A Genetic Algorithm for Continuous Design Space Search
Khaled Rasheed
[email protected]
Haym Hirsh
[email protected]
Computer Science Department Rutgers University New Brunswick, NJ 08903 USA (908) 445{2001, FAX {5691
Andrew Gelsey
[email protected]
Abstract
Genetic algorithms (GAs) have been extensively used as a means for performing global optimization in a simple yet reliable manner. However, in some realistic engineering design optimization domains the simple, classical implementation of a GA based on binary encoding and bit mutation and crossover is often inecient and unable to reach the global optimum. In this paper we describe a GA for continuous designspace optimization that uses new GA operators and strategies tailored to the structure and properties of engineering design domains. Empirical results in the domains of supersonic transport aircraft and supersonic missile inlets demonstrate that the newly formulated GA can be signi cantly better than the classical GA in both eciency and reliability.
keywords: genetic algorithms, design optimization http://www.cs.rutgers.edu/shehata/papers.html
To appear in AI in Engineering
1 Introduction Genetic Algorithms (GAs) [Goldberg 1989] are search algorithms that mimic the behavior of natural selection. GAs attempt to nd the best solution to some problem (e.g., the maximum of a function) by generating a collection (\population") of potential solutions (\individuals") to the problem and through mutation and recombination (crossover) operations, better solutions are hopefully generated out of the current set of potential solutions. This process continues until an acceptably good solution is found. GAs have many advantages over other search techniques, including the ability to deal with qualitatively di erent types of domains, such as continuous variable domains, discrete or quantized variable domains, or mixed-type variable domains. The classical approach has been to convert all the variables involved in the search to the domain of binary integers and encode each individual in the solution population as a bit string. Subsequently, the mutation and recombination operations are bit manipulation operations. This representation has been successful in solving numerous problems in various domains of science and engineering. This paper studies the application of GAs in realistic continuous-variable engineering design domains. In such domains (such as the two explored in Section 3) a design is represented by a number of continuous design parameters, so that potential solutions are vectors (points) in a multidimensional vector space. Determining the quality (\ tness") of each point involves the use of a simulator that computes relevant physical properties of the artifact and summarizes them into a single gure of merit. One problem faced in the application of GAs to such problems is that the tness function can su er from a number of shortcomings. First, not all points in the space are legitimate designs | some points in the search space (\unevaluable points") cause the simulator to crash, and others (\infeasible points"), although evaluable by the simulator, do not correspond to physically realizable designs. (This can result, for example, from the use of a penalty term to handle design constraints.) Further, even for feasible points in the design space the simulator will often take a non-negligible amount of time, ranging from a fraction of a second to, in some cases, many days. Another problem is imposed by the structure of many design spaces. Consider, for example, the hypothetical two-dimensional design space depicted in Figure 1. In this example the classical crossover operation using two parent points that are both inside the evaluable (or the feasible) slab-like region may well produce two child points that are out of the evaluable (feasible) region. Further, when the crossover operation does not observe parameter boundaries (i.e., the crossover point does not fall on the boundary between the encodings of two parameters) the expected behavior will be even worse. More generally, when evaluable points form slab-like regions that are not axis parallel (as was the case in this work) the classical crossover operator may not yield satisfactory results [Wright 1990]. A nal problem is posed by the sparsity of the evaluable and feasible regions, which causes diculties for the classical mutation operator. Flipping a few bits at random is likely to generate an unevaluable or infeasible point, and thus it is not likely to perform any better than random search. This paper presents a GA speci cally tailored to realistic continuous-variable engineering design domains of this sort. First, each element of the population is represented by a vector of real-valued numbers. Crossover is performed by picking a point along the line connecting
Figure 1: Classical (Bit String) Crossover behavior two points. Mutation is performed by perturbing parameters values, with the amount of the perturbation decreasing over the course of the run. Finally, a point is only evaluated if it is not too close to a previously generated point that proved to be bad. The remainder of this paper rst presents a more detailed description of our GA. We then present a number of experiments concerning the use of our GA on two realistic engineering tasks. We conclude the paper with a discussion of related e orts and future work.
2 GA Architecture The foundation of our GA is the representation of individuals as vectors of real values. Each individual in the GA population represents a parametric description of an artifact, such as an aircraft or a missile, with each parameter taking on a value in some continuous interval. The tness of each individual is based on the sum of a proper measure of merit computed by a simulator (such as the takeo mass of an aircraft), and a penalty function if relevant (such as to impose legal limits on the permissible noise of an aircraft). Operators are applied to elements of the population via some selection scheme. Here selection was performed by
rank (rather than via the actual value of the tness function on each individual) because of the wide range of tness values caused by the use of a penalty function | rank selection prevents the rst discovered evaluable/feasible points from dominating the population. A steady state GA model is used, in which existing points in the population are replaced by newly generated points via some replacement strategy. The replacement strategy used here takes into consideration both the tness and the proximity of the points in the GA population, with the goal of selecting for replacement a point that both has low tness and is relatively close to the point being introduced. These aspects of our GA architecture are not original. What is innovative about our GA is our recombination operators and our policy for evaluating the tness of individuals. Our description of these comprises the remainder of this section.
2.1 Line Crossover
Line crossover (LC) may be viewed as a generalized version of the linear crossover operator introduced in [Wright 1990]. After two candidates for crossover have been selected, crossover is performed by joining a line between the two candidates and choosing a point at random along this line or its extensions from either side. The informal intuition behind this operator is that a line joining two points that represent relatively good designs is likely to contain points that represent good designs as well, and a line joining one point that represents a bad design to one point that represents a good design will contain points that represent good designs, either between the points or in the extensions. Thus, for example, in the case of search spaces whose evaluable regions have a slab structure (such as the one in Figure 2), the line joining two evaluable points is likely to remain in the evaluable region and may yield better points in the search space.
2.2 Guided Crossover
Guided Crossover (GC) is an operator used in the latter part of a search. It works as follows: 1. One candidate point is selected from the GA population using the normal selection rule (by rank) and called candidate1. 2. The second candidate point is also selected from the GA population but in a di erent way: for each point X in the GA population other than candidate1 a quantity Q(X,candidate1) is computed. Where (A) ? fitness(B )) Q(A; B ) = (fitness distance(A; B )2
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A choice for X that maximizes Q(X,candidate1) is taken to be candidate2. 3. candidate1 and candidate2 are swapped if necessary, to make candidate1 the point that has the higher tness among the two.
Figure 2: Line Crossover behavior 4. The result of the crossover is a point along the line joining candidate1 to candidate2 which is selected at random from the small region around candidate1 (the better point) as follows: Result = L candidate1 + (1 ? L) candidate2 where L is in the interval [0.9,1.1] The guided crossover operator is greedy and consequently it is used instead of line crossover only a small percentage of the time (in the current implementation it is used 10% of the time) and only in the second half of the search. The intuition behind guided crossover is that it endows the GA with a way to get very close to the optimum once it is already near it | an advantage usually claimed for gradientbased methods over GAs | without the costly computation of gradients using expensive simulation-based evaluations in high-dimension spaces.
2.3 Shrinking Window Mutation
Shrinking window mutation may be viewed as a simpli ed version of the Dynamic Mutation operator introduced in [Janikow and Michalewicz 1991]. Whenever line crossover is performed, the resulting point is not directly introduced to the GA population as a new individual. Instead, a mutation operation is done rst, in which the parameters composing the point are perturbed by a random amount whose amplitude is a function of the value of the parameter and the stage of the optimization. Thus, in the early stages of the optimization the perturbation is allowed to be relatively large, and as the number of iterations increases the amplitude of the changes decreases. The intuition behind this operator is to add the avor of simulated annealing to the search, and to help in exploring the search space by preventing premature clustering. Note that mutation is not used with guided crossover, since it is intended for use late in the search when elements of the population are already close to the optimum.
2.4 Screening Module
Evaluating an individual can be time consuming, and thus it can be bene cial to only select for evaluation points that seem promising. The screening module decides whether a point is likely to correspond to a good design without invoking any simulator to do this by extrapolating from points evaluated earlier in the search. In particular, the screening module uses a simple nearest-neighbor approach that maintains a relatively large random sample of the points encountered in the search so far (in the current implementation it is 1000 points, which is about ten times the size of the GA population | the size of the sample should in general be selected based on the speed of the simulator and domain knowledge, if available). Before a candidate point generated by crossover and possibly mutation is evaluated, the module nds the nearest neighbor of the point among the sample. If this nearest neighbor has a tness that is better than some threshold, the point is evaluated and added to the GA population, otherwise the point is just discarded. A good choice of the threshold is very important for the success of the whole search. In the aircraft domain we have used the tness of the worst member of the current GA population as this threshold and in the missile inlet domain we used the tness of the second worst. In general, a more expensive evaluation function increases the need for the search to be more focused and therefore a higher threshold may be more appropriate.
3 Experimental Results To evaluate our GA we applied it to design problems in two domains, the conceptual design of supersonic transport aircraft, and the design of inlets for supersonic missile inlets. This section discusses our results in these two domains.
3.1 Supersonic Transport Aircraft Design
Our rst domain concerns the conceptual design of supersonic transport aircraft. We summarize it brie y here; it is described in more detail in [Gelsey et al. 1996]. Figure 3
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engineScale=1.53896 wing_area(m^2)=420.038 wing_aspect_ratio=1.4604 fuselage_taper_length(m)=37.4374 wing_t_over_c=0.0286792 fuel_annulus_width(m)=0.00418291 passenger_cabin_radius(m)=1.3716 structure_annulus_width(m)=0.3048 fuselage_mid_length(m)=27.432 fuselage_diameter(m)=3.36117 wing_sweep(rad)=1.22642 wing_root_chord(m)=33.9185 wing_span(m)=24.7674 v_tail_sweep(rad)=0.785398 v_tail_root_chord(m)=11.0671 v_tail_taper_ratio=0.33 v_tail_semi_span(m)=3.67982 v_tail_t_over_c=0.03 nacelle_length(m)=11.5824 nacelle_inlet_diameter(m)=0.85344 engine_diameter(m)=1.13751 wing_sweep_over_design_mach_angle=1.17114 wing_taper_ratio=1.33194e-06
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Figure 3: Supersonic transport aircraft designed by our system (dimensions in feet)
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Mach Altitude (ft.) Duration (min.s) comment 0.227 0 5 \takeo " 0.85 40,000 85 subsonic cruise (over land) 2.0 60,000 180 supersonic cruise (over ocean) capacity: 70 passengers. Figure 4: Mission speci cation for aircraft in Figure 3
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Figure 5: Axisymmetric convergent-divergent exhaust nozzle ( ow from left to right) shows a diagram of a typical airplane automatically designed by our software system to y the mission shown in Figure 4. The GA attempts to nd a good design for a particular mission by varying major aircraft parameters such as wing area, aspect ratio, and engine size. The GA evaluates candidate designs using a multidisciplinary simulator. In our current implementation, the GA's goal is to minimize the takeo mass of the aircraft, a measure of merit commonly used in the aircraft industry at the conceptual design stage. Takeo mass is the sum of fuel mass, which provides a rough approximation of the operating cost of the aircraft, and \dry" mass, which provides a rough approximation of the cost of building the aircraft. The simulator computes the takeo mass of a particular aircraft design for a particular mission as follows: 1. Compute \dry" mass using historical data to estimate the weight of the aircraft as a function of the design parameters and passenger capacity required for the mission. 2. Compute the landing mass m(t nal) which is the sum of the fuel reserve plus the \dry" mass. 3. Compute the takeo mass by numerically solving the di erential equation dm = f (m; t) dt which indicates that the rate at which the mass of the aircraft changes is equal to the rate of fuel consumption, which in turn is a function of the current mass of the aircraft and the current time in the mission. At each time step, the simulator's aerodynamic model is used to compute the current drag, and the simulator's propulsion model is used to compute the fuel consumption required to generate the thrust which will compensate for the current drag. A complete mission simulation requires about 1/5 second of CPU time on a DEC Alpha 250 4/266 desktop workstation. The aircraft simulation model used is based on both implicit and explicit assumptions and engineering approximations and since it is being used by a numerical optimizer rather than a
human domain expert, some design parameter sets may correspond to aircrafts that violate these assumptions and therefore may not be physically realizable even though the simulator does not detect this fact. For this reason a set of constraints has been introduced to safeguard the optimization process against such violations. In particular, a penalty function approach was used to incorporate the e ect of constraint violations into the optimization: the penalty function was added to the takeo mass value returned by the simulator and the resulting sum was the quantity that the optimizer actually minimizes (which therefore also serves as the tness value assigned to each point of the GA population). The speci c penalty function was simply a large constant multiplied by the sum of the amounts of constraint violation for all the violated constraints. The presence of constraints induced a partition of the search space into three mutually exclusive regions: 1. Unevaluable points: These are points that represent designs that violate the model assumptions so much that the simulator cannot complete the simulation process to produce any signi cant information. For such points a very large ctitious takeo mass is generated as the value of the objective function. 2. Infeasible evaluable points: These are points that represent unrealizable aircrafts but the type and extent of model violation is moderate enough for the simulator to complete its work and report the constraint violation information. As described above, a penalty function is added to the takeo mass returned by the simulator to account for the model violation. 3. Feasible points: The simulator succeeds in evaluating the take o mass for such points and no violations occur in the process. The penalty function for these points is zero.
3.1.1 Results In the rst experiment to test the e ectiveness of our GA we used an eleven-dimensional design space in which the optimizer varied the following aircraft conceptual design parameters over a continuous range of values: 1. exhaust nozzle convergent length 2. exhaust nozzle divergent length 3. exhaust nozzle external length 4. exhaust nozzle radius(r7) 5. engine size 6. wing area 7. wing aspect ratio 8. fuselage taper length (how \pointed" the fuselage is)
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Figure 6: Performance comparison of the Classical and Adapted GA in the Aircraft Design Domain 9. e ective structural thickness over chord (a nondimensionalized measure of wing thickness) 10. wing sweep over design mach angle (a nondimensionalized measure of wing sweep) 11. wing taper ratio (wing tip chord divided by wing root chord) Figure 5 illustrates the structure of the four bar linkage nozzle model used, which contributes the rst four parameters, and Figure 3 illustrates the rest of the parameters. The results of our experiments are shown in Figure 6. Five random populations of 100 points each were generated, and for each population the GA was allowed to proceed for 30000 iterations (an iteration denotes one call to the simulator, which takes, on average, 0.2 seconds). The graph plots percent deviation from the global optimum as the GA run progresses for each of the ve starting populations, with one set of curves corresponding to our new GA and a second corresponding to the traditional binary-encoding GA (bit-string representation for individuals, the classical bit-string crossover operator, the classical bit mutation operator, and no screening module), with both GAs using the same selection and replacement strategies described in the previous section.
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Figure 7: E ect of di erent adaptations in the Aircraft Design Domain The gure clearly demonstrates how the introduced modi cations strongly improved the GA performance. Note that since the evaluation function is the output of a numerical simulator we refer to the best design found throughout all optimization attempts as the global optimum, where in fact we have no formal proof that it is indeed global. This \global optimum" was found using a GA run of 100000 iterations. All ve runs of the classical implementation of the GA failed to get within 10% deviation from the global optimum (note that this is a minimization problem) and the average deviation for these runs was 12.72%. On the other hand, all ve runs of the adapted implementation of the GA were within 1% deviation from the global optimum and the average deviation for these runs was 0.61%. Therefore the steady state deviation from the global optimum was improved by a factor of 20. This superiority of the adapted GA is no surprise. As the line crossover operator is likely to produce child points that stay within the slab-like evaluable (or the feasible) region provided that their parent points were also in the evaluable (feasible) region, unlike the classical crossover operator which is more likely to produce child points outside the evaluable (feasible) region in this domain. Moreover, the screening module plays an important role in pruning the search and saving the time that would otherwise be lost in examining bad points, and the guided crossover operator does the job of ne tuning the optima.
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Figure 8: E ect of screening and guided crossover only in the Aircraft Design Domain To calibrate the results of our GA a collection of conventional optimization techniques, representative of most categories of optimization paradigms, were applied to this minimization problem, but they performed more poorly in this domain. The best conventional method was found to be a multistart CFSQP [Lawrence et al. 1995], a state-of-the-art implementation of the Sequential Quadratic Programming method, a quasi-Newton method that solves a nonlinear constrained optimization problem by tting a sequence of quadratic programs. 1000 multistarts of CFSQP with a total cost of 851,237 iterations failed to reach the global optimum found by 100,000 iterations of the adapted GA, and out of those 1000 multistarts only 26 got to within 1% deviation from the global optimum.
3.1.2 Analysis Our GA re ects a number of changes to the classical GA model, and we therefore conducted experiments to investigate the magnitude that each change contributes to the overall improvement. These experiments were conducted in a twelve dimension superspace of the previously described one, in which one more dimension was added (Fuel Annulus width) and a small change in the constraint functions was made. The space was about ten times as sparse and much more dicult to search. The results are summarized in Figure 7 where each curve represents the average of ve runs from di erent random starting populations of
Table 1: Fixed Parameters No. Parameter De nition 1 D cowl diameter 2 rf centerbody radius of constant cross section region 3 xg length of inlet for computation 4 xl length of inlet for computation (= xg ) 5 xn length of inlet for computation (= xg ) 6 rm external diameter 120 points each. The gure shows that each adaptation in isolation yields a more successful GA. The gure also suggests that LC contributed much more than GC and SM in this domain. However Figure 8 indicates that MS and GC, when added on top of classical crossover and mutation, considerably improved performance. SM made the search more ecient (the parts of the curves pertaining to the rst half of the search support this conclusion) and GC especially improved the steady state performance (the noticeable drop in the curve at the midpoint of the search, the point when GC comes to action, supports this conclusion).
3.2 Supersonic Missile Inlet Design
Our second domain concerns the design of inlets for supersonic and hypersonic jets. We sumarize it brie y here; it is described in more detail in [Zha et al. 1996]. For this problem we have access to two simulators, GASP and NIDA. GASP is a very robust but computationally expensive simulator which requires about one week per simulation. NIDA is a much faster but less robust simulator that usually takes less than 10 seconds per simulation. Because of the large di erence in run time between the two simulators, and to keep the research time within acceptable limits, most of the experiments and all of the optimization in this domain were based on the use of NIDA, although we use GASP as a \sanity check" on the nal results of optimizations using NIDA. NIDA was developed at United Technology Research Center (UTRC) as an inlet analysis/design tool [Haas et al. 1992]. It uses a 1D aerodynamic model with the method of characteristics for the supersonic part upstream of the throat, and empirical correlations based on experimental data downstream of the throat for the region of the terminal shock wave/turbulent boundary layer interaction and subsonic di user. Unfortunately, NIDA suffers from a number of serious shortcomings. There are numerous small discontinuities in the function it computes and in its rst derivative, and there are numerous unevaluable points that cause NIDA to crash or print an error message. These discontinuities are sometimes in the middle of regions of good designs. Figure 9 shows a cross section of the search space de ned by NIDA, which illustrates these problems. Each point along this curve is for a di erent value of the axial location of the throat, xd. The blank regions of the curve are unevaluable points. The gure clearly shows how dicult it is to do optimizations in this domain. The missile inlet in this design work is an axisymmetric mixed compression inlet which cruises at Mach 4 at 60000 feet altitude. Minimum manufacture cost for this inlet is the objective for optimization, and therefore, techniques such as boundary layer bleed and variable
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Table 2: Parameters to Optimize No. Parameter De nition 1 1 initial cone angle 2 2 nal cone angle 3 xd axial location of throat 4 rd radial location of throat 5 xe axial location of end of \constant" cross section 6 3 internal cowl lip angle 7 Hej height at end of constant cross section 8 Hfk height at beginning of constant internal cross section Note: All angles measured positive in counter-clockwise direction geometry are not used | the performance of the inlet thus relies solely on the aerodynamic design of the rigid geometry, such as the extent of external and internal compression, contraction ratio, inlet start throat area, throat location, shock train length, and divergence of subsonic di user. Figure 10 shows the model of the missile geometry which is composed of six xed parameters and eight design parameters given in Table 1 and Table 2, respectively. The missile inlet is axisymmetric and the coordinates are given in terms of axial (x) and radial (r) positions.
3.2.1 Results As in the aircraft domain, our experiments consist of runs with our GA and the classical binary-represented GA starting from ve random initial populations. In this domain each population consists of 80 random individuals. For each population the GA was allowed to proceed for 30000 iterations (an iteration denotes one call to the simulator, which, on average, takes six seconds). The results are shown in Figure 11. The gure clearly demonstrates how the adaptation hastened the convergence of all ve runs to the global optimum region. With the new GA all ve runs got within a little over 2% from the global optimum in at most 15000 iterations whereas the runs with the classical GA were in the vicinity of 4% away from the global optimum given the same number of iterations. we compared our GA to two other optimization methods in this domain, CFSQP and Random Probes. The fact that the evaluation function is very expensive made it impractical to do a statistical comparison. An anecdotal comparison is illustrated in Figure 12. The gure shows the result of running the new GA, CFSQP and Random Probes two times each. Each run was allowed to proceed for about 50 CPU hours. The gure shows that the new GA did better than the other two methods.
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Figure 11: Performance comparison of the Classical and Adapted GA in the Missile Inlet Design Domain
3.2.2 Analysis
To investigate the magnitude of each of our GA componenets we conducted further experiments analogous to those in the aircraft domain. The results are summarized in Figure 13, where each curve represents the average of ve runs from di erent random starting populations. The gure shows that each element of our GA imparts some improvement on its own. The gure also suggests that SM contributed much more than GC in this domain. However Figure 14 indicates that GC, when added on top of classical crossover and mutation and SM, considerably improved the steady state performance (the noticeable drop in the curve at the midpoint of the search, the point when GC comes to action, supports this conclusion).
4 Final Remarks This paper has presented a new GA tailored to continuous design space optimization. Key ideas include encoding each individual in the population by a vector of real-valued numbers, performing crossover on two individuals by selecting a point along the line connecting them, performing mutation by perturbing parameters values (with the amount of the perturbation decreasing over the course of the run), and screening generated points so that only those
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Figure 12: Performance comparison of the optimization methods used in the Missile Inlet Design Domain that are near previously seen good points are actually evaluated. Experimental results demonstrated the superiority of our GA over the classical GA in the domains of aircraft design optimization and missile inlet design. Although a great deal of work has been done in the area of numerical optimization algorithms [Gill et al. 1981, Vanderplaats 1984, Peressini et al. 1988, More and Wright 1993, Papalambros and Wilde 1988], not much has been published about the particular diculties of attempting to optimize functions de ned by large \real-world" numerical simulators. A number of research e orts have combined AI techniques with numerical optimization [Tong et al. 1992, Powell 1990, Powell and Skolnick 1993, Bouchard et al. 1988, Bouchard 1992, Sobieszczanski-Sobieski et al. 1985, Agogino and Almgren 1987, Williams and Cagan 1994, Hoeltzel and Chieng 1987, Cerbone 1992], and although a GA was used in some of these e orts [Powell 1990, Bouchard 1992, Powell and Skolnick 1993], this typically represented the use of an o -the-shelf GA, with no attempt to adapt it to the domain as done in this research. Our GA has demonstrated superior reliability as a global optimizer with a high potential for e ectively searching a design search space in a reasonable amount of time. However, the GA based search can be further improved in many ways. This includes automating the adjustment of GA parameters (such as population size, mutation rate, etc.) along the
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Figure 13: E ect of di erent adaptations in the Missile Inlet Design Domain way as the optimization progresses, complementing the screening module in speeding up the search and making it more reliable. We also plan to explore the use of more sophisticated machine-learning techniques to extrapolate from past evaluations as part of the screening module.
Acknowledgments
We thank our aircraft design expert, Gene Bouchard of Lockheed, for his invaluable assistance in this research. We also thank all members of the HPCD project, especially Donald Smith, Keith Miyake, and Mark Schwabacher. This research was partially supported by NASA under grant NAG2-817 and is also part of the Rutgers-based HPCD (Hypercomputing and Design) project supported by the Advanced Research Projects Agency of the Department of Defense through contract ARPA-DABT 63-93-C-0064.
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