Single-Solution Particle Swarm Optimization in .NET framework

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Single-Solution Particle Swarm Optimization
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Initial empirical studies of the basic PSO and basic variations as discussed in this chapter have shown that the PSO is an e cient optimization approach for the benchmark
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165 Single-Solution Particle Swarm Optimization
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t Figure 166 Stochastic Particle Trajectory for w = 09 and c1 = c2 = 20 problems considered in these studies Some studies have shown that the basic PSO improves on the performance of other stochastic population-based optimization algorithms such as genetic algorithms [88, 89, 106, 369, 408, 863] While the basic PSO has shown some success, formal analysis [136, 851, 863, 870] has shown that the performance of the PSO is sensitive to the values of control parameters It was also shown that the basic PSO has a serious defect that may cause stagnation [868] A variety of PSO variations have been developed, mainly to improve the accuracy of solutions, diversity and convergence behavior This section reviews some of these variations for locating a single solution to unconstrained, single-objective, static optimization problems Section 1652 considers approaches that di er in the social interaction of particles Some hybrids with concepts from EC are discussed in Section 1653 Algorithms with multiple swarms are discussed in Section 1654 Multi-start methods are given in Section 1655, while methods that use some form of repelling mechanism are discussed in Section 1656 Section 1657 shows how PSO can be changed to solve binary-valued problems Before these PSO variations are discussed, Section 1651 outlines a problem with the basic PSO
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16 Particle Swarm Optimization
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Guaranteed Convergence PSO
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The basic PSO has a potentially dangerous property: when xi = yi = y, the velocity update depends only on the value of wvi If this condition is true for all particles and it persists for a number of iterations, then wvi 0, which leads to stagnation of the search process This point of stagnation may not necessarily coincide with a local minimum All that can be said is that the particles converged to the best position found by the swarm The PSO can, however, be pulled from this point of stagnation by forcing the global best position to change when xi = yi = y The guaranteed convergence PSO (GCPSO) forces the global best particle to search in a con ned region for a better position, thereby solving the stagnation problem [863, 868] Let be the index of the global best particle, so that y = y GCPSO changes the position update to x j (t + 1) = yj (t) + wv j (t) + (t)(1 2r2 (t)) (1643) (1642)
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which is obtained using equation (161) if the velocity update of the global best particle changes to v j (t + 1) = x j (t) + yj (t) + wv j (t) + (t)(1 2r2j (t)) (1644)
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where (t) is a scaling factor de ned in equation (1645) below Note that only the global best particle is adjusted according to equations (1643) and (1644); all other particles use the equations as given in equations (161) and (162) The term x j (t) in equation (1644) resets the global best particle s position to the position yj (t) The current search direction, wv j (t), is added to the velocity, and the term (t)(1 2r2j (t)) generates a random sample from a sample space with side lengths 2 (t) The scaling term forces the PSO to perform a random search in an area surrounding the global best position, y(t) The parameter, (t) controls the diameter of this search area, and is adapted using if #successes(t) > s 2 (t) 05 (t) if #f ailures(t) > f (1645) (t + 1) = (t) otherwise where #successes and #f ailures respectively denote the number of consecutive successes and failures A failure is de ned as f ( (t)) f ( (t + 1)); (0) = 10 was found y y empirically to provide good results [863, 868] The threshold parameters, s and f adhere to the following conditions: #successes(t + 1) #f ailures(t + 1) > > #successes(t) #f ailures(t + 1) = 0 #f ailures(t) #successes(t + 1) = 0 (1646) (1647)
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The optimal choice of values for s and f is problem-dependent Van den Bergh et al [863, 868] recommends that s = 15 and f = 5 be used for high-dimensional search
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