Algorithms and data structures Simulated annealing Tabu search Genetic algorithms Ant algorithms Neural networks

Simulated annealing SA heuristic considers some neighboring state s* of the current state s, and probabilistically decides between moving the system to state s* with probability or staying in-state s When T tends to zero, the probability P ( e , e ′ , T ) must tend to zero if e ′ > e and to a positive value otherwise. A B

Simulated annealing state = initial_state k = 0 while k < kmax : T = temperature(k ∕ kmax) snew = pick_a_random_neighbour(state) if P(E(state), E(snew), T) ≥ rand(): state = snew k += 1 T(k) = t0e-ak

SA – sample application Graph coloring: Initial_state, state – valid coloring pick_a_random_neighbour (s) – a coloring that was obtained by changing a color (or a number of colors dependent on T) in state s to another colors P: E(s) –> number of colors in s random factor e(T)

Tabu search Improving changes (moves) are accepted. The potential solutions that have been previously visited within a certain short-term period or have violated a rule, are marked as "tabu" (forbidden).

Tabu search bestCandidate = state = initial_state tabuList = [initial_state] while (not stoppingCondition()): sNeighborhood = getNeighbors(bestCandidate) bestCandidate = sNeighborHood.firstElement for sCandidate in sNeighborHood: if (not tabuList.contains(sCandidate) and \ (E(sCandidate) > E(bestCandidate)) ) : bestCandidate = sCandidate if E(bestCandidate) > E(state): state = bestCandidate tabuList.push(bestCandidate) if tabuList.size > maxTabuSize : tabuList.removeFirst()

TS – sample application Graph coloring: Initial_state, state – valid coloring E(s) –> number of colors in s

Genetic algorithm Initial population of randomly generated individuals In each generation, the fitness of every individual in the population is evaluated; The more fit individuals are stochastically selected from the current population, and each individual's genome is modified (recombined and possibly randomly mutated) to form a new generation. The new generation of candidate solutions is then used in the next iteration of the algorithm. The fitness is usually the value of the objective function for the optimization problem being solved.

GA - sample appliction Salesman route: Recombination: for parent-individuals some parts of solution are exchanged between parent-individuals Mutation: a part of soluton (a single edge in the way?) is randomly replaced by another route

Ant colony algorithm "simulated ants" walking around the graph representing the problem ants (initially) wander randomly, and upon finding food return to their colony while laying down pheromone trail If other ants find such a path, they are likely not to keep travelling at random, but instead to follow the trail, returning and reinforcing it if they eventually find food

ACA – sample application Salesman route It must visit each city exactly once; A distant city has less chance of being chosen (the visibility); The more intense the pheromone trail laid out on an edge between two cities, the greater the probability that that edge will be chosen; Having completed its journey, the ant deposits more pheromones on all edges it traversed, if the journey is short; After each iteration, trails of pheromones evaporate.

Neuron model Discrete y = 1 if xiwi >= P or 0 otherwise Contineous y = 2/(1-e-net), where net = xiwi y w1 wk ..... x1 xk

Neural network O1 O2 Output layer Hidden layer(s) Input layer I1 I2 I3

Topology Input/output sizes depend on imput size/categories count The number and size of hidden layers should be determined experimentally There are possible inter-layes connections, slow-down connections, loop-back connections, different transition functions, convolutions etc.

Interesting properties Supervised learning Preparation of training set – correct answers: Back propagation algorithm Interesting properties Error resistance Intrinsic recall

A Classifier O1 O2 Output layer Hidden layer(s) ..... Input layer I1

Sample applications Hand written text recognition and image classification Diagnistic skin deeseases Risk odf bank loan estimation Detection of the natural deposits Real estates values apraisal Fraud detection