Problems. Just for fun. Nothing to win!

Open question Comment


Dynamic functions

What happens if the solution point moves over the time ? Can PSO "track" it quck enough ?. Advance: Carlisle's and Dozier's work.


El Farol

Population members must coordinate their problem solutions to find a population-level optimum. See Adaptive versions.


Distributed human PS.

People at computers on a network, given p_i (best previous position) and p_g (best previous position in the neighbourhood) feedback -- how do they approach problem solving? How do they do at it?


Online neural nets.

The population members have to explore the search space collecting data about it, for instance, where obstacles are, as well as collectively optimizing a set of weights to describe it.



Technically, it is possible to use PSO to "optimize" PSO parameters, for a given set of problems. But is it worthwhile ? Advance: recursive PSO.



Cultural algorithms are special PSO cases. It may also be possible to describe a meta-model including PSO, GA (Genetic Algorithms), ACO (Ant Colony), etc.


Swarm intelligence

It can be proved that ACO algorithm is formally equivalent to a sequential one with just one ant at a time. Is it the same for PSO, or is there a real "swarm intelligence" ? (See also Question 2).


Generalized PSO (1)

In classical PSO, a particle just consider the best neighbour. Is it also interesting to consider others neighbours, for example with smaller "weights" ? Advance: Kennedy's and Mendes's work on FIPS (Fully Informed Particle Swarm)


Generalized PSO (2)

In classical PSO, a particle remember just its best previous position. Is it interesting to remember (and to use) more than one previous position ? Advance: TRIBES


Mathematical explanation

The current mathematical "explanation" (moves in a 5D space) does not take interactions into account. If possible (see Question 7), a better one would be interesting.


 Strategies  There is no special reason for all particles do use the same search srategy. For example, it may be better for a "good" particle to follow its own way more than for a "bad" one. What could be the best set of possible strategies and the best set of meta-rules to choose among them? Advance: TRIBES


 Proximity areas  When you choose   a point at random with a formula like  rand(0,phi)(p_i(d) - x(d)) , you implicitely use a hyperparallelepid with a uniform distribution in it. Wouldn't be another distribution better? Hyperspherical? Non uniform? Gaussian? Advance: Gaussian PSO, a lot of hybrid PSO, TRIBES


 Information and "guided" randomness
 The move (and also the particle generation in adaptive version) do use randomness. Is it possible to take advantage of information theory to "guide" this randomness towards promising areas? Advance: SunnySpell
In most of languages, the rand() function is quite bad. For example, in ANSI C, it is just the choice of an integer amongst 32767. What happens if you use a better pseudo-random number generator? Variant: what happens when using chaos? On the contrary, what happens when reducing the number of possible pseudo-random values?
Class function and performance maps
For a given problem, if you systematically try a lot of different parameter sets (including swarm size), you obtain some performance maps. With some (apparently) quite different problems, like say Sphere and Rosenbrock, the structures of these maps are very similar. It would be interesting to understand why. Advance: PSO-equivalence
Adaptive weighted information link
Usually each particle trusts equally its neighbours and chooses the best. What happens if the confidence coefficients are modified during the process (more or less like in neural networks)?
17 A straigth line? Standard PSO. Sphere function. During the process, plot the curve (ln(error) vs number_of_fitness_evaluations).  The best polynomial regression curve is a linear one. Prove it, and find the coefficients.
18 Bell curve Standard PSO. Dimension 1. For a given particle, keep p and g constant. Plot the histogram of the successive positions x(t).  It is a gaussian-like (but not gaussian) curve. Prove it.
19 NFLT? It seems that PSO is a kind of algorithm for which the No Free Lunch Theorem does not hold (maybe because of the velocity).  Is is true or not?
20 Clamping or not? When a particle tends to leave the search space you may either clamp it on the bound (or inside the search space) or do nothing special (and not evaluate the position).  In the last case, it usualy comes back. Two questions: 1) sometimes it apparently never comes back. Why?  2) What is the most effective method, on a given benchmark set, say the one of CEC 2005
2008: several works show that clamping is definitely better
21Discrete/binary (non combinatorial)There are a lot of very different binary PSOs (or, more generally, discrete PSOs). Is there possible to define a "canonical" one, based on a mathematical analysis?
22Combinatorial PSOThere are a lot of very different combinatorial PSOs. Is there possible to define a "canonical" one, based on a mathematical analysis?
23InitializationIt is well known for years that the classical "uniform" initialization is not very good.  Better ones  (more regular) do exist. What could be the "best" one?
24StrategiesIn Standard PSO, all particles have the same behavior.  Some attempts seem to show that using at least two kind of particles improves the algorithm. For example "normal" particles, and "scout" particles.  But there are a lot of other possible methods
A "standard" unbiased PSO
In 2010, W. Spears et al. mathematically explained (Biases in Particle Swarm Optimization, International Journal of Swarm Intelligence Research) why the most common PSO versions are rotationally variant, and why they easier find the solution when it lies on a coordinate axis or on a diagonal. PSO versions without this drawback do exist for years but are either quite complicated or very different from the standard one. Or both. So, the challenge is to design a PSO version that is both "near to" the current standard PSO 2007, and at least equivalent in terms of performance on classical benchmarks (CEC 2005, for example), in order to suggest a possible new standard.
2011: SPSO 2011, which makes use of hyperspheres, seems to be a possible solution.
Random Number Generator
For a given problem, and a given algorithm, the results can be significantly different when using different RNGs. Moreover, for a given problem, two algorithms A and B, and two RNG, RNG1 and RNG2, we may have the following situation:
- with RNG1, A is better than B
- with RNG2, B is better than A
How to compare A and B? Are the published comparisons reliable?

List-based optimisers
Some experiments show that there is no relationship between the "quality" of the RNG and the performance of a given algorithm. Moreover, the RNG can be replaced by a short list of predefined numbers, used cyclically. The result is sometimes even better. Is there a non empirical way to define such "good" lists?