Optimization is the process of finding the point that minimizes a function. More specifically:

A

*local*minimum of a function is a point where the function value is smaller than or equal to the value at nearby points, but possibly greater than at a distant point.A

*global*minimum is a point where the function value is smaller than or equal to the value at all other feasible points.

Generally, Optimization
Toolbox™ solvers find a local optimum.
(This local optimum can be a global optimum.) They find the optimum
in the *basin of attraction* of the starting
point. For more information, see Basins of Attraction.

In contrast, Global Optimization Toolbox solvers are designed to search through more than one basin of attraction. They search in various ways:

`GlobalSearch`

and`MultiStart`

generate a number of starting points. They then use a local solver to find the optima in the basins of attraction of the starting points.`ga`

uses a set of starting points (called the population) and iteratively generates better points from the population. As long as the initial population covers several basins,`ga`

can examine several basins.`particleswarm`

, like`ga`

, uses a set of starting points.`particleswarm`

can examine several basins at once because of its diverse population.`simulannealbnd`

performs a random search. Generally,`simulannealbnd`

accepts a point if it is better than the previous point.`simulannealbnd`

occasionally accepts a worse point, in order to reach a different basin.`patternsearch`

looks at a number of neighboring points before accepting one of them. If some neighboring points belong to different basins,`patternsearch`

in essence looks in a number of basins at once.`surrogateopt`

begins by quasirandom sampling within bounds, looking for a small objective function value.`surrogateopt`

uses a*merit function*that, in part, gives preference to points that are far from evaluated points, which is an attempt to reach a global solution. After it cannot improve the current point,`surrogateopt`

resets, causing it to sample widely within bounds again. Resetting is another way`surrogateopt`

searches for a global solution.

If an objective function *f*(*x*) is
smooth, the vector –∇*f*(*x*) points
in the direction where *f*(*x*) decreases
most quickly. The equation of steepest descent, namely

$$\frac{d}{dt}x(t)=-\nabla f(x(t)),$$

yields a path *x*(*t*) that
goes to a local minimum as *t* gets
large. Generally, initial values *x*(0) that
are close to each other give steepest descent paths that tend to the
same minimum point. The *basin of attraction* for
steepest descent is the set of initial values leading to the same
local minimum.

The following figure shows two one-dimensional minima. The figure
shows different basins of attraction with different line styles, and
it shows directions of steepest descent with arrows. For this and
subsequent figures, black dots represent local minima. Every steepest
descent path, starting at a point *x*(0),
goes to the black dot in the basin containing *x*(0).

The following figure shows how steepest descent paths can be more complicated in more dimensions.

The following figure shows even more complicated paths and basins of attraction.

Constraints can break up one basin of attraction into several
pieces. For example, consider minimizing *y* subject
to:

*y*≥ |*x*|*y*≥ 5 – 4(*x*–2)^{2}.

The figure shows the two basins of attraction with the final points.

Code for Generating the Figure

The steepest descent paths are straight lines down to the constraint
boundaries. From the constraint boundaries, the steepest descent paths
travel down along the boundaries. The final point is either (0,0)
or (11/4,11/4), depending on whether the initial *x*-value
is above or below 2.