Bohachevsky, I.O., Johnson, M.E., and Stein, M.L. (1986): Generalized Simulated Annealing for Function Optimization. Technometrics 28(3), pp. 209--218.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....currently takes the form of traditional optimization algorithms; the terms discipline level design and subspace optimization can thus be used interchangeably in that sense. The optimization strategy used is a hybrid technique, described in detail in [6] in which the simulated annealing [9, 10] and generalized reduced gradient [11] methods are performed sequentially. During subspace optimization, response surface approximations are used in lieu of complete system analyses to provide information about nonlocal states, which is required to solve the system design problem at the discipline ....

I. O. Bohachevsky, M. E. Johnson, and M. L. Stein. Generalized Simulated Annealing for Function Optimization. Technometrics, 28(3), August 1986.

....a point recognized by the authors. Their FORTRAN code is in the public domain [76] 4.11. Military Optimal disbursement of resources is a common problem to large systems and is especially critical in defense. A study in optimal deployment of missile interceptors [77] used an SQ algorithm [78], permitting the acceptance criteria to get stricter as the temperature decreases, by multiplying the difference of saved and generated cost functions by the value of the saved cost function raised to an ad hoc power. Tracking problems, in air, on the seas, and under water, present optimization ....

I.O. Bohachevsky, M.E. Johnson, and M.L. Stein, Generalized simulated annealing for function optimization, Technometrics 28 (3), 209-217 (1986).

....In this way, they usually demand a large number of empirical studies to f me tune the annealing schedule parameters to specific problems or cost functions. Along with the above annealing schedules there are others aiming to provide practical adaptive properties as Generalized Simulated Annealing [20], Adaptive Simulated Annealing [21] or Natural Optimization [22] In spite of the arduose efforts performed in providing an efficient annealing schedule valid for a wide range of problems, it remains as an open question and much experimentation must still be performed. 4. Thermodynamic ....

M. E. Johnson I. O. Bohachevsky, M. L. Stein, "Generalized simulated annealing for function optimization", Technometrics, 28(3), 1986.

.... vector in n with components (y 1 ; y 2 ; yn ) The change in state is taken as a move in hyperspace to y Deltay, so that the change in energy is f(y Deltay) Gamma f (y) A number of methods by which to choose the direction and length of the step Deltay have been proposed [24] [25], 26] including a modification of the downhill simplex method of Nelder and Mead [27] A simplex is a geometrical figure of n 1 vertices in n dimensional space, for example a triangle in 2 dimensional space or a tetrahedron in 3 dimensional space. The simplex is initialized with a set of n ....

....in Appendix B. A problem occurs with the simplex algorithm when y is bounded, especially when the global function minimum is located on or very near to the boundary (as is often the case in continuous variables used in fuzzy models) The standard procedure when not using the simplex adaptation [25] is to simply reject any step that generates y outside its bounds. However, if the simplex adaptation is used and vertices that lie outside the bounds are rejected, then a global minimum that lies on a boundary will never be enclosed by the simplex and thus will not be successfully located. To ....

I.O. Bohachevsky, M.E. Johnson, and M.L. Stein, "Generalized simulated annealing for function optimization," Technometrics, vol. 28, no. 3, pp. 209--217, 1986.

....(GRG) method. It is a gradient based technique that is suitable for constrained optimization [8] The discrete optimization is performed by simulated annealing. The theory on which this method is founded, as well as its formulation and usefulness in discrete optimization, has been documented [9, 10]. The reasons it was selected for the mixed CSSO framework are related to certain characteristics of the method. The algorithm pos4 sesses the ability to escape local minima, increasing the likelihood that the global mimimum will be found. Conversely, the GRG method, like all gradient based ....

I. O. Bohachevsky, M. E. Johnson, and M. L. Stein. Generalized Simulated Annealing for Function Optimization. Technometrics, 28(3), August 1986.

....is an n dimensional vector in n with components y 1 y 2 aba y n . The change in state is taken as a move in hyperspace to y 6 Dy, so that the change in energy is f y Dy f y . A number of methods by which to choose the direction and length of the step Dy have been proposed [127, 16, 25], including a modification of the downhill simplex method of Nelder and Mead [88] A simplex is a geometrical figure of n 6 1 vertices in n dimensional space, for example a triangle in 2 dimensional space or a tetrahedron in 3 dimensional space. The simplex is initialised with a set of n 6 1 ....

....trial and error. A problem occurs with the simplex algorithm when y is bounded, especially when the global function minimum is located on or very near to the boundary (as is often the case in continuous variables used in fuzzy models) The standard procedure when not using the simplex adaptation [16] is to simply reject any step that generates y outside its bounds. However, if the simplex adaptation is used and vertices that lie outside the bounds are rejected, then a global minimum that lies on a boundary will never be enclosed by the simplex and thus will not be successfully located. To ....

I.O. Bohachevsky, M.E. Johnson, and M.L. Stein. Generalized simulated annealing for function optimization. Technometrics, 28(3):209--217, 1986.

....probability and the generation of random points to try to take into account the local structure of the objective function. Aluffi Pentini et al. [5] proposed a method to compute global minimizer(s) by following the paths of a system of stochastic differential equations. Bohachevsky et al. [7] presented a generalised simulated annealing algorithm in which the approach followed is basically that of a random direction method, in which, at each step, a random point is generated on the surface of a sphere with a pre fixed radius, centered on the current point. However, firstly these ....

M. E. Bohachevsky, M. E. Johnson and M. L. Stein, Generalized Simulated Annealing for Function Optimization, Technometrics, Vol.28, pp.209-217, 1986.

....= min x2X f(x) where the feasible region X ae R n is a continuous domain and the objective function f is a continuous function. There is a wide literature on practical and theoretical results about SA algorithms for continuous global optimization. From the practical point of view we cite [1] [3], 5] 7] 8] 10] 12] 15] 16] 22] 24] 25] 26] In particular, the most thoroughly investigated SA software is the ASA code introduced and developed in [12] 15] and available at the web site http: www.ingber.com #ASAREPRINTS. From the theoretical point of view convergence ....

I.O. Bohachevsky, M.E. Johnson, M.L. Stein, Generalized simulated annealing for function optimization, Technometrics, 28, 209-217 (1986)

....Michael C. Rohl Fachbereich Statistik Universitat Dortmund Claus Weihs Fachbereich Statistik Universitat Dortmund August 1999 Abstract We propose a computer intensive method for linear dimension reduction which minimizes the classification error directly. Simulated annealing (Bohachevsky et al. 1986) as a modern optimization technique is used to solve this problem effectively. This approach easily allows to incorporate user requests by means of penalty terms. Simulations demonstrate the superiority of optimal classification to classical discriminant analysis (McLachlan 1992) Special ....

....values are discrete. On the other hand one needs more function evaluations than common gradient algorithms. In physics it is well known that freezing and crystallizing of liquids overcomes local energy minima. This strategy serves as the prototype for a computer program: Simulated Annealing (Bohachevsky et al. 1986). To model the natural procedure, we need a configuration space (a discrete or continuous domain) a mechanism which describes how to get from one configuration to another and a cooling schedule describing how to decrease the temperature T (T 0 T 1 : Tn : In a concrete ....

I. O. Bohachevsky, M. E. Johnson, M. L. Stein, Generalized Simulated Annealing for Function Optimization, Technometrics, 28, 3, 209-217 (1986).

....Dimension Reduction Michael C. Rohl, Claus Weihs Lehrstuhl fur Computergestutzte Statistik, Universitat Dortmund, D 44221 Dortmund, Germany Abstract: We describe a computer intensive method for linear dimension reduction which minimizes the classification error directly. Simulated annealing (Bohachevsky et al. 1986)) is used to solve this problem. The classification error is determined by an exact integration. We avoid distance or scatter measures which are only surrogates to circumvent the classification error. Simulations (in two dimensions) and analytical approximations demonstrate the superiority of ....

....algorithms. The computerintensive method achieves minimal misclassification error if adequately implemented. 3.2 Simulated Annealing The freezing and crystallizing of liquids overcomes local energy minima. This physical strategy serves as the prototype for a computer program: Simulated Annealing (Bohachevsky (1986)) To model the natural procedure, we need a configuration space (a discrete or continous domain) a mechanism which describes how to get from one configuration to another and a cooling schedule describing how to decrease the temperature T (T 0 T 1 : T n : At each temperature ....

BOHACHEVSKY, I.O., JOHNSON, M.E., STEIN, M.L. (1986): Generalized Simulated Annealing for Function Optimization, Technometrics, 28, 209-217.

....probability, increasing energy will be accepted thus allowing a certain hill climbing . Originally, simulated annealing was proposed for problems from combinatorial optimization [10, 24] Today, a number of proposals for generalizing the idea to continuous global optimization are known (e.g. [1, 9, 15, 20, 21, 23]) Algorithm 1 is a rather informal description of simulated annealing. simulated annealing(x in , x out , T max , T min , k max ) x : x in ; T : T max ; while T # T min do for k = 1 to k max do x # : generate(x,T) if j(x # ) j(x) then x : x # elif rand y(j(x # ) j(x) then ....

....: cool(T ) endwhile x out : x; Algorithm 1: Sketch of a simulated annealing algorithm for minimizing j : D # R For a concrete realization, a number of ingredients must be discussed: 1. Acceptance criterion (rand y(j(x # ) j(x) 6 rand is a random number distributed uniformly over [0, 1]. As discussed above originally y(c) exp( c k B T ) was chosen [16] which is known as the Metropolis criterium today. But a number of other criteria can be used as well [22] 2. State generation (generate) For each x # D, a neighborhood D(x,T ) # D is defined. Sometimes it is given ....

Ihor O. Bohachevsky, Mark E. Johnson, and Myron L. Stein. Generalized simulated annealing for function optimization. Technometrics, 28(3):209--217, 1986.

....optimum decomposition of probabilistic networks and suggested control schedules, including the radius of the search space as an additional control parameter. Although the annealing algorithm has been successfully applied to provide high quality approximate solutions to many NP complete problems (Bohachevsky, Johnson Stein 1986, Geman Geman 1984, Lundy 1985, Thomas 1986) convergence of the algorithm in less than exponential time cannot be guaranteed for general problems (Lundy Mees 1986) The purpose of this paper, however, has been to investigate the applicability of the algorithm under bounded resources. The ....

Bohachevsky, I. O., Johnson, M. E. & Stein, M. L. (1986). Generalized simulated annealing for function optimization, Technometrics 28(3): 209--217.

....the group densities in the original d space. Then, the densities are projected on a d 0 dimensional subspace and the overall error rate is evaluated by integration. By means of direct minimization of the error rate we choose that subspace which minimizes the overall error. Simulated Annealing (Bohachevsky et al. 1986)) is used as the optimization algorithm. When the data set is small , we rely on LDA or DDA in the original space. But after the optimal subspace is selected and the data is projected into it, less parameters have to be estimated, and QDA is applied. Finally, the error rate is evaluated by leave ....

I. O. Bohachevsky, M. E. Johnson, M. L. Stein (1986), Generalized Simulated Annealing for Function Optimization, Technometrics, 28, 3, 209-217.

....Statistics, Rice University, Houston, TX. z National Center for Supercomputer Applications, University of Illinois, Champaign, IL. 2 A variety of methods have been proposed in the literature (see [10, 11] for descriptions of many of these methods) One technique is that of simulated annealing [2, 4, 6, 31] which is a stochastic search procedure based on the concept of annealing from chemical physics. The main advantages of this method is its ability to avoid getting trapped in local extrema in its search for a global solution. It also does not require calculation of derivatives, so it can be ....

I. O. Bohachevsky, M. E. Johnson, and M. L. Stein. Generalized simulated annealing for function optimization. Technometrics, 28:209--217, 1986.

.... points, and generate new sample points based on both constraint satisfaction and objective values of existing sample points [164] SA performs an adaptive stochastic search, and the acceptance rates of search points are determined by a combination of constraint satisfaction and objective value [32, 208,259]. Our new global search method handles constraints using the transformational approach. Constrained optimization problems are transformed into Lagrangian functions using Lagrange multipliers. Saddle points of Lagrangian functions are, then, searched using local and global search methods. 3.2 ....

I. O. Bohachevsky, M. E. Johnson, and M. L. Stein. Generalized simulated annealing for function optimization. Technometrics, 28:209--217, 1986.

....Randomly perturbing this state provides a random walk about the domain of the function. Using the Boltzmann distribution, the state can be progressed out of local minima such that the global minimum is located. The method has been successfully employed for a number of optimisation tasks [8, 15, 139]. It has also been shown to provide a regression technique for non linear least squares fitting problems [139] Press et al. 111] present a variation on traditional simulated annealing which is cited as more efficient than the methods given above. They propose the use of a geometric simplex of ....

I. O. Bohachevsky, M. E. Johnson, and M. L. Stein. Generalized simulated annealing for function optimization. Technometrics, 28(3):209--217, 1986.

....D optimum design is a complicated optimization problem. Most of numerical algorithms for construction of the exact D optimum designs such as 11 Fedorov [22] Gali and Kiefer [27] and Mitchell [28] are the local optimization algorithms. However, by using simulated annealing, Bohachevsky et al. [29] and Hains [30] constructed the exact D optimum designs for various models. If there is only one factor in model as discussed in the [1] the D optimum design with finite support points may be not very difficult to construct. Because of the nonlinearity of the model (7) with three factors, the ....

Bohachevsky, I.O., Jonson, M.E., and, Stein, M.L., Generalized simulated annealing for function op timization, Technometrics, 28 (1986), 209-217.

....By slowly decreasing the value of the distribution will approach the delta function at the minimizing value of . The clue then is to choose a temperature scheme where one avoids getting trapped in local minima for too many iterations, or where too many uphill climbs are accepted. In Bohachevsky et al. 1986) it was suggested that the temperature should be chosen so that p 2 [0:5; 0:9] Here also a generalized algorithm was proposed where p was calculated according to p = exp ffi(N [ Gamma N [ 1 ] N [ Gamma N min )g, where fi is approximately 3.5 and N min is an estimate of the normally ....

Bohachevsky, I. O., Johnson, M. E., and Stein, M. L. (1986), Generalized simulated annealing for function optimization, Technometrics, 28(3):209--217.

....annealing. Usually, the sampling distribution is chosen to be a uniform distribution on bounded regions, e.g. fixed (Wille and Vennik 1985; Khachaturyan 1986; Wille 1987) or adapted hypercubes (Vanderbilt and Louie 1984; Haines 1987; Corana, Marchesi, Martini, and Ridella 1987) and fixed (Bohachevsky, Johnson, and Stein 1986) or adapted hypersphere surfaces (Bertocchi and Sergi 1992) As it is not possible with those distributions to reach each state in M when trapped in a local minimum a mechanism must be provided that allows the possibility of transitioning to regions with worse objective function values. This is ....

.... c 2 (0; 1) Vanderbilt and Louie (1984) Wille and Vennik (1985) Wille (1987) Corana et al. 1987) Bertocchi and Sergi (1992) subtractive T t = maxf0; T 0 Gamma t DeltaT g Haines (1987) linear T t = T 0 = t 1) Szu and Hartley (1987a) Szu and Hartley (1987b) function value T t = ff f(X t ) fi Bohachevsky et al. 1986) Empirical results with this so called cooling schedule T t indicate that the time until convergence is of exponential order. Thus, other schedules are used in practical applications (see table 1) which provide faster but possibly nonglobal convergence. This problem can be circumvented by an ....

Bohachevsky, I., M. Johnson, and M. Stein (1986). Generalized simulated annealing for function optimization. Technometrics 28 (3), 209--217.

.... Distillation Column Sequencing [30,90] 30,70] Reactor Separator Recycle System [100 120] 80,100] Complex Chemical Reactor Network [40,110] 30,100] Heat Exchanger Network Synthesis [10,60] 10,40] Speed Reducer Weight Minimization [5,10] 10,20] Phase and Chemical Reaction Equilibrium [7,10] [4,13] The second class of application problems we have studied are the satisfiability (SAT) problem described in the DIMACS benchmark suite. NOVEL, as a general method for global optimization, shows competitive results when compared to the best existing methods designed for these problems. This paper ....

....many regions of attraction before ending up as a greedy search. SA was originally designed to solve combinatorial optimization problems [50, 14] for an extensive survey, see [1, 2] The application of SA and related techniques to solve continuous global optimization problems can be found in [95, 13, 16, 65, 53, 47]. Recently, Romeijn and Smith [67] used SA to solve constrained continuous global optimization problems. Their results are comparable in quality to existing solutions on a collection of classical test problems. Clustering Methods. By cluster analysis on random sample points, these algorithms try ....

I. O. Bohachevsky, M. E. Johnson, and M. L. Stein. Generalized simulated annealing for function optimization. Technometrics, 28:209--217, 1986.

....reduce this time can be divided into two classes; one is parallel processing and the other is careful tuning of the SSA s control parameters, such as initial temperature, cooling rate, inner loop stop criterion and outer loop stop criterion. Different acceptance probabilities have also been tried [19, 20]. However, little work has done on the impact of SA s generation probabilities, which are directly related to SA s neighbourhood system as well as its search behaviour in the configuration space, on SA s performance. The following sections will analyze this impact and give a general model of SA. ....

....i Deltac 1 j r (65) and other definitions are the same as (9) 14) Proof: The same as Corollary 5.1. Note that 8X;Y 2 S, f (d XY ) 1 T n d 2 XY T 2 n Q.E.D. Although we adopted (3) as the acceptance probability in the above analyses, others can also be used, such as [20] aXY (T n ) min ( 1; exp c j X (c X Gamma c Y ) T n ) j 0 or those described in [23, 19] Xin Yao: Simulated Annealing with Extended Neighbourhood 19 6 Conclusion This paper gives a probabilistic analysis of the impact of SA s neighbourhood on SA s performance and shows that an ....

I. O. Bohachevsky, M. E. Johnson, and M. L. Stein. Generalized simulated annealing for function optimization. Technometrics, 28:209--217, 1986.

....uses restarts to bring a search out of a local minimum when little improvement can be made locally [20, 35, 36] More advanced methods rely on probability to indicate whether the search should ascend from a local minimum. Simulated annealing is one of these methods that accepts up hill movements [22, 30, 2, 3, 21, 13]) based on some probability. Other stochastic methods rely on probability to decide which intermediate points to be interpolated as new starting points, such as random recombinations and mutations in evolutionary algorithms [8] All these algorithms are weak in either their local search [14] or ....

I. O. Bohachevsky, M. E. Johnson, and M. L. Stein. Generalized simulated annealing for function optimization. Technometrics, 28:209--217, 1986.

.... the experimental evaluation of different optimization techniques, we compare the following global search techniques: V1: adaptive random search [12] V2: adaptive random search combined with a locally operating downhill simplex algorithm [13] V3: simulated annealing for continuous functions [3, 7], V4: multi start algorithm [31, 32] V5: grid simplex algorithm [25] and V6: pure probabilistic search [2, 31] The average number of function evaluations required for the detection of the global maximum, and the average runtime on a monoprocessor system is shown in Table 2, a) In our ....

M. E. Bohachevsky, M. E. Johnson, and M. L. Stein. Generalized simulated annealing for function optimization. Technometrics, 28(3):209--217, 1986.

.... have several tuning parameters (mutation rate, crossover rate) that can effect its performance (convergence rate, quality of solution) In contrast, simulated annealing typically uses a single parameter tied to the cooling schedule (see either [Bohachevsky, Johnson, and Stein, 1995) or [Bohachevsky, Johnson, and Stein, 1986)] Convergence theorems dictate the value of the parameters in theory, although in applications, empirical adjustments are necessary [Bohachevsky, Johnson, and Stein, 1986) Employing the conventional wisdom associated with the settings of tuning parameters for the genetic algorithm indicates ....

.... typically uses a single parameter tied to the cooling schedule (see either [Bohachevsky, Johnson, and Stein, 1995) or [Bohachevsky, Johnson, and Stein, 1986) Convergence theorems dictate the value of the parameters in theory, although in applications, empirical adjustments are necessary [Bohachevsky, Johnson, and Stein, 1986)] Employing the conventional wisdom associated with the settings of tuning parameters for the genetic algorithm indicates that realistically sized problems are unapproachable [Parsons, Forrest, and Burks, 1995) We started with a simple DNA sequence assembly problem addressed with a genetic ....

I.O. Bohachevsky, M.E. Johnson, and M.L. Stein. Generalized simulated annealing for function optimization. Technometrics, 28:209--217.

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Bohachevsky, I.O., Johnson, M.E., and Stein, M.L. (1986): Generalized Simulated Annealing for Function Optimization. Technometrics 28(3), pp. 209--218.

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I. Bohachevsky, M. Johnson and M. Stein, Generalized simulated annealing for function optimization, Technometrics, 1986, 28(3), 209-217.

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I.O. Bohachevsky, M.E. Johnson, and M.L. Stein. Generalized simulated annealing for function optimization. Technometrics, 28(3):209--217, 1986.

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I.O. Bohachevsky, M.E. Johnson, and M.L. Stein, "Generalized simulated annealing for function optimization," Technometrics, vol. 28, no. 3, pp. 209--217, 1986.

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Bohachevsky, I. O., Johnson, M. E. , and Stein, M. L., "Generalized Simulated Annealing for Function Optimization", TECHNOMETRICS, Aug. 1986, Vol. 28,No. 3, p.209-217

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Bohachevsky, I., Johnson, M., and Stein, M., "Generalized Simulated Annealing for Function Optimization," Technometrics, Vol. 28, August 1986, pp. 209-217.

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Bohachevsky and Johnson and Stein. Generalized Simulated Annealing for Function Optimization. Technometrics. August, 1986.

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Mark E. Johnson Ihor O. Bohachevsky and Myron L. Stein. Generalized simulated annealing for function optimization. TECHNOMETRICS, Vol. 28, No. 3, pp. 209--217, 1986.

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I.O. Bohachevsky, M.E. Johnson and M.L. Stein, "Generalized Simulated Annealing for Function Optimization," Technometrics, 28:3, pp. 209-215,

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