Event Will Never Forget Essay Example for Free

Event Will Never Forget Essay Comparison of Di erent Neighbourhood Sizes in Simulated Annealing Xin Yao Department of Computer Science University College, University of New South Wales Australian Defence Force Academy Canberra, ACT, Australia 2600 Abstract Neighbourhood structure and size are important parameters in local search algorithms. This is also true for generalised local search algorithms like simulated annealing. It has been shown that the performance of simulated annealing can be improved by adopting a suitable neighbourhood size. However, previous studies usually assumed that the neighbourhood size was xed during search. This paper presents a simulated annealing algorithm with a dynamic neighbourhood size which depends on the current emperature value during search. A method of dynamically deciding the neighbourhood size by approximating a continuous probability distribution is given. Four continuous probability distributions are used in our experiments to generate neighbourhood sizes dynamically, and the results are compared. combinatorial optimisation. A method of generating dynamic neighbourhood sizes by approximating continuous probability distributions is given in this section. Section 4 compares the experimental results of using di erent continuous probability distributions to generate dynamic neighbourhood sizes. Finally, Section 5 concludes with some remarks and directions of future research. 2 General Simulated Annealing Although SA can be used in both continuous and discrete cases, this paper only considers combinatorial optimisation by SA unless otherwise indicated explicitly. A combinatorial optimisation problem can be informally described as nding an optimal con guration X from a nite or in nite countable con guration space S . Each con guration X 2 S can be represented by its n (gt; 0) components, i. e. , X = (x1; x2; ; xn ), where xi 2 Xi , i = 1; 2; ; n. An excellent discussion of combinatorial optimisation and its complexity can be found in Garey and Johnsons book 8]. A general model of SA, which is applicable to both continuous and discrete problems, can be described by Figure 1, where function generate (X; Tn) is decided by the generation robability gXY (Tn ), which is the probability of generating con guration Y from con guration X at temperature Tn , function accept (X; Y; Tn) is decided by the acceptance probability aXY (Tn ), which is the probability of accepting con guration Y after it has been generated at temperature Tn , and function update (Tn ) decides the rate of the temperature decrease. These three functions determine the convergence o f general SA 5, 6, 9], but parameters in general SA, such as the initial temperature, initial con guration, inner-loop stop criterion, and outer1 Introduction Simulated Annealing (SA) algorithms can nd very good near optimal solutions to a wide range of hard problems, but at the high computational cost. Various methods have been proposed to speed up its convergence, which can roughly be divided into three categories: (1) Optimising functions and parameters in SA 1]; (2) Combining SA with other search algorithms 2, 3]; and (3) Parallelising SA 4]. This paper falls into the above rst category. Section 2 of this paper describes a general SA algorithm 5, 6] which uni es di erent variants of the classical one 7]. Section 3 presents SA with a dynamic neighbourhood size and its application in Published in Proc. of Fourth Australian Conf. on Neural Networks, ed. P. Leong and M. Jabri, pp. 216{219, 1993, Melbourne, Australia. generate initial con guration X at random; generate initial temperature T0; REPEAT REPEAT Y = generate(X; Tn); IF accept(X; Y; Tn) THEN X = Y ; UNTIL `inner-loop stop criterion satis ed; Tn+1 = update (Tn ); n = n + 1; UNTIL `outer-loop stop criterion satis ed Figure 1: General simulated annealing. loop stop criterion, can have signi cant impact on its nite-time behaviour. That is, the computation time in practice depends on the three functions as well as these parameters. Most research on SA has concentrated on the update and accept function and various algorithmic parameters, only limited attention has been paid to the generate function. However, the generate function decides an important part | the neighbourhood structure and size | of a local search algorithm regardless of whether it is a deterministic one or a stochastic one like SA. The neighbourhood NX of a con guration X is de ned by con guration. The xed-size neighbourhood clearly does not conform with the basic search strategy behind SA. It is appealing to have a neighbourhood size which can adjust itself in the di erent search stages. Fast SA 12] can be regarded as an example of SA with a dynamic neighbourhood size, but it is only used in the continuous case. The application of dynamic neighbourhood size in combinatorial optimisation, to our best knowledge, has not been well-studied. 3 Dynamic Neighbourhood Size in Simulated Annealing This section gives a method of dynamically deciding the neighbourhood size in SA according to the temperature parameter 5, 6]. In the high temperature stages, SA algorithms have high acceptance probability for both good and ad moves, i. e. , exploration plays a major role in search, and thus a large neighbourhood size is used to enhance such exploration. In the low temperature stages, exploitation plays a major role in search, and thus a smaller neighbourhood is more suitable. In the following discussion, we say that the Hamming distance between two con guration X = (x1 ; x2; ; xn ) and Y = (y1; y2 ; ; yn ) is if there are exactly di erent elements between them. Let f (x) be the continuous density function which is used to generate the Hamming distance between the current con guration and the next one. Denote the set of con gurations which are distant from the current con guration X as SX ( ), SX ( ) = fY 2 S; gXY (Tn) gt; 0g where X 62 NX , and X 2 NY i Y 2 NX . NX = fY jY (1) ing search once de ned for a problem. Goldstein and Waterman 10] and Cheh et al. 11] carried out some experiments on comparing SA with di erent neighbourhood sizes, but the sizes are still xed once decided. A limitation of SA with a xed neighbourhood size is its inability to perform search at di erent scales in di erent stages of search. As indicated in our previous study 5], SA can be viewed as an attempt to combine exploration of a space and exploitation of a sub-space into the same algorithm, i. e. , coarse-grained search in the high temperature stages explores the con guration space and tries to locate promising regions, while ned-grained search in the low temperature stages exploits the promising regions and tries to nd a good near optimal gXY (Tn ) = 1=jNX j, where jNX j is the size of NX , i. . , the number of con gurations in NX , and is the same for all X in S . Moreover, jNX j is xed dur- Previous research on SA normally assumed that j Y 2 S; dXY = g (2) The probability of generating con guration Y , which is dXY distant from con guration X , is dened as 1 = jS (1 )j P rob dXY ? 2 lt; X dXY Z dXY + 1 2 f (x)dx = jS (1 )j 1 X dXY dXY ? 2 f (dXY ) jSX (dXY )j 2 gXY (Tn ) dXY + 1 2 (3) Suppose the maximum Hamming distance allowed for one move is dmax 1 , then the normalised generation function is f (dXY ) / jSX (dXY )j gXY (Tn ) = (4) FX (Tn ) where FX (Tn ) = X X f (d ) max XZ jSX (dXZ )j dXZ =1 Z 2S 4 Experimental Results We adopt the Traveling Salesman Problem (TSP) as a benchmark to evaluate our SA algorithms because of its clear mathematical de nition and high computational complexity. Goldstein and Waterman 10] and Cheh et al. 11] have experimented with TSPs using di erent but xed neighbourhood sizes and found that a small neighbourhood size is better than a large neighbourhood size. That is, the SA algorithm performs the best when dXY = 1. TSPs with 40 cities are used in our experiment and are generated at random. The same initial conguration, inner-loop stop criterion, out-loop stop criterion, and temperature decreasing rate are used in our experiments in order to evaluate the impact of the neighbourhood size on the performance of SA algorithms. Our experiments, albeit preliminary, have demonstrated that SA with a dynamic neighbourhood size outperforms SA with a xed neighbourhood size. Table 1 gives the results of four typical runs of two kinds of SA algorithms. Table 2 gives the results of using di erent distributions to generate neighbourhood sizes. roblem instance 1 2 3 4 initial value 15080 12260 13760 15820 NorSA 2540 2140 2560 2300 CSA 3120 2520 2880 2460 Table 1: Comparison of SA with a xed neighbourhood size (CSA) and SA with a dynamic neighbourhood size (NorSA). Normal distribution is used to generate the neighbourhood size. (5) Theorem 3. 1 ( 5]) Suppose the acceptance function in an SA algorithm is aXY (Tn ) = min 1; exp ? ? cY T cX n : (6) and the generation function is (4), where f (x) in (4) can be anyone of the following, (a) the Normal function N (0; Tn), i. e. , 1 exp ? d2 XY f (dXY ) = p 2Tn 2 Tn (b) the exponential function E (Tn ), i. . , f (dXY ) = ! 1 exp ? dXY Tn Tn (c) the Cauchy function C (Tn ), i. e. , 1 T f (dXY ) = 2 n 2 dXY + Tn (d) the stable function with index 1 13], i. e. , 2 f (dXY ) = q exp ? 2d1 XY 2 d3 XY 1 5 Concluding Remarks Neighbourhood size is an important parameter in local search algorithms, but only a xed size was adopted in previous application of SA to combinatorial optimisation problems. This paper proposes a method of using a dynamic neighbourhood size in SA based on our analysis of SA search. Preliminary experiments have demonstrated the advantage of a dynamic neighbourhood size in SA. The idea of a dynamic neighbourhood size could also be introduced into other local search algorithms. It is, in fact, related to a more profound 3 Then the SA algorithm converges to global minima if the cooling rate is Tn = ln n + n0 ; n = 1; 2; (7) where and n0 are positive constants. It is set to n, the number of elements in a con guration, in our experiments. 1 problem instance initial value CauSA NorSA ExpSA StableSA 1 17800 2480 2540 2640 3760 2 15500 3000 3340 3180 4420 3 16600 3300 2920 3460 4500 4 14780 3000 2980 3280 3760 References 1] P. J. M. van Laarhoven and E. H. L. Aarts, Simulated Annealing: Theory and Applications, D. Reidel Publishing Co. , 1987. 2] D. H. Ackley, A Connectionist Machine for Genetic Hillclimbing, Kluwer Academic Publishers, Boston, 1987. 3] X. Yao, Optimization by genetic annealing, In M. Jabri, editor, Proc. of ACNN91, pages 94{97, Sydney, 1991. 4] D. R. Greening, Parallel simulated annealing techniques, Physica D, 42:293{306, 1990. 5] X. Yao, Simulated annealing with extended neighbourhood, International J. of Computer Math. , 40:169{189, 1991. 6] X. Yao and G. -J. Li, General simulated annealing, J. of Computer Sci. Tech. 6:329{ 338, 1991. 7] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, Optimization by simulated annealing, Science, 220:671{680, 1983. 8] M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman Co. , San Francisco, 1979. 9] S. Anily and A. Federgruen, Ergodicity in parameteric nonstationary Markov chains: an application to annealing methods, Ope r. Res. , 35:867{874, 1987. 10] L. Goldstein and M. Waterman, Neighborhood size in the simulated annealing algorithm, Amer. J. of Math. and Management Sci. , 8:409{423, 1988. 11] K. M. Cheh, J. B. Goldberg, and R. G. Askin, A note on the e ect of neighborhood structure in simulated annealing algorithm, Computers and Oper. Res. , 18:537{547, 1991. 12] H. H. Szu and R. L. Hartley, Nonconvex optimization by fast simulated annealing, Proc. of IEEE, 75:1538{1540, 1987. 13] W. Feller, An Introduction to Probability Theory and Its Applications, volume 2, John Wiley Sons, Inc. , 2nd edition, 1971. 4 Table 2: SA with a dynamic neighbourhood size which is generated by the Cauchy function (CauSA), Normal function (NorSA), Exponential function (ExpSA), and Stable function with index 1=2 (StableSA). esearch issue in search theory, i. e. , the issue of exploration versus exploitation or global search versus local search. Although local search based on some heuristics can be quite e cient under many circumstances, the problem of local optima is very hard to deal with. Some kind of global search has to be used if a global optimum or near optimum is required. However, the computational cos t of global search is often prohibitively high for most real-world applications due to the vast search space. It is bene cial to combine global and local search together. An open question here is how to decide when global or local search should be performed. It is also di cult to draw the line strictly between local and global search in practice. Dynamic neighbourhood size offers a way to deal with the problem by transferring from global search to local search smoothly based on a control parameter, temperature in SA. However, more work has to be done on deciding which kind of generation functions is most suitable for an application, i. e. what is the optimal rate of reducing the neighbourhood size. As indicated before, Fast SA 12] o ers a big improvement over classical SA 7] due to the adoption of Cauchy distribution. An interesting topic is to investigate whether the discrete version of Fast SA can o er similar improvement over classical SA. Our preliminary experiments seem to give a negative answer. Acknowledgement | The author is grateful to Drs. B. Marksjo and R. Sharpe for their support of his work while he was with CSIRO Division of Building, Construction and Engineering.

The K+ Channel Revisited :: Chemical Neuroscience Chemistry Papers

The K+ Channel Revisited Many scientists enter the field of neuroscience with a desire to understand how the human brain functions to create our actions. Some take a large approach and study reactions to stimulus in live animals. This approach leaves us still wanting to know about things that aren't a result of an outside stimulus. Other scientists take the minimalist approach and start by studying the exact mechanisms of individual cells of the brain. But, what good does that do us for everyday life? There is also the approach of taking our knowledge of behavior from diseases of the brain. Would we have thought about how our brain regulates our moods if it weren't for the fact that some people experience depression, in which the regulation of mood goes amiss? Many of the disease have led us to a deep understanding of the chemical interactions in our brain and body. At a very broad level, and ignoring religious discussions, we can say that every part of our existence is a result of chemical interactions . Besides giving us our substance, chemistry also gives us a mode of communication throughout our bodies. In our brain a large chemical component is that of the action potential that is conducted along a neuron as a result in the changing permeability of the cell. The axons of our neurons are the pathway for the communication that exists in our nervous system. This communication takes the form of an electric signal, also called an action potential. The action potential occurs due to a change in voltage across the membrane of the axon. The change in voltage is achieved by a change in the permeability of the neurons to the ions, Na+, Ca+, and K+.(1)The cell starts with a large concentration of potassium ions, K+, inside the cell, and a large concentration of sodium ions, Na+, outside the cell. The action potential propagates down the axon due to openings and closing of different channels allowing changing of the permeability to the differing ions (10). Channels are proteins that span the membrane of the axon. These proteins have a structure so that they can be allow ions to flow through pores that are only open at the appropriate times. Some of the channels are opened and closed by other chemicals, while some are initiated by a change in the membrane potential.

Jamaica’ crime rate statistic 2013 Essay

Jamaica has one of the world’s highest per-capita murder rates, and a 2010 state of emergency threw the harsh glare of publicity on the violent gang and drug culture in the capital, Kingston. Violent crime can be a real problem in Kingston and other parts of the country, but typically such crimes involve attacks by Jamaicans on other Jamaicans and revolve around drugs, gangs, politics, poverty, or revenge. Most crimes targeting visitors in tourist areas like Montego Bay, Negril and Ocho Rios are property-oriented — pickpocketing and petty theft, for example. Armed robberies do occasionally involve tourists, and can turn violent if victims resist. Credit-card skimming is an ongoing problem in Jamaica. Some scammers will make a copy of your credit-card information when you give your card to a restaurant server or shopkeeper. ATMs also may be rigged to steal your card information, or individuals may observe you at the ATM and try to steal your password. Avoid using credit cards or ATMs whenever possible; carry just enough cash for what you need that day. If you do need to use a credit card, keep an eye on the person handling your card. If you need to get cash, use the ATM at your hotel. Sexual assaults by hotel employees in resort areas on Jamaica’s north coast have occurred with some frequency, as well. Police in Jamaica are generally short on manpower and training. You will see an increased police presence in areas of Montego Bay and Ocho Rios frequented by tourists, but if you are a victim of crime you may find the response of the local police to be lacking — or nonexistent. Locals generally have little trust in the police, and while visitors are unlikely to be mistreated by police, the Jamaican Constabulary Force is widely viewed as corrupt and ineffectual. Tourists are advised to avoid traveling in notoriously high-threat areas of Kingston including, but not limited to, Mountain View, Trench Town, Tivoli Gardens, Cassava Piece and Arnett Gardens. In Montego Bay, avoid the areas of Flankers, Canterbury, Norwood, Rose Heights, Clavers Street and Hart Street. Several of the latter neighborhoods are adjacent to Montego Bay’s Sangster International Airport. Road Safety The north coastal road linking popular tourist destinations such as Montego Bay, Ocho Rios, and Negril is much improved in recent years. However, most roads are poorly maintained and have poor signage. Smaller roads may not be paved, and often are narrow, winding, and crowded with pedestrians, bicycles, and livestock. Driving is on the left, and Jamaica’s roundabouts (traffic circles) can be confusing for drivers used to driving on the right. Seat-belt use is required and recommended even for taxi passengers, given the hazardous driving conditions. Use of public transportation is not recommended, since public buses are often overcrowded and can become venues for crime. Take a cab from your hotel or use transportation from vendors that are part of JUTA — the Jamaica Union of Travelers Association.

The American Holocaust And American Americans - 1256 Words

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