Optimal Design Using Chaotic Descent Method

Vojin Jovanovic Design & Manufacturing Institute, Stevens Institute of Technology, Hoboken, NJ 07030 e-mail: [email protected]

Kazem Kazerounian Professor, Dept. of Mech. Eng., University of Connecticut, Storrs, CT 06269 e-mail: [email protected]

1

Optimal Design Using Chaotic Descent Method This paper describes a method for searching of global minima in design optimization problems. The method is applicable to any general nonlinear function. It is based on utilizing sensitive fractal areas to locate all of the solutions along one direction in a variable space. The search begins from an arbitrary chosen point in the variable space and descends towards a better design along a randomly chosen direction. Descent depends on finding points that belong to a fractal set which can be used to locate all of the solutions along that direction. The process is repeated until optimal design is obtained. To examine the behavior of the algorithm appropriate examples were selected and results discussed. 关S1050-0472共00兲00703-0兴

Introduction

Finding global solutions to a nonlinear optimization problem is a difficult task still not resolved to a satisfactory level by the general theory of optimization. The problem is difficult due to the fact that, in general, the behavior of the objective function throughout the space is unknown. The existing nonlinear programming methods are based on exploiting the local properties of the objective function and converging to a solution that in many cases is only a local minimum. However, several local minima may exist at which the corresponding function values may differ substantially. Most of the time we can only find the function’s local characteristics which is the reason for many well developed local optimization methods 关1,2兴. Only in some cases where the behavior of the function is known, i.e., convex, quadratic function, the chance of finding the global optima is greatly enhanced. Situations where practical results may be achieved are described in Wilde 关3兴. The problem of designing algorithms that obtain global solutions is difficult since in general there is no a local criterion for deciding whether a local solution is global. During the past three decades on the subject of global optimization or related problems a variety of deterministic and stochastic methods for finding global solutions have been developed. Deterministic methods include enumerative techniques, cutting planes, branch and bound, decomposition based approaches, bilinear programming, interval analysis, interior point methods, and approximate algorithms for large-scale problems. Stochastic methods include simulated annealing, pure random search techniques, and the clustering method. Most of these methods, due to the problem’s intrinsic complexities, are based on iterative numerical techniques that are all tuned up for fastest possible convergence and satisfaction of different criteria. However, it appears that the iteration itself, as a dynamic iterative process, has been neglected in the optimization literature even though the iteration is in the heart of nonlinear programming. In Jovanovic and Kazerounian 关4,5兴 and Jovanovic 关6兴 it was reported that the process is very much dependent on the sensitivity of numerical computations which was shown to be useful. The analysis was based on the theory of iteration of analytic functions developed by mathematicians Julia 关7兴 and Fatou 关8兴 motivated by a one page paper by Cayley 关9兴. In this paper, we will show how this important theory can be utilized in search for global minima even though the search relies only on local characteristics of the objective function. We will demonstrate how to utilize sensitive fractal regions to iterate towards global minima. Some of the concepts from the theory of iterations of functions Contributed by the Design Automation Committee for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received May 1998. Associate Technical Editor: M. A. Ganter.

Journal of Mechanical Design

will be introduced with the assistance of computer graphics which is one reason for recent popularity of this theory. Throughout the paper the analyticity of the functions 共satisfaction of CauchyRiemann equations兲 will be emphasized since this is the main reason for obtaining some important results about iterations. Presumably, readers will be familiar with complex numbers as building blocks of analytic functions. The chaotic descent method and all the reasoning used to develop it will be presented. The theoretical discussion will be supported with numerical experiments. However, before proceeding further, we would like to emphasize that no method today can guarantee finding global minima for the reasons described above; therefore, the method described here is not an exception. One can be assured in success only if the exhaustive search in the whole feasible domain is performed. The exhaustive search is usually unacceptable for obvious reasons, but here is where the method is aiming to bring an improvement. The procedure attempts to perform the exhaustive search in an efficient manner therefore changing a priori negative connotation of ‘‘exhaustive.’’

2

Iteration of Complex Analytic Functions

As a first step, it is necessary to define the notion of iteration of a complex analytic function f (z) where z is a complex number. The iteration is a repeated application of function f on a complex variable z. Specifically, if we select a point z 0 in the complex plane C and apply f repeatedly on z 0 , we are constructing, in turn, the points z 0 , z 1 ⫽ f (z 0 ), z 2 ⫽ f (z 1 ) . . . . In this fashion iteration can be viewed simply as a feedback process for which behavior depends on the operator f . When this operator is linear, it is quite easy to predict the outcome of the iteration process. However, the situation is quite different when f is a nonlinear operator. In these instances, the outcome of iterations is unpredictable and its study reveals quite intricate properties. We begin with the simplest possible nonlinear operator f (z) ⫽z 2 . There are two possible outcomes of its iteration process. Namely, with repeated application of f (z) starting with a point z 0 inside of the unit circle 兵 z: 兩 z 兩 ⫽1 其 , the iteration converges to 0. When starting with a point outside of the unit circle, the iteration converges to infinity. These two final states of the process are called the fixed points of f because f (0)⫽0 and f (⬁)⫽⬁. In general, fixed points are those z that satisfy f (z)⫽z while periodic points are z that satisfy f n (z)⫽z where f n represents n-fold composition of f , i.e. f o f o . . . f (z). Therefore, f n (z) is the nth iterate f ( f ( f ( . . . ( f (z))) of z. The least positive integer n for which f n (z)⫽z is called the period of the periodic orbit, where, given z 0 that belongs to C, the orbit of z 0 under f is the sequence of points z 0 , z 1 , z 2 , . . . where z n ⫽ f (z n⫺1 ) for n⫽1,2,3, . . . . Furthermore, the fixed point z * is

Copyright © 2000 by ASME

SEPTEMBER 2000, Vol. 122 Õ 265

1 2 3 4

attractive if 0⬍ 兩 f ⬘ (z * ) 兩 ⬍1, superattractive if f ⬘ (z * )⫽0, repelling if 兩 f ⬘ (z * ) 兩 ⬎1 neutral if 兩 f ⬘ (z * ) 兩 ⫽1.

The ‘‘ ⬘ ’’ above is the notation for a derivative of f . Periodic points of period n are similarly classified by replacing f with f n in the above definition. The reason for this terminology is as follows: if z 0 is an attracting or superattracting fixed point, then there is an open neighborhood U of z 0 having the property that f n (z)→z 0 as n→⬁ for each z in U. The set of all points whose orbits converge to z 0 is called the basin of attraction of z 0 . In view of these definitions, using our example f (z)⫽z 2 we can classify the two fixed points 0 and ⬁ as superattracting. The fixed point z 0 ⫽1 is repelling. This point belongs to the unit circle for which it is interesting to examine the dynamics of f . The circle has the striking property that it is both forward and backward invariant under f . That is, each point of the circle has its entire history and future lying on it. It can be shown that all the points on the unit circle are repelling periodic points. The closure of all repelling points is called Julia set and denoted with J( f ). The complement of the Julia set is called Fatou set or stable set and it is denoted with F( f ). We will adopt the following definition of a chaotic system: a completely chaotic system must exhibit unpredictability, indecomposability, and recurrence. A dynamical system is unpredictable if it exhibits sensitive dependence on initial conditions, i.e., given any initial state z 0 , there must be a nearby state w 0 whose orbit diverges from that of z 0 . The dynamical system is indecomposable if there is an orbit that eventually enters any pre-assigned region, no matter how small, in the plane. Finally, a dynamical system exhibits recurrence if, given an initial condition z 0 , there is another initial condition w 0 arbitrarily close to z 0 that is periodic. Dynamics of f on the unit circle are chaotic.

3

Complex Newton-Raphson Method

Perhaps the most widely used of all root-locating formulas is the Newton-Raphson method. The method can be derived from the Taylor series expansion and is represented by Eq. 共1兲 x n⫹1 ⫽x n ⫺

f 共xn兲 . f ⬘共 x n 兲

(1)

Geometrically, this formula gives an estimate of a root of a nonlinear function f . We take its linear approximation at some initial point x n and solve for an intersection with the x-axis. The intersection x n⫹1 we use later as a new initial point for a new linear approximation. Eventually, if we are ‘‘sufficiently close’’ to a root of the original function, the iteration will settle in that root. This process brings very rapid convergence of the method but also creates problems widely stated in numerical textbooks. Experiments show that the Newton-Raphson 共NR兲 has the tendency to oscillate around a local maximum or minimum where such oscillations may persist. Also, it is possible that for an initial guess that is close to one root, the iteration can jump to a location several roots away. This tendency to move away from the area of interest is due to the fact that near zero slopes are encountered. However, in view of the previous section, it is possible to make sense of this behavior. Additionally, these problems may actually be of help when one searches for all the roots to a nonlinear equation. Let us begin with a simple example. It is required to solve equation. f 共 x 兲 ⫽x 3 ⫺x⫽0

(2)

For this simple function there are three solutions x 1 ⫽1, x 2 ⫽1, and x 3 ⫽0. They can be obtained by using the NR method to construct x n⫹1 ⫽x n ⫺(x 3n ⫺x n )/(3x 2n ⫺1). Now, instead of considering only real numbers, we can substitute z for x to obtain z n⫹1 ⫽z n ⫺(z 3n ⫺z n )/(3z 2n ⫺1). Initial points for the iteration can be chosen anywhere in the complex plane. The NR method can be 266 Õ Vol. 122, SEPTEMBER 2000

Fig. 1 Basin of attraction for NR method for f „ z …Ä z 3 À z

viewed as an iteration of analytic function N(z n )⫽z n ⫺(z 3n ⫺z n )/(3z 2n ⫺1). For a better understanding of the iterative process of this function a picture of its basins of attraction is depicted in Fig. 1. The horizontal and vertical axes in the figure are the real and imaginary axes of the complex plane respectively. The three shades of gray represent the basins of attraction of three roots of the equation that we want to solve z 1 ⫽1, z 2 ⫽⫺1, and z 3 ⫽0. Note the ragged boundary between each of the basins of attraction and all three shades of gray on it. The boundary represents the points that belong to the Julia set of N. Zooming into the boundary in the vicinity of a point that belongs to it will reveal a snowflake picture of self-similar regions of the whole domain. In fact, no matter how close we zoom in we ‘‘see’’ the whole complex plane throughout the ‘‘iterative eyes’’ of the function N. This is a consequence of a very special property that all complex functions must satisfy; the satisfaction of CauchyRiemann equations. As a result, there exists a theorem known as Montel’s theorem which states that the iterates of a complex analytic function f starting from any neighborhood U of a point p in the complex plane must cover the whole complex plane, with a possible exception of one point, when the family of iterates of f fails to be normal at p 关10兴. The event of omitting a point rarely occurs. It can be shown that if p belongs to the Julia set of the function, the consequence of the Montel’s theorem will hold. Applied to our example above, the iterates of N must cover the whole complex plane without missing any of the roots of the equation that we desire to solve. Moreover, the pictures of basins of attraction we see will assume fractal shapes as it was demonstrated in Fig. 1. Therefore, if one desires to solve an equation for all the roots, it appears beneficial to locate a point that belongs to the Julia set and proceed with iterates of any neighborhood of that point. In Jovanovic and Kazerounian 关4,5兴 we described a method that determines such a point. More detailed discussion with a shortcut can be found in Jovanovic 关6兴. Once the point is located we know that we will be ‘‘in the iterative sense’’ infinitely close to all the roots of the equation. Another important property of the Julia set should be noted; a point on the boundary of one of the domains of attraction must be on the boundary of all of them. Mathematically, this statement is represented with Eq. 共3兲. Transactions of the ASME

J共 N 兲 ⫽ ⳵ A 共 z 1 兲 ⫽ . . . ⫽ ⳵ A 共 z n 兲

(3)

where ⳵ A is the boundary of the domain of attraction of root z n . This result, a consequence of the Montel’s theorem, shows that the Julia set contains all the ‘‘gates’’ to all the roots of the particular equation. This is a very important finding which enables the application of the theory discussed in this paper. The above exposition demonstrates that the work of Julia, Fatou and Montel has made the Newton-Raphson method a more powerful tool than it was previously realized.

4

Chaotic Descent

Many engineering problems could be transformed into a problem with an objective function that needs to be minimized with or without constraints. Such a problem is a central one in the theory of optimization. In fact a goal of engineering design is to achieve an optimal solution based on the criteria known at a particular instance. To achieve such an objective, a number of numerical procedures exist that are able to locate optimal solutions. However, locating global optimal solutions remains a formidable task. In the literature a number of methods appeared to address the problem. Among promising ones are genetic algorithms and simulated annealing. The content of this paper is an attempt to contribute to this difficult problem using ideas presented in the previous section. The following discussion is a short introduction to what was developed in Jovanovic 关6兴. To describe the main outline of a procedure based on Julia sets that can efficiently search for optimal design, let us start with the following general problem. It is required to locate an optimal design of some objective function F(x 1 ,x 2 ) where x 1 and x 2 are optimization variables. A two-variable problem is convenient to graphically describe the procedure, but the treatment is the same for a n-variable problem. To solve this problem with existing optimization methods one would use a local optimizer 共conjugate gradient method for example兲 to obtain most likely a local minimum. In order to obtain a global solution one needs to exhaustively search the whole feasible space. This however can be a source of difficulty since quite often the feasible space is not known in advance. The exhaustive search is therefore unavoidable since the knowledge of the objective function is usually quantitative 共only local characteristics are known兲 as opposed to qualitative ones 共global behavior兲. Due to this fact many exhaustive procedures where proposed in order to efficiently search the variable space. The one described here is an attempt to contribute to this body of work. Its uniqueness is based on the ability to locate all of the roots of a nonlinear equation via sensitive regions. To begin let us imagine a representation of the function in a three dimensional space x 1 , x 2 , and x 3 where x 3 ⫽F(x 1 ,x 2 ). F certainly consists of hills and valleys, some of which are our global minima. To aid in the discussion an illustration of a contour plot of F in x 1 x 2 plane is depicted in Fig. 2. The axis x 3 is perpendicular to the plane of the figure. The curves in Fig. 2 are equicostal lines of F. Let us now pick a point in the x 1 x 2 plane, label it A, and draw a circle around it. We then select a point B on the circle and connect A and B to form a direction in x 1 x 2 plane. Now, imagine a plane that is normal to the plane of the figure that contains the direction AB. The plane cuts through F(x 1 ,x 2 ) and is folded into the plane of the figure so that its content can be observed. As it can be seen, the plane contains a one-dimensional curve F(xជ ⫹ ␣ ⌬xជ ) which is a function that depends only on distance ␣ measured from A. Since this is a one-dimensional function, we are able to find all of its roots by using the Newton-Raphson method as it was elaborated in the previous section. We take the derivative of F ␣⬘ (xជ ⫹ ␣ ⌬xជ ) and solve equation F ␣⬘ ( ␣ )⫽0 for all of its roots ␣ i . Based on the discussion in the previous sections, we can locate all of the ␣ i , However, we are only interested in real ones and those that represent the minima of F. By examining the second derivative at Journal of Mechanical Design

Fig. 2 One-dimensional search through the variable space

real ␣ i , we memorize only those that satisfy the minimum condition. We continue in the same fashion by drawing another cut through C, offset by ⌬ ␪ from B, and we repeat the procedure. Now, when ⌬ ␪ goes to 0 and the initial angle ␪ 0 increases to ␪ 0 ⫹ ␲ we will cover the whole x 1 x 2 plane with our onedimensional cuts. Whatever set of minima we found along the one-dimensional cuts will contain all of the global minima. Essentially, we were searching for sensitive fractal areas along the lines in order to locate ␣ i . The procedure just described, even though it locates all of the global minima, is not practical since one would have to cut the variable space along infinitely many directions to locate all the global minima of F. To improve the practicality of the procedure let us further examine the plot of equicostal lines of F 共Fig. 3兲. Let us imagine at this point that we are trying to solve the problem of minimizing F with a classical method, the conjugate gradient 共CG兲 method for example. By choosing some initial point close to a minimum, CG will converge to that minimum. Assigning a color to all the initial points that lead to the minimum

Fig. 3 Cutting the basins of attraction formed by CG method

SEPTEMBER 2000, Vol. 122 Õ 267

would generate a basin of attraction. Besides this one there are most likely other basins of attraction to which we can assign different colors. Therefore, by starting the above procedure from some point to cut through F, we will intersect some of the basins of attraction. Seeing this we can do the following: when we cut through the variable space and locate all the stationary minima along one direction, we can supply these locations to the CG method and let it converge to a local minimum. With this possibility it should be clear that we do not have to have our ⌬ ␪ small because basins of attraction formed by CG are generally not small. However, the question remains-what ⌬ ␪ do we choose? One choice is to start searching with ⌬ ␪ ⫽180° and to continuously halve it to get to smaller ⌬ ␪ s. When do we stop? Simply when no new minima are encountered for a specified number of steps. Another more convenient choice is to randomly choose a direction in the ␲ segment, because random tries will uniformly distribute directions over the whole segment. This would be effective since one may find immediately a direction which contains basins of global minima, just through luck. To summarize, the following algorithm can be devised using the second choice above. We start from a point in the variable space and, along a randomly chosen direction by using NR together with Julia sets, locate a stationary point that leads to a best local minimum. Then, we proceed along a new random direction from that minimum to locate the next one. It should be clear that eventually this process must stop at a global minimum, because, as pointed out before, we are performing an exhaustive search. Due to its nature of descending via chaotic regions, the search was termed chaotic descent. We would like to emphasize that the process described above is strictly decreasing the value of F. We only allow one-dimensional cuts to proceed from those points at which F had a value smaller than in the previous step. Until then, we do nothing and just continue searching.

5

Numerical Experiments and Analysis

To test the algorithm we need an optimization problem that possesses the characteristics which frequently make searching for global minima a difficult task. Such difficulties are numerous local minima within the feasible region, and irregular boundaries of the feasible region. These difficulties are more than sufficient to seriously jeopardize success of locating the global minimum. The reason for this is usual nonlinearity of objective functions due to which all of the optimization procedures are necessarily numerical and can determine only the local behavior of the function. Therefore, to determine the global behavior of the objective function one has no choice but to exhaustively search the whole variable space. These are the main ingredients of any numerical procedure including the chaotic descent. However, this procedure searches exhaustively in a very efficient manner. The main problem in optimization is related to the uncertainty of the boundaries of the feasible region and the distribution of the local minima. This implies that very often it is unknown where to start the exhaustive search and how fine a distribution of initial points to generate. To add to the complexity, one is usually faced with a goal of optimizing a function that is highly nonlinear and subject to many equality and inequality constraints. A frequent technique in such situations is to use the penalty function approach that transfers the problem of optimizing a constrained function to that of optimizing an unconstrained one. The technique consists of adding an outside penalty term to the objective function which detracts from good performance, as measured by the objective function, when an associated constraint is violated. Usual choices for the penalty term are squared constraints themselves. This technique, however, does not remove the uncertainty of locations of feasible regions, but it makes it unimportant. Because, it permits one to start searching anywhere in the variable space without being concerned whether the starting location belongs to the feasible region or not. Nevertheless, the question 268 Õ Vol. 122, SEPTEMBER 2000

Fig. 4 3-D view of the objective function F „ x 1 , x 2 …

where to start the search remains unanswered. The chaotic descent does not attempt to answer this question, but it avoids it by permitting one to take one step further after the application of the penalty function technique. The method allows one to start efficient exhaustive searching anywhere without compromising the ability of finding global minima if such can be found. The reason is the method’s ability to bound and quickly determine all of the solutions along one line through the variable space. The adjective ‘‘efficient’’ is to be understood relative to this claim only which represents the essence of the chaotic descent. To quantify the success of the method’s ability of reaching a global minimum is not a trivial task. The emphasis is on the success of reaching an optimal solution in the presence of many local ones. We selected the following two problems which in our opinion provide ample of difficulties mentioned above. The first one consists of minimizing F 共 x 1 ,x 2 兲 ⫽1⫺cos共 x 1 兲 cos共 x 2 兲 ⫹.5 冑共 x 1 ⫺n 1 ␲ 兲 2 ⫹ 共 x 2 ⫺n 2 ␲ 兲 2 (4) subject to 共 x 21 ⫹x 22 兲 2 ⫺300 x 21 x 2 ⭐0.

The first part of F, 1⫺cos(x1) cos(x2), provides a wavy surface with many local minima 共exactly 441 in the feasible region兲, the other part is a cone located at one of the local minima making it a global minimum. The slope of the surface of the cone is .5. This function ensures that the chaotic descent will be exposed to many local minima. Note that local minima are located at 共n 1 ␲ , n 2 ␲兲 where n 1 and n 2 are pairs of even or odd integers. To show just how wavy this function is, its 3-D representation is depicted in Fig. 4. The algorithm from the previous section has been implemented in Turbo C⫹⫹ with double precision on 486 66MHz personal computer, Windows 95 platform. For a local minimizer, a very fast Fletcher-Reeves version of the CG method with Polak-Ribiere improvement was used. The objective function was implemented using the penalty function approach. The calculation of derivatives was done using a central difference approximation with a leading error of order h 2 . Table 1 shows the outcome of the algorithm starting from different initial positions in variable space. The global minimum was taken to be at 共7␲, 5␲兲. Where d is the number of the direction taken, l is the number of local minima evaluated, t l is the total time for evaluation of local minima, t is the total time of the algorithm and t n is the time per direction for locating a Julia point and all of the roots along that direction. All of the values in Table 1 were observed at the moment when the global minimum was found. The results in Table 1 show that the method is very fast for Transactions of the ASME

Table 1 Times for finding the global minimum of F

this particular function. In the two-dimensional case, it took on average 14 to 15 onedimensional cuts to locate the global minimum. The most time was spent on evaluating local minima, on average, a total of .65 seconds. Searching for the Julia points 共infinity in this case兲 and all the roots along one direction amounted to a total of .03 seconds per try. The speed of the whole routine was helped by the very fast Fletcher-Reeves local minimizer. It took no more than a total of 1.1 seconds to fall into the global minimum. Of course, a designer should keep in mind that since in practical problems the location of the global minimum is unknown, he or she should allow more time for the method to span as much space as possible. A decision for the termination of the algorithm solely depends on the designer since the chaotic descent is an exhaustive procedure as any other and it does not ensure finding a global minimum unless the whole variable space is spanned. However, as pointed out before, the spanning of the space is done in a very efficient manner. It should also be mentioned that throughout the testing there was no single occasion that the global minimum was not found. This certainly adds to the likelihood of finding a global minimum with this algorithm. The knowledge of the feasible region was not needed since a penalty function approach was used. One should also note that selected initial points for the algorithms were absolutely arbitrary. This is really the most important advantage of the chaotic descent. To see this, it is sufficient to try to solve this problem with the existing methods. In the general case, when no additional hints about the behavior of the objective function or constraints 共like convexity etc.兲 are known, these methods search the whole variable space, just as chaotic descent does. However, they need a search region to be specified in advance which in many nonlinear programming problems cannot be easily determined. Therefore, the question that remains is what region to cover. One can start from a point in space and try to cover the space in expanding circles with some finite grid until everything is covered. But this space is infinite and without knowing the stopping criterion, this is a very inefficient way to search which does not have a good chance of success. It is also a time-consuming process. For example, in the two-dimensional case for above function, the time per local minimization was about .003 seconds. Therefore, the total time for the search will be .003 seconds multiplied by the number of points we intend to examine in a region which we do not know if it contains any minima and with a grid which we do not know if it is fine enough. Therefore, the total time could be very large without increasing the likelihood of finding the optimal solution. To test the algorithm in design we selected a four-bar linkage five-precision-point problem with prescribed timing. For this problem we are in position to know what the global solution is and therefore monitor the performance of the algorithm. The formulation of the problem was adopted from Paradis and Willmert 关11兴. Its mathematical description follows. The problem consists of designing a four-bar mechanism that generates a coupler curve with five prescribed positions. The desired coupler curve to be generated is shown in Fig. 6 while five prescribed states are shown in Table 2. The problem can be transformed to that of optimization. The objective function is the sum of the squares of x and y differences between the given coupler curve and the one generated by the mechanism 共Eq. 共5兲兲. The Journal of Mechanical Design

Fig. 6 Desired global solution

selected constraints force the mechanism to be a crankrocker with the link of length x 2 described as the crank. Minimize q

F共 X 兲⫽

兺 共 XX ⫺XG 兲 ⫹ 共 Y Y ⫺Y G 兲 i

i⫽1

i

2

i

i

2

(5)

where XX i ⫽x 2 cos ␥ i ⫹x 5 cos ␺ i ⫹x 7 Y Y i ⫽x 2 sin ␥ i ⫹x 5 sin ␺ i ⫹x 6

␺ i ⫽x 8 ⫹x 9 ⫹ ␩ i ⫺␧ i ␧ i ⫽tan⫺1

␩ i ⫽cos⫺1

x 2 sin共 ␥ i ⫺x 8 兲 x 1 ⫺x 2 cos共 ␥ i ⫺x 8 兲

x 21 ⫹x 22 ⫹x 23 ⫺x 24 ⫺2x 1 x 2 cos共 ␥ i ⫺x 8 兲 2x 3 冑x 21 ⫹x 22 ⫺2x 1 x 2 cos共 ␥ i ⫺x 8 兲

␥ i ⫽x 10⫹⌬ ␥ i

i⫽1,2, . . . 5.

subject to constraints ⫺x i ⭐0 i⫽1, . . . 5 x 2 ⫺x i ⭐0 i⫽1,3,4 x 1 ⫹x 2 ⫺ 共 x 3 ⫹x 4 兲 .75⭐0 x 2 ⫹x 3 ⫺ 共 x 1 ⫹x 4 兲 .75⭐0 x 2 ⫹x 4 ⫺ 共 x 1 ⫹x 3 兲 .75⭐0 This optimization problem consists of 10 variables for whose geometrical meaning is depicted in Fig. 5. x 10 is the angle variable corresponding to the first position of the crank angle. This optimization function has many local minima which makes it suitable for the testing of the procedure. The outcome of several runs of the algorithm is shown in Table 3 where d, l, t l state and t are as explained before. The results are Table 2 Given points on desired curve

SEPTEMBER 2000, Vol. 122 Õ 269

the algorithm is taking very long time to locate a global minimum it means that a current calculated position in variable space is very distant from the basins of attraction of global solution so that the probability of choosing the best descent direction is small. However, eventually the global solution must be found since, given enough time, the whole variable space will be spanned. For the examined problem the algorithm never failed to locate the solution within 25 minutes.

6

Fig. 5 Four-bar Table 3 Times for finding the global minimum of F

obtained on Pentium II Windows NT environment and represent a necessary number of iterations and needed time for the algorithm to reach the global solution for each run. The results show that the time needed to find a global solution significantly varied 共from about 120 seconds to 1532 seconds of CPU time兲 which validates the choice for the random direction approach while descending towards the global minima. The time is very much dependent on the probability of choosing the best direction in 10-dimensional variable space to pass through the basin of attraction of the global solution. For this function we speculate that, considering the trigonometric functions involved, basins of attraction are small patches in the space which are not easy to reach. Nevertheless, in this case the time needed for obtaining the global solution was reasonable which is the consequence of the algorithm’s ability to very quickly locate all of the solutions along one direction in variable space. Due to that fact the algorithm collapses the uncertainty of finding the global minima from n to n⫺1 dimensions. The ultimate goal would be to have no uncertainty in any dimension; whether this could be achieved in general it is difficult to say. However, even a partial reduction of uncertainty along only one dimension appears to improve chances of locating the global minima. Finally, the time of finding the global solution will always depend on the size of its basin of attraction. The improvement to the classical exhaustive procedure should be noticeable if comparison is attempted. At those instances where

270 Õ Vol. 122, SEPTEMBER 2000

Conclusion

In this paper we presented a method which enhances likelihood of locating global solutions in design optimization problems. The method incorporates a novel idea based on using sensitivity of numerical computations to locate all of the roots of the nonlinear equation. Specifically, we first locate a point that belongs to the Julia set and then iterate from its neighborhood using the NewtonRaphson method to locate all of the solutions to the equation. To make this procedure useful for multivariable problems, we span the variable space with one dimensional lines along which we try to locate better minima for a given objective function until we descent to a global one. The performance of the algorithm was observed on the optimization problems where the global solution was known in advance. We discussed the results in view of the practical value of the procedure. For further reports it remains to compare the procedure with existing methods and to establish its usefulness in locating global minima. Further research should be focused on collapsing uncertainty in more than one dimension which would certainly improve the speed of finding global minima. One approach to this problem is to find a way to apply the fractal conjecture given in Jovanovic 关6兴.

References 关1兴 Jacoby, S. L. S., Kowalik, J. S., and Pizzo, J. T., 1972, Iterative Methods for Nonlinear Optimization Problems, Prentice-Hall, Englewood Cliffs, NJ. 关2兴 Dennis, J. E., and Schnabel, R. B., 1996, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Society for Industrial & Applied Mathematics, Philadelphia, PA. 关3兴 Wilde, D. J., 1978, Globally Optimal Design, Wiley-Interscience, New York, NY. 关4兴 Jovanovic, V. T., and Kazerounian, K., 1998, ‘‘Using Chaos to Obtain Global Solutions in Computational Kinematics,’’ ASME J. Mech. Des., 120, No. 2, pp. 299–304. 关5兴 Jovanovic, V. T., and Kazerounian, K., 1998, ‘‘On the Metrics and Coordinate-SystemInduced Sensitivity.’’ Int. J. Numer. Methods Eng., 42, No. 4, pp. 729–747. 关6兴 Jovanovic, V., 1997, ‘‘Identifying, Utilizing and Improving Chaotic Numerical Instabilities in Computational Kinematics,’’ Ph.D. Thesis, The University of Connecticut, Storrs, CT. 关7兴 Julia, G., 1918, ‘‘Memoire sur l’iteration des fonctions rationelles,’’ J. Math. Pure Appl., 8, pp. 47–245. 关8兴 Fatou, M. P., 1919, ‘‘Sur les equations fonctionelles,’’ Bull. Soc. Math. Fr., 47, pp. 161–271. 关9兴 Cayley, A., 1879, ‘‘The Newton-Fourier Imaginary Problem,’’ Am. J. Math., 2, pp. 97. 关10兴 Devaney R. L., 1989, An Introduction to Chaotic Dynamical Systems, Perseus, a division of Harper/Collins, Reading, MA. 关11兴 Paradis, M. J., and Willmert, K. D., 1983, ‘‘Optimal Mechanism Design using the Gauss Constrained Method,’’ ASME J. Mech. Trans. Auto. Design, 105, pp. 187–196.

Transactions of the ASME