Optimize Using Particle Swarm

This example shows how to optimize using theparticleswarmsolver.

The objective function in this example is De Jong’s fifth function, which is included with Global Optimization Toolbox software.

dejong5fcn

This function has 25 local minima.

Try to find the minimum of the function using the defaultparticleswarmsettings.

fun = @dejong5fcn; nvars = 2; rngdefault% For reproducibility[x,fval,exitflag] = particleswarm(fun,nvars)

Optimization ended: relative change in the objective value over the last OPTIONS.MaxStallIterations iterations is less than OPTIONS.FunctionTolerance.

x =1×2-31.9521 -16.0176

fval = 5.9288

exitflag = 1

Is the solutionxthe global optimum? It is unclear at this point. Looking at the function plot shows that the function has local minima for components in the range[-50,50]. So restricting the range of the variables to[-50,50]helps the solver locate a global minimum.

lb = [-50;-50]; ub = -lb; [x,fval,exitflag] = particleswarm(fun,nvars,lb,ub)

Optimization ended: relative change in the objective value over the last OPTIONS.MaxStallIterations iterations is less than OPTIONS.FunctionTolerance.

x =1×2-16.0079 -31.9697

fval = 1.9920

exitflag = 1

This looks promising: the new solution has lowerfvalthan the previous one. But isx一个真正的全球解决方案吗?尽量减少again with more particles, to better search the region.

options = optimoptions('particleswarm','SwarmSize',100); [x,fval,exitflag] = particleswarm(fun,nvars,lb,ub,options)

Optimization ended: relative change in the objective value over the last OPTIONS.MaxStallIterations iterations is less than OPTIONS.FunctionTolerance.

x =1×2-31.9781 -31.9784

fval = 0.9980

exitflag = 1

This looks even more promising. But is this answer a global solution, and how accurate is it? Rerun the solver with a hybrid function.particleswarmcalls the hybrid function afterparticleswarmfinishes its iterations.

options.HybridFcn = @fmincon; [x,fval,exitflag] = particleswarm(fun,nvars,lb,ub,options)

Optimization ended: relative change in the objective value over the last OPTIONS.MaxStallIterations iterations is less than OPTIONS.FunctionTolerance.

x =1×2-31.9783 -31.9784

fval = 0.9980

exitflag = 1

particleswarmfound essentially the same solution as before. This gives you some confidence thatparticleswarmreports a local minimum and that the finalxis the global solution.

Related Topics

• What Is Particle Swarm Optimization?
• Particle Swarm Optimization Algorithm