## Multiple Solutions

### About Multiple Solutions

You obtain multiple solutions in an object by calling `run`

with
the syntax

[x,fval,exitflag,output,manymins] = run(...);

`manymins`

is a vector of solution objects; see `GlobalOptimSolution`

. The `manymins`

vector is in
order of objective function value, from lowest (best) to highest (worst). Each
solution object contains the following properties (fields):

`X`

— a local minimum`Fval`

— the value of the objective function at`X`

`Exitflag`

— the exit flag for the local solver (described in the local solver function reference page:`fmincon`

`exitflag`

,`fminunc`

`exitflag`

,`lsqcurvefit`

`exitflag`

, or`lsqnonlin`

`exitflag`

`Output`

— an output structure for the local solver (described in the local solver function reference page:`fmincon`

`output`

,`fminunc`

`output`

,`lsqcurvefit`

`output`

, or`lsqnonlin`

`output`

`X0`

— a cell array of start points that led to the solution point`X`

**Note**

`manymins`

contains only those solutions corresponding to
positive local solver exit flags. If you want to collect all the local
solutions, not only the ones corresponding to positive exit flags, use the
`@savelocalsolutions`

output function. See Output Functions for GlobalSearch and MultiStart.

There are several ways to examine the vector of solution objects:

In the MATLAB

^{®}Workspace Browser. Double-click the solution object, and then double-click the resulting display in the Variables editor.Using dot notation.

`GlobalOptimSolution`

properties are capitalized. Use proper capitalization to access the properties.For example, to find the vector of function values, enter:

fcnvals = [manymins.Fval]

fcnvals = -1.0316 -0.2155 0

To get a cell array of all the start points that led to the lowest function value (the first element of

`manymins`

), enter:smallX0 = manymins(1).X0

Plot some field values. For example, to see the range of resulting

`Fval`

, enter:histogram([manymins.Fval],10)

This results in a histogram of the computed function values. (The figure shows a histogram from a different example than the previous few figures.)

### Change the Definition of Distinct Solutions

You might find out, after obtaining multiple local solutions,
that your tolerances were not appropriate. You can have many more
local solutions than you want, spaced too closely together. Or you
can have fewer solutions than you want, with `GlobalSearch`

or `MultiStart`

clumping
together too many solutions.

To deal with this situation, run the solver again with different
tolerances. The `XTolerance`

and `FunctionTolerance`

tolerances
determine how the solvers group their outputs into the `GlobalOptimSolution`

vector.
These tolerances are properties of the `GlobalSearch`

or `MultiStart`

object.

For example, suppose you want to use the `active-set`

algorithm
in `fmincon`

to solve the problem in Example of Run with MultiStart. Further
suppose that you want to have tolerances of `0.01`

for
both `XTolerance`

and `FunctionTolerance`

.
The `run`

method groups local solutions whose objective
function values are within `FunctionTolerance`

of
each other, and which are also less than `XTolerance`

apart
from each other. To obtain the solution:

% Set the random stream to get exactly the same output % rng(14,'twister') ms = MultiStart('FunctionTolerance',0.01,'XTolerance',0.01); opts = optimoptions(@fmincon,'Algorithm','active-set'); sixmin = @(x)(4*x(1)^2 - 2.1*x(1)^4 + x(1)^6/3 ... + x(1)*x(2) - 4*x(2)^2 + 4*x(2)^4); problem = createOptimProblem('fmincon','x0',[-1,2],... 'objective',sixmin,'lb',[-3,-3],'ub',[3,3],... 'options',opts); [xminm,fminm,flagm,outptm,someminsm] = run(ms,problem,50);

MultiStart completed the runs from all start points. All 50 local solver runs converged with a positive local solver exit flag.

someminsm

someminsm = 1x5 GlobalOptimSolution Properties: X Fval Exitflag Output X0

In this case, `MultiStart`

generated five distinct
solutions. Here “distinct” means that the solutions
are more than 0.01 apart in either objective function value or location.