Regrouping Particle Swarm Optimization: A New Global Optimization Algorithm with Improved Performance Consistency Across Benchmarks George I. Evers Advisor: Dr. Mounir Ben Ghalia Electrical Engineering Department The University of Texas – Pan American

Outline From Physics to PSO Visual Illustration of Stagnation & the Regrouping Method III. RegPSO Formulation IV. Graph of Solution Quality V. Statistical Comparison with Basic PSO VI. Summary VII. Future Work

How PSO Derives from Standard Physics Equations From Physics to PSO

From Physics to PSO Displacement Formula of Physics: assuming constant acceleration over the time period

From Physics to PSO Iterative Version: Using 1 time unit between iterations: t = (k + 1) – k = 1 iteration per update t2 = 1 iteration2 per update For practical purposes, t drops out of the equation.

From Physics to PSO Subscript “i” Used for Particle Index: (All particles follow the same rule.)

From Physics to PSO Particles are physical conceptualizations accelerating according to social and cognitive influences.

From Physics to PSO Cognitive Acceleration The cognitive acceleration is proportional to (i) the distance, , of a particle from its personal best, and (ii) the cognitive acceleration coefficient, .

From Physics to PSO Social Acceleration The social acceleration is proportional to (i) the distance, , of a particle from its global best, and (ii) the social acceleration coefficient, .

From Physics to PSO Total Acceleration The overall acceleration can therefore be written as Substitution then leads from to

From Physics to PSO Total Acceleration In place of constant , a pseudo-random number with an expected value of is generated per dimension to add an element of stochasm to the algorithm. In this manner becomes

From Physics to PSO Simulating Friction To prevent velocities from growing out of control, only a fraction of the velocity is carried over to the next iteration. This is accomplished by introducing an inertia weight, , which is set less than 1. In this manner becomes

From Physics to PSO Velocity and Position Updates The previous equation is separated into two more succinct equations, allowing velocities and positions to be recorded and analyzed separately.

The Main Obstacle: Premature Convergence/ Stagnation II. Visual Example of Stagnation & The Regrouping Method

Rastrigin Benchmark Used to Illustrate Stagnation

Swarm Initialization Velocities are randomly initialized to lie between [-vmax, vmax] per dimension. Particles 1 and 3 are selected to visually illustrate how velocities and positions are updated.

First Velocity Updates The velocities of the previous iteration are reduced by the inertia weight to produce the inertial components in red. First Velocity Updates Particle 6 found the best function value, which it communicates to its friends. Social acceleration is shown in blue.

First Position Updates Particles moved along the resultant velocity vectors to their new positions (Page Up, Page Down to see this). First Position Updates Particle 1 found a new personal best, but particle 3 did not.

Second Velocity Updates Particle 3 is pulled cognitively toward its personal best and socially toward the global best while experiencing inertia. Second Velocity Updates Particle 1 is at its personal best, so it experiences only inertia and social acceleration.

Second Position Updates Particles again moved along their resultant velocity vectors to new positions (Page Up, Page Down to see this). Second Position Updates Particle 3 found a new personal best, while particle 1 did not.

Swarm Snapshots Having seen how particles iteratively update their positions, the following slides show the swarm state each 10 iterations to track the progression from initialization to eventual solution.

Swarm Initialization at Iteration 0 Rewinding to monitor collective behavior from the beginning…. Swarm Initialization at Iteration 0 Particles are randomly initialized within the original initialization space.

Swarm Collapsing at Iteration 10 Particles are converging to a local minimizer near [2,0] via their attraction to the global best in that vicinity.

Exploratory Momenta at Iteration 20 Momenta and cognitive accelerations keep particles searching prior to settling down.

Convergence in Progress at Iteration 30 Personal bests move closer to the global best and momenta wane as no better global best is found. Particles continue converging to the local minimizer near [2,0].

Momenta Waning at Iteration 40 Momenta continue to wane as particles are repeatedly pulled toward (a) the global best very near [2,0] and (b) their own personal bests in the same vicinity.

Mostly Converged at Iteration 50 Most particles are improving their approximation of the local minimizer found, while two particles still have some momenta.

Momenta Waning at Iteration 60 The final two particles are collapsing upon the global best while the remaining particles are refining the solution.

Momenta Waning at Iteration 70 All particles are in the same general vicinity.

Cognitive Acceleration at Iteration 80 At least one particle still has some exploratory momentum.

Premature Convergence Detected at Iteration 102 All particles have converged to within 0.011% of the diameter of the initialization space. It is important to allow particles to refine each solution before regrouping since they have no prior knowledge of which solution is the global minimizer.

Options for Dealing with Stagnation Terminate the search rather than wasting computations while stagnated. Allow the search to continue and hope for solution refinement. Restart particles from new positions and look for a better solution. Somehow flag solutions already found so that each restart finds new solutions, and continue restarting until no better solutions are found. Reinvigorate the swarm with diversity to continue the current search for the global minimizer.

“Regrouping” Definition Regroup: “to reorganize (as after a setback) for renewed activity” – Merriam Webster’s online dictionary

Regrouping at Iteration 103 Regrouping is more efficient than restarting on the original initialization space.

Exploration at Iteration 113 “Gbest” PSO continues as usual within the new regrouping space. Particles move toward the global best with new momenta, personal bests, and positions/perspectives.

Swarm Migration at Iteration 123 The swarm is migrating toward a better region discovered by an exploring particle near [1,0].

Differences of Opinion at Iteration 133 Some particles are refining a local minimizer near [1,0] while others continue exploring in the vicinity.

Solution Comparison at Iteration 143 Cognition pulls some particles back to the local well containing a local minimizer near [1, 0].

Solution Comparison at Iteration 153 Cognition and momenta keep particles moving as momenta wane.

Unconvinced of Optimality on Horizontal Dimension at Iteration 163 There is still some uncertainty on the horizontal dimension.

New Well Agreed Upon at Iteration 173 All particles agree that the new well is better than the previous.

Waning Momenta at Iteration 183 Momenta wane.

Premature Convergence Detected Again at Iteration 219 Regrouping improved the function value from approximately 4 to approximately 1, and premature convergence is detected again.

Swarm Regrouped Again at Iteration 220 The swarm is regrouped a second time.

Best Well Found at Iteration 230 The well containing the global minimizer is discovered.

Swarm Migration at Iteration 240 The swarm migrates to the newly found well.

Convergence at Iteration 250 Particles swarm to the newly found well due to its higher quality minimizer.

Cognition at Iteration 260 Momenta carry particles beyond the well.

Convergence at Iteration 270 Solution refinement of the global minimizer is in progress.

Regrouping PSO (RegPSO) Formulation III. RegPSO Formula

Regrouping PSO (RegPSO) Detection of Premature Convergence

Regrouping PSO (RegPSO) Regrouping the Swarm

Regrouping PSO (RegPSO) High-Level Pseudo Code Do Run Gbest PSO until premature convergence. Regroup the swarm. Re-calculate the velocity clamping value based on the range of the new initialization space. Re-initialize velocities. Re-initialize personal bests. Remember the global best. Until Search Termination

Effectiveness of RegPSO Demonstrated Graphically IV. Graphical Comparison of Mean Function Values

Effectiveness of RegPSO Demonstrated Statistically V. Statistical Comparison

Regrouping PSO (RegPSO) Compared to Gbest, Lbest PSO

Summary By regrouping the swarm within an efficiently sized regrouping space when premature convergence is detected, RegPSO considerably improves performance consistency, as demonstrated with a suite of popular benchmarks.

Future Work Theoretical Improvements Give the algorithm the ability to progress from regrouping to a solution refinement phase. Testing NP hard problems Applications to real-world problems