a growing field of practical problems in real world domains and applications. Keywords: Rule, Exploration, Genetic Algorithms 1. Existing Work Genetic algorithms (GA) optimize a function using a controlled stochastic process that mimics the process of natural selection and evolution. Genetic algorithms are most useful when applied to problems that cannot be solved analytically and are too large to solve iteratively. While educated guesses can approximate workable values, each of the parameters for many problems must be precise in order to get accurate motion. Also, with such a large number of combinations, it is impractical to iteratively test a problem until good values are found. Complicated optimization problems have been shown to be best-solved using genetic algorithms 3. As attempts are made to solve more complex problems in chemistry, construction, aerodynamics and other areas it is often found that practical, real world, sufficing answer are needed in a reasonable amount of time. The complexity of these problems includes very high dimensionality, discontinuous search space and highly interdependent variables. Much of current efforts focus on improving GA’s to more effectively handle these various complex facets. These objectives are the hallmarks of current GA research that focus on trying to find better answers, more reliably, in less time. Any one of these goals is suitable for advancing current techniques. The danger in attempting to make a GA more streamlined is that it may converge to local minima or maxima rather than a global one. This provides a difficult balance for GA's to straddle, finding as good an answer as quickly as possible without becoming stuck in local optima. One approach for maximizing the function of a GA in a shorter amount of time is the Micro-GA 1. This GA uses very small populations, as low as five, in an attempt to converge quickly. Micro GA's have been demonstrated to be faster than traditional GA's but have not been extensively tested on the large complex domain problems that are becoming more pervasive today. The ultra small nature of the micro-GA may make it more likely to rapidly converge The existing work on GA's shows this trend in trying to explore more area in the search space, but still converge on an answer in reasonable time. The micro-GA is one approach that drives as hard as possible for convergence. The heuristic rules developed in this paper attempt to further optimize a GA to balance its rate of exploration and convergence. 2. Previous Work The heuristics in this paper are built on top of a Meta-Control Mechanism for GA's known as Iterated Genetic algorithms (IGA). The concept of an IGA is to run a standard GA in the normal way for short durations instead of an entire run, then in between each iteration select which individuals will be used to start the next iteration. In this way the IGA tries to control the flow of individuals from one generation, or iteration, to the next. A specific, IGA, GAITER 2 performs runs on each generation using an already optimized GA called GADO or Genetic Algorithm for Continuous Design Optimization 3. One of the primary strengths of GADO is its ability to rapidly converge. After each generation has occurred, GAITER clusters all of the individuals together into groups based on quality of the answers then redistributes a number of individuals from different clusters to the next generation combined with a number of random individuals. The number of random individuals in the Original GA varies based on a set-triggering rule. The original rule for varying the number of individuals to be reseeded is based on the occurrence of stagnation. If the population of one iteration changes less than then a set tolerance, normally two percent then increase the number of random individuals by twenty, with a maximum of all but twenty individuals randomly generated. In addition if there is a change of more than two percent twenty fewer random individuals are seeded to the next iteration. In this way GAITER attempts to alter its amount exploration and convergence based on the results it finds within the search space. In the original work GAITER found answers in as little as eight percent of the time of GADO. 3. Thesis The original GAITER was so successful because of its use of heuristic rules to guide the stochastic elements of the GA. The rules for clustering and the decision rule for how many random individuals to reseed into the next generation both provide more information to the GA enabling it to make more intelligent decisions about exploration and convergence. Offering increased performance in complicated search spaces, an algorithm called Negative Linear Reseeding (NLR) is a new decision rule for how many individuals to bring to the next generation that is based on additional information. In addition to the measure of merit of the preceding generation the algorithm reseeds based on the number of generations that have already occurred. Most genetic algorithms start out by exploring the search space, and after finding a promising possibility converging on that region of search space in order to refine the answer. In general, this principle works well in small dimensions because an algorithm can sample a significant portion of the search space before choosing a section to focus on. However, when a problem has a high number of dimensions there are too many trends to explore any significant portion of them. So the genetic algorithm has a low chance of converging on a global optima. Instead of following this standard search pattern, Negative Linear Reseeding follows an inverted scheme. NLR starts out by converging on every promising trend so that it can quickly discover how soon that path stagnates at a local optimum. This way the algorithm has a clearer idea of the trends in the search space. As time progresses, our new algorithm gradually explores more of the search space which is important since the algorithm has more of an opportunity to stagnate at the end of a run. NLR works well in high dimensional, complex search spaces where more exploration is needed to find better answers, particularly to break out of the ever-present local optima that the GA tends to become trapped in. If a condition indicating stagnation occurs (change less then two percent) Negative Linear Reseeding in accordance with equation (1) will add a random number of individuals inversely proportional to the number of runs 2058 that have occurred and the total number of individuals in the population. Else it will reseed zero random individuals. ReseedPop = MaxPopulation * ( (MaxGenerations+1-currentIteration) / MaxGenerations) ) (1) Following equation (1), the algorithm scales the number of individuals to be reseeded to encourage more exploration when it is more likely that stagnation at a local optima has occurred and more exploration is needed. 4. Experiments Negative Linear Reseeding algorithm was tested against the original GAITER in two problem domains. Both domains embody the difficult problems for GA's including high dimensionality and incomplete search spaces. The first experimental domain discovers the maximum gape of a snake’s jaw. The snake jaw problem involves eight dimensions that describe the angles between bones involved in describing the snake’s mouth. In this case the complexity of solving the problem is compounded by the fact that the variables are strongly related to each other and also some points are unrealizable, as a snake simply cannot move its jaw to certain positions. This sort of complexity is the most difficult for a GA to successfully navigate. In addition Negative Linear Reseeding was tested on a fluid model of a swimming fish. In this fitness function the parameters describe the motion of the fish as it interacts with vortex points within the water, then a comparison is done between the calculated fish model and a video of a fish to determine how close to the real world the models parameters are. This complex fluid dynamics model involves sixteen parameters to describe the actions of the fish. This even higher dimensionality problem is more complex and another good test of the robustness of our algorithms changes to GAITER's selection process. The snake algorithm was evaluated on four separate instances of test sets, for both GAITER and Negative Linear Reseeding. In each instance 252 different sets of conditions for the snake were tested. From these large numbers of instances we can form an accurate statistics with which to compare the average performance of GAITER versus Negative Linear Reseeding. The standard metric for comparison for this type of problem is to compare the number of evaluations it took to reach a set percentage of the maximum answer that was found by any algorithm for that particular test. Points of note for these tests is that Negative Linear Reseeding seems to be decreasing the number of evaluations by approximately two percent in our test data. In addition we note a twenty-two percent reduction in variance in Negative Linear Reseeding at the higher range of measures of merit (90, 95, 98) this seems to indicate that the Negative Linear Reseeding algorithm is more consistent than the original GAITER. It also seems prudent to examine this large number of test cases for the best answers that are found, as this is another goal of optimizing genetic algorithms. So we also examine the number of Best answers found by each of the instances of the two algorithms. The important information here is that it appears as if Negative Linear Reseeding is reaching that 100% answer more frequently and also when it reaches the answer it appears to reach it earlier then the GAITER runs. The 100% best measures of merit are obviously not vastly superior to the 98% that all instances are accessing but the fact that Negative Linear Reseeding does appear to be pushing further and faster is encouraging for our original hypothesis. Figure 1. NLR improvement over basic 2059 Results for the fish show that, on average, we are reaching 98% of the best solution 26% sooner than GAITER. While, at the same time it is taking 1% longer to reach 75% of the best solution. These results suggest two possible explanations. First is that the algorithm is performing like predicted. Because of our inverted search's lack of exploration in the beginning, it spends many of its early evaluations focusing on less than optimum sections of the search space. Hence its poor performance getting to middle solutions. However, later on in the run it can explore when exploration is more profitable. The second explanation is that exploration is superior to convergence so constant exploration should be performed in lieu of convergence. Further experiments that evaluated the advantage of the explanation show sores performance. Since the largest improvement came during the time when GAITER was only carrying over a few parameter sets, we performed a test that simulated this environment. When the process was making progress we carried over the all the parameter sets just like in the convergence to exploration idea. However, when the process was stagnating, we carried over only 5% of the parameter sets just like at the end of the convergence to exploration process. This provided for an environment of constant exploration. The results of this test showed very poor performance from the only exploration idea. This suggests that exploration at the end, but not always, is important to discovering good solutions, and it is even more important if preformed at the end of a run as opposed to the beginning. 5. Conclusion Our original hypothesis was that we could improve the performance of genetic algorithms with heuristic rules incorporating additional information. Additionally, we could improve genetic algorithms performance in complex search spaces by encouraging even more exploration in the form of more random individuals as the number of iterations increased. The experiments demonstrate a marked improvement in our algorithm compared to the original GAITER, this suggests both that Negative Linear Reseeding could be used to improve the performance of genetic algorithms, but more importantly it provides support for our hypothesis. This suggests that future research into the optimization of genetic algorithms should pursue the ideas of additional rules based on more information, and try to explore more in complex landscapes to reduce stagnation. 6. References 1. Microgenetic algorithms for stationary and nonstationary function optimization Krishnakumar, Kalmanje AA (Univ. of Alabama) Proc. SPIE Vol. 1196, p. 289-296, Intelligent Control and Adaptive Systems, Guillermo Rodriguez; Ed. (SPIE Homepage) 02/1990 2. Exploration or Convergence? Another Meta-Control Mechanism for GAs C.W. Liew and Mayank Lahiri Proceedings of Eighteenth International Florida AI Society Conference (FLAIRS), May 2005. 3. Khaled Rasheed. "GADO: A Genetic Algorithm for Continuous Design Optimization". Technical Report DCS- TR-352, Department of Computer Science, Rutgers University, New Brunswick, NJ, 1998. Ph.D. Thesis. 2060

DOCUMENT INFO

Shared By:

Categories:

Tags:
search space, simulated annealing, common lisp, local search, objective function, artificial intelligence, tabu search, the user, ftp site, genetic algorithms, the network, usage time, search algorithm, slow convergence, ant colony

Stats:

views: | 8 |

posted: | 2/12/2010 |

language: | English |

pages: | 4 |

OTHER DOCS BY bfb53718

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.