文件列表:

PSO.c (10482, 2012-01-06)
Results.pdf (101205, 2011-01-12)
alea.c (8076, 2012-01-06)
cec2005.c (1626, 2011-02-01)
cec2005data.c (3503, 2009-01-05)
cec2005pb.c (1905, 2012-01-07)
cellular_phone.c (1213, 2011-03-26)
lennard_jones.c (648, 2010-03-24)
main.c (10122, 2012-01-08)
main.h (4792, 2012-01-07)
perf.c (8902, 2012-01-07)
perf_pressure_vessel.c (533, 2011-04-13)
pressure_vessel.c (760, 2011-04-13)
problemDef.c (9867, 2012-01-08)
tools.c (1210, 2011-01-02)

Standard PSO 2011 Contact for remarks, suggestions etc.: pso@writeme.com Updates: 2012-01-06 BW[2] is now for randomness options. In particular, one can use quasi-random numbers for initialisation (Sobol or Halton sequences) Halton sequences seem to be better. 2011-05-25 Added option 3 for BW[1]: when P=G, then just look around X 2011-04-19 Fixed a small bug in the initialisation of the velocity 2011-05-16 Fixed a small bug in Compression spring (third constraint) 2011-02-01 Added Pressure Vessel 2011-01-21 temporary experimental option (may be removed later): variable ring topology 2011-01-21 aleaIndex() as a function in alea.c (was directly coded in main.c) 2011-01-15 A whole family of stable distributions (CMS method) May also replace the Box-Muller method for Gaussian distribution 2011-01-12 Penalized function (code 23) 2011-01-07 BW[1]. Specific methods in case P=G 2011-01-05 BW[0]. Random swarm size Initial version: 2011-01-01 Note 1: there are a few optional "bells and whistles" (BW) Not really part of the standard, but may improve the performance for some problems (or the contrary ...) Example: the random swarm size "around" a mean value Note 2: the code is not optimised, in order to be more readable. -------------------------------- Contributors Works and comments of the following persons have been taken into account while designing this standard. Sometimes this is for including a feature, and sometimes for leaving one out. Auger, Anne Blackwell, Tim Bratton, Dan Clerc, Maurice Croussette, Sylvain Dattasharma, Abhi Eberhart, Russel Hansen, Nikolaus Helwig, Sabine Keko, Hrvoje Kennedy, James Krohling, Renato Langdon, William Li, Wentao Liu, Hongbo Miranda, Vladimiro Poli, Riccardo Serra, Pablo Spears, William Stickel, Manfred Yue,Shuai -------------------------------- Motivation Quite often, researchers claim to compare their version of PSO with the "standard one", but the "standard one" itself seems to vary! Thus, it is important to define a real standard that would stay unchanged for at least a few years. This PSO version does not intend to be the best one on the market (in particular, there is no adaptation of the swarm size or of the coefficients). This is simply similar to the original version (1995), with few improvements based on some recent works. ------ The above comment was for Standard PSO 2007. It is still valid. However, all "standard" PSOs (2007 and before) are not satisfying for at least one reason: on a given problem, the performance depends on the system of coordinates. This phenomenon is sometimes called "rotation sensitivity". This happens because for each particle and at each time step, the set of all "next possible positions" (NPP) is itself coordinate dependent: typically, a combination of two D-rectangles whose sides are parallel to the axes, with uniform probability distribution inside each rectangle. This combination is itself a D-rectangle, but with a non uniform distribution (more dense near the centre). So the obvious idea is to define an NPP domain that is _not_ coordinate dependent. There are a lot of reasonable simple shapes. Let X be the current position, P the best previous one, and G the best previous best in the neighbourhood (informants). We could for example: - define a D-sphere centred somewhere between X, P, G (like in some PSO variants) - define a D-cylinder along X-P (resp. X-G) - define a D-Ice_cream_cone (base on X, and the ball around P (resp. G)) - etc. Experiments suggest that a simple hypersphere H(Gr,||Gr-X||) is a good compromise, where Gr is the "centre of gravity" of X, P', and G', where P' is X+c*(P-X) and G'=X+c*(G-X) with c>1. ****** This is the core of this standard. It combines the same information that the canonical PSO (i.e. X, P, and G), with the same parameters (w and c), but in a way that is both simple and not coordinate dependent. ****** However, note that perfect non-sensitivity to rotation is impossible (except for very particular problems) whenever the search space itself is not a sphere. ------------------------------------ From SPSO-07 to SPSO-2011 - normalisation of the search space if possible, which is then [0,normalise]^D Reason : for some problems, the range of values is not the same for all variables (the search space is not a hypercube). In such a case, using spheres may lead to bad performance. In the definition of the problem, it is then better to set pb.SS.normalise to a positive value (typically 1). - new definition of the next possible position domain : hypersphere H(Gr,||Gr-X||) Reason: non sensitivity to rotation (as much as possible) - also, a distribution option has been added (Gaussian distribution instead of the uniform one) Reason: global convergence property. It does not always mean "better performance", but quite often, it indeed works better. - deterministic formula for swarm size has been removed. Now the user gives the value (default 40), or a mean value and for each run the swarm size is randomly chosen "around" this value Reason: the size given by the formula in SPSO 2007 (depending on the dimension) often used to come out as far from optimal. - velocity initialisation is slightly modified Reason: so that no particle leaves the search space at the very first step. Note: There do exist far better initialisation schemes than the random one used for positions in SPSO 2007 (it has not been modified here), but position initialisation is not really an inherent part of a search strategy. Actually, for fair comparison of two iterative algorithms, initial positions should be the same. - a (non) confinement option has been added (the old historical "let them fly" method) Reason : All other methods are more or less arbitrary. There is a drawback, though. When the search space is discrete (at least along some dimensions) the convergence may be extremely slow when using some "bells and whistles". See problem 11 (Network), which is partially binary. --------------------------------- Metaphors swarm: A team of communicating people (particles) At each time step Each particle chooses a few informants , selects the best one from this set, and takes into account the information given by the chosen particle (the informant). If it finds no particle better than itself, it may choose the informant at random. ----------------------------------- Parameters/Options All parameters have been hard coded with default values, but you can of course modify them. S := swarm size K := maximum number of particles _informed_ by a given one w := first cognitive/confidence coefficient c := second cognitive/confidence coefficient ----------------------------------- Equations For each particle Equation 1: v(t+1) = w*v(t) + V(t) Equation 2: x(t+1) = x(t) + v(t+1) where V(t) := Gr(t)- x(t) + r*alea_sphere(Gr,||Gr-X||) Gr(t) := (x(t)+p'(t)+l'(t))/3 v(t) := velocity at time t x(t) := position at time t p'(t) := x(t)+c*(p(t)-x(t)) l'(t) := x(t)+c*(l(t)-x(t)) p(t) := previous best position of the particle. Best position of the particle that has been found so far l(t) := local best. Best position amongst the previous best positions of the informants of the particle w := inertia weight c := acceleration coefficient Note 1: It is tempting to define a weighted sum Gr(t) := w1*x(t)+w2*p(t)+w3*l(t) In practice, a lot of weighting schemes have been tried. None of them was really better than the (1/3,1/3,1/3) used here (on average on a large benchmark. For a specific problem, a particular scheme may indeed be better). Note 2: When the particle has no informant better than itself, it implies p(t) = l(t) The simplest way is to do nothing special. We then have Gr(t) := (x(t)+2*p'(t))/3 The standard rule here is Gr(t) := (x(t)+ p'(t))/2 A BW is to simply say that Gr=G. Another one is to choose a different l(t) at random amongst the other particles with a probability that depends on S, K, and on the dimension D. It usually improves the performance (but not always). ----------------------------------- Use Define the problem (you may add your own one in problemDef() and perf()) You may modify some parameters. Run and enjoy! ================================================================================ A few results over 100 runs (success rate and Mean best). For reproducible results the seed of the random number generator KISS is set to 1234567890, and is "warmed up" 10000 times before the first run fo each function. As 100 runs is not big enough, the success rate is indeed very sensitive to the RNG and to the seed. < see the file Results.pdf > Comments: 1) The performance may be better because the problem is biased (typically the solution point lies on (0,0,...,0)) or because, by chance, the swarm size (which is automatically computed in SPSO 2007 by using a formula depending one the dimension) is a "good" one (for each problem, there exists an optimal swarm size). 3) The performance with the Gaussian distribution is slightly better than the one with the uniform distribution, except without any confinement and some BW.

近期下载者：

相关文件：

评论：[我要评论] [举报此文件]

收藏者：