## Find Global or Multiple Local Minima

### Function to Optimize

This example illustrates how `GlobalSearch` finds a global minimum efficiently, and how `MultiStart` finds many more local minima.

The objective function for this example has many local minima and a unique global minimum. In polar coordinates, the function is

f(r,t) = g(r)h(t),

where

`$\begin{array}{c}g\left(r\right)=\left(\mathrm{sin}\left(r\right)-\frac{\mathrm{sin}\left(2r\right)}{2}+\frac{\mathrm{sin}\left(3r\right)}{3}-\frac{\mathrm{sin}\left(4r\right)}{4}+4\right)\frac{{r}^{2}}{r+1}\\ h\left(t\right)=2+\mathrm{cos}\left(t\right)+\frac{\mathrm{cos}\left(2t-\frac{1}{2}\right)}{2}\text{.}\end{array}$` The global minimum is at r = 0, with objective function 0. The function g(r) grows approximately linearly in r, with a repeating sawtooth shape. The function h(t) has two local minima, one of which is global. ### Single Global Minimum Via GlobalSearch

1. Write a function file to compute the objective:

```function f = sawtoothxy(x,y) [t r] = cart2pol(x,y); % change to polar coordinates h = cos(2*t - 1/2)/2 + cos(t) + 2; g = (sin(r) - sin(2*r)/2 + sin(3*r)/3 - sin(4*r)/4 + 4) ... .*r.^2./(r+1); f = g.*h; end```
2. Create the problem structure. Use the `'sqp'` algorithm for `fmincon`:

```problem = createOptimProblem('fmincon',... 'objective',@(x)sawtoothxy(x(1),x(2)),... 'x0',[100,-50],'options',... optimoptions(@fmincon,'Algorithm','sqp','Display','off'));```

The start point is `[100,-50]` instead of `[0,0]`, so `GlobalSearch` does not start at the global solution.

3. Validate the problem structure by running `fmincon`:

```[x,fval] = fmincon(problem) x = 45.7332 -107.6469 fval = 555.5422```
4. Create the `GlobalSearch` object, and set iterative display:

`gs = GlobalSearch('Display','iter');`
5. Run the solver:

```rng(14,'twister') % for reproducibility [x,fval] = run(gs,problem) Num Pts Best Current Threshold Local Local Analyzed F-count f(x) Penalty Penalty f(x) exitflag Procedure 0 200 555.5 555.5 0 Initial Point 200 1463 1.547e-15 1.547e-15 1 Stage 1 Local 300 1564 1.547e-15 5.858e+04 1.074 Stage 2 Search 400 1664 1.547e-15 1.84e+05 4.16 Stage 2 Search 500 1764 1.547e-15 2.683e+04 11.84 Stage 2 Search 600 1864 1.547e-15 1.122e+04 30.95 Stage 2 Search 700 1964 1.547e-15 1.353e+04 65.25 Stage 2 Search 800 2064 1.547e-15 6.249e+04 163.8 Stage 2 Search 900 2164 1.547e-15 4.119e+04 409.2 Stage 2 Search 950 2356 1.547e-15 477 589.7 387 2 Stage 2 Local 952 2420 1.547e-15 368.4 477 250.7 2 Stage 2 Local 1000 2468 1.547e-15 4.031e+04 530.9 Stage 2 Search GlobalSearch stopped because it analyzed all the trial points. 3 out of 4 local solver runs converged with a positive local solver exit flag. x = 1.0e-07 * 0.0414 0.1298 fval = 1.5467e-15```

You can get different results, since `GlobalSearch` is stochastic.

The solver found three local minima, and it found the global minimum near `[0,0]`.

### Multiple Local Minima Via MultiStart

1. Write a function file to compute the objective:

```function f = sawtoothxy(x,y) [t r] = cart2pol(x,y); % change to polar coordinates h = cos(2*t - 1/2)/2 + cos(t) + 2; g = (sin(r) - sin(2*r)/2 + sin(3*r)/3 - sin(4*r)/4 + 4) ... .*r.^2./(r+1); f = g.*h; end```
2. Create the problem structure. Use the `fminunc` solver with the `Algorithm` option set to `'quasi-newton'`. The reasons for these choices are:

• The problem is unconstrained. Therefore, `fminunc` is the appropriate solver; see Optimization Decision Table (Optimization Toolbox).

• The default `fminunc` algorithm requires a gradient; see Choosing the Algorithm (Optimization Toolbox). Therefore, set `Algorithm` to `'quasi-newton'`.

```problem = createOptimProblem('fminunc',... 'objective',@(x)sawtoothxy(x(1),x(2)),... 'x0',[100,-50],'options',... optimoptions(@fminunc,'Algorithm','quasi-newton','Display','off'));```
3. Validate the problem structure by running it:

```[x,fval] = fminunc(problem) x = 1.7533 -111.9488 fval = 577.6960```
4. Create a default `MultiStart` object:

`ms = MultiStart;`
5. Run the solver for 50 iterations, recording the local minima:

```% rng(1) % uncomment to obtain the same result [x,fval,eflag,output,manymins] = run(ms,problem,50) MultiStart completed some of the runs from the start points. 9 out of 50 local solver runs converged with a positive local solver exit flag. x = -142.4608 406.8030 fval = 1.2516e+03 eflag = 2 output = struct with fields: funcCount: 8586 localSolverTotal: 50 localSolverSuccess: 9 localSolverIncomplete: 41 localSolverNoSolution: 0 message: 'MultiStart completed some of the runs from the start points.↵↵9 out of 50 local solver runs converged with a positive local solver exit flag.' manymins = 1×9 GlobalOptimSolution array with properties: X Fval Exitflag Output X0```

You can get different results, since `MultiStart` is stochastic.

The solver did not find the global minimum near `[0,0]`. It found 10 distinct local minima.

6. Plot the function values at the local minima:

`histogram([manymins.Fval],10)` Plot the function values at the three best points:

```bestf = [manymins.Fval]; histogram(bestf(1:3),10)``` `MultiStart` started `fminunc` from start points with components uniformly distributed between –1000 and 1000. `fminunc` often got stuck in one of the many local minima. `fminunc` exceeded its iteration limit or function evaluation limit 40 times.

Watch now