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Enter on line 2 above 0 1 2 3 4 5 6 7 If wages from LOWEST paying job are 75,001 - 100,000 100,001 - 110,000 110,001 and over Enter on line 2 above 8 9 10

where 1 0 is the average pheromone concentration on the edges of the last best path before the global update, and 2 0 after the global update Unfortunately, the optimal values of 1 and 2 are not known Again for the TSP, empirical analyses showed that ACS worked best when ( 1 1)/( 2 1) 04, which gives nk = 10 [215] It is important to note that the above is for a speci c algorithm, used to solve a speci c class of problems The value of nk should rather be optimized for each di erent algorithm and problem nt , maximum number of iterations: It is easy to see that nt plays an important role in ensuring quality solutions If nt is too small, ants may not have enough time to explore and to settle on a single path If nt is too large, unnecessary computations may be done 0 , initial pheromone: During the initialization step, all pheromones are either initialized to a constant value, 0 (for MMAS, 0 = max ), or to random values in the range [0, 0 ] In the case of random values, 0 is selected to be a small positive value If a large value is selected for 0 , and random values are selected from the uniform distribution, then pheromone concentrations may di er signi cantly This may cause a bias towards the links with large initial concentrations, with links that have small pheromone values being neglected as components of the nal solution There is no easy answer to the very important question of how to select the best values for the control parameters While many empirical studies have investigated these parameters, suggested values and heuristics to calculate values should be considered with care These are given for speci c algorithms and speci c problems, and should not be accepted in general To maximize the e ciency of any of the algorithms, the values of the relevant control parameters have to be optimized for the speci c problem being solved This can be done through elaborate empirical analyses Alternatively, an additional algorithm can be used to learn the best values of the parameters for the given algorithm and problem [4, 80].

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The algorithm restarts each nw consecutive epochs for which no reduction in error could be achieved, at which point the algorithm nds a new direction to search The E function to calculate the derivative, E (w) = w , computes the derivative of E with respect to each weight for each of the patterns The derivatives over all the patterns are then summed, ie PT E E = (379) wi wi,p p=1 where wi is a single weight

If wages from HIGHEST paying job are $0 50,001 100,001 150,001 270,001 - $50,000 - 100,000 - 150,000 - 270,000 and over Enter on line 7 above $450 800 900 1,050 1,200

LeapFrog is an optimization approach based on the physical problem of the motion of a particle of unit mass in an n-dimensional conservative force eld [799, 800] The potential energy of the particle in the force eld is represented by the function to be minimized in the case of NNs, the potential energy is the MSE The objective is to conserve the total energy of the particle within the force eld, where the total energy consists of the particle s potential and kinetic energy The optimization method simulates the motion of the particle, and by monitoring the kinetic energy, an interfering strategy is adapted to appropriately reduce the potential energy The LeapFrog NN training algorithm is given in Algorithm 35 The reader is referred to [799, 800] for more information on this approach

We shall not cease from exploration And the end of all our exploring Will be to arrive where we started And know the place for the rst time T S Eliot

Particle swarm optimization (PSO), which is a stochastic population-based search method (refer to 16), can be used to train a NN In this case, each particle represents a weight vector, and tness is evaluated using the MSE function (refer to Section 167 for more detail on NN training using PSO) What should be noted is that weights and biases are adjusted without using any error signals, or any gradient information Weights are also not adjusted per training pattern The PSO velocity and position update equations are used to adjust weights and biases, after which the training set is used to calculate the tness of a particle (or NN) in PT feedforward passes Evolutionary algorithms can also be used in a similar way to train NNs

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