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## Global Optimization Toolbox Solver Characteristics

### Solver Choices

This section describes Global Optimization Toolbox solver characteristics. The section includes recommendations for obtaining results more effectively.

To achieve better or faster solutions, first try tuning the recommended solvers by setting appropriate options or bounds. If the results are unsatisfactory, try other solvers.

Desired SolutionSmooth Objective and ConstraintsNonsmooth Objective or Constraints
Explanation of “Desired Solution”Choosing Between Solvers for Smooth ProblemsChoosing Between Solvers for Nonsmooth Problems
Single local solutionOptimization Toolbox™ functions; see Optimization Decision Table`fminbnd`, `patternsearch`, `fminsearch`, `ga`, `particleswarm`, `simulannealbnd`, `surrogateopt`
Multiple local solutions`GlobalSearch`, `MultiStart``patternsearch`, `ga`, `particleswarm`, `simulannealbnd`, or `surrogateopt` started from multiple initial points `x0` or from multiple initial populations
Single global solution`GlobalSearch`, `MultiStart`, `patternsearch`, `particleswarm`, `ga`, `simulannealbnd`, `surrogateopt``patternsearch`, `ga`, `particleswarm`, `simulannealbnd`, `surrogateopt`
Single local solution using parallel processing`MultiStart`, Optimization Toolbox functions`patternsearch`, `ga`, `particleswarm`, `surrogateopt`
Multiple local solutions using parallel processing`MultiStart``patternsearch`, `ga`, or `particleswarm` started from multiple initial points `x0` or from multiple initial populations
Single global solution using parallel processing`MultiStart``patternsearch`, `ga`, `particleswarm`, `surrogateopt`

### Explanation of “Desired Solution”

To understand the meaning of the terms in “Desired Solution,” consider the example

f(x)=100x2(1–x)2x,

which has local minima `x1` near 0 and `x2` near 1:

The minima are located at:

```fun = @(x)(100*x^2*(x - 1)^2 - x); x1 = fminbnd(fun,-0.1,0.1) x1 = 0.0051 x2 = fminbnd(fun,0.9,1.1) x2 = 1.0049```

Description of the Terms

TermMeaning
Single local solutionFind one local solution, a point x where the objective function f(x) is a local minimum. For more details, see Local vs. Global Optima. In the example, both `x1` and `x2` are local solutions.
Multiple local solutionsFind a set of local solutions. In the example, the complete set of local solutions is `{x1,x2}`.
Single global solutionFind the point x where the objective function f(x) is a global minimum. In the example, the global solution is `x2`.

### Choosing Between Solvers for Smooth Problems

#### Single Global Solution

1. Try `GlobalSearch` first. It is most focused on finding a global solution, and has an efficient local solver, `fmincon`.

2. Try `MultiStart` next. It has efficient local solvers, and can search a wide variety of start points.

3. Try `patternsearch` next. It is less efficient, since it does not use gradients. However, `patternsearch` is robust and is more efficient than the remaining local solvers To search for a global solution, start `patternsearch` from a variety of start points.

4. Try `surrogateopt` next. `surrogateopt` attempts to find a global solution using the fewest objective function evaluations. `surrogateopt` has more overhead per function evaluation than most other solvers. `surrogateopt` requires finite bounds, and accepts integer constraints, linear constraints, and nonlinear inequality constraints.

5. Try `particleswarm` next, if your problem is unconstrained or has only bound constraints. Usually, `particleswarm` is more efficient than the remaining solvers, and can be more efficient than `patternsearch`.

6. Try `ga` next. It can handle all types of constraints, and is usually more efficient than `simulannealbnd`.

7. Try `simulannealbnd` last. It can handle problems with no constraints or bound constraints. `simulannealbnd` is usually the least efficient solver. However, given a slow enough cooling schedule, it can find a global solution.

#### Multiple Local Solutions

`GlobalSearch` and `MultiStart` both provide multiple local solutions. For the syntax to obtain multiple solutions, see Multiple Solutions. `GlobalSearch` and `MultiStart` differ in the following characteristics:

• `MultiStart` can find more local minima. This is because `GlobalSearch` rejects many generated start points (initial points for local solution). Essentially, `GlobalSearch` accepts a start point only when it determines that the point has a good chance of obtaining a global minimum. In contrast, `MultiStart` passes all generated start points to a local solver. For more information, see GlobalSearch Algorithm.

• `MultiStart` offers a choice of local solver: `fmincon`, `fminunc`, `lsqcurvefit`, or `lsqnonlin`. The `GlobalSearch` solver uses only `fmincon` as its local solver.

• `GlobalSearch` uses a scatter-search algorithm for generating start points. In contrast, `MultiStart` generates points uniformly at random within bounds, or allows you to provide your own points.

• `MultiStart` can run in parallel. See How to Use Parallel Processing in Global Optimization Toolbox.

### Choosing Between Solvers for Nonsmooth Problems

Choose the applicable solver with the lowest number. For problems with integer constraints, use `ga`.

1. Use `fminbnd` first on one-dimensional bounded problems only. `fminbnd` provably converges quickly in one dimension.

2. Use `patternsearch` on any other type of problem. `patternsearch` provably converges, and handles all types of constraints.

3. Try `surrogateopt` for problems that have time-consuming objective functions. `surrogateopt` searches for a global solution. `surrogateopt` requires finite bounds, and accepts integer constraints, linear constraints, and nonlinear inequality constraints.

4. Try `fminsearch` next for low-dimensional unbounded problems. `fminsearch` is not as general as `patternsearch` and can fail to converge. For low-dimensional problems, `fminsearch` is simple to use, since it has few tuning options.

5. Try `particleswarm` next on unbounded or bound-constrained problems. `particleswarm` has little supporting theory, but is often an efficient algorithm.

6. Try `ga` next. `ga` has little supporting theory and is often less efficient than `patternsearch` or `particleswarm`. `ga` handles all types of constraints. `ga` and `surrogateopt` are the only Global Optimization Toolbox solvers that accept integer constraints.

7. Try `simulannealbnd` last for unbounded problems, or for problems with bounds. `simulannealbnd` provably converges only for a logarithmic cooling schedule, which is extremely slow. `simulannealbnd` takes only bound constraints, and is often less efficient than `ga`.

### Solver Characteristics

SolverConvergenceCharacteristics
`GlobalSearch`Fast convergence to local optima for smooth problemsDeterministic iterates
Gradient-based
Automatic stochastic start points
Removes many start points heuristically
`MultiStart`Fast convergence to local optima for smooth problemsDeterministic iterates
Can run in parallel; see How to Use Parallel Processing in Global Optimization Toolbox
Gradient-based
Stochastic or deterministic start points, or combination of both
Automatic stochastic start points
Runs all start points
Choice of local solver: `fmincon`, `fminunc`, `lsqcurvefit`, or `lsqnonlin`
`patternsearch`Proven convergence to local optimum; slower than gradient-based solversDeterministic iterates
Can run in parallel; see How to Use Parallel Processing in Global Optimization Toolbox
No gradients
User-supplied start point
`surrogateopt`Proven convergence to global optimum for bounded problems; slower than gradient-based solvers; generally stops by reaching a function evaluation limit or other limitStochastic iterates
Can run in parallel; see How to Use Parallel Processing in Global Optimization Toolbox
Best used for time-consuming objective functions
Requires bound constraints, accepts linear constraints and nonlinear inequality constraints
Allows integer constraints; see Mixed-Integer Surrogate Optimization
No gradients
Automatic start points or user-supplied points, or a combination of both
`particleswarm`No convergence proofStochastic iterates
Can run in parallel; see How to Use Parallel Processing in Global Optimization Toolbox
Population-based
No gradients
Automatic start population or user-supplied population, or a combination of both
Only bound constraints
`ga`No convergence proofStochastic iterates
Can run in parallel; see How to Use Parallel Processing in Global Optimization Toolbox
Population-based
No gradients
Allows integer constraints; see Mixed Integer ga Optimization
Automatic start population or user-supplied population, or a combination of both
`simulannealbnd`Proven to converge to global optimum for bounded problems with very slow cooling scheduleStochastic iterates
No gradients
User-supplied start point
Only bound constraints

Explanation of some characteristics:

• Convergence — Solvers can fail to converge to any solution when started far from a local minimum. When started near a local minimum, gradient-based solvers converge to a local minimum quickly for smooth problems. `patternsearch` provably converges for a wide range of problems, but the convergence is slower than gradient-based solvers. Both `ga` and `simulannealbnd` can fail to converge in a reasonable amount of time for some problems, although they are often effective.

• Iterates — Solvers iterate to find solutions. The steps in the iteration are iterates. Some solvers have deterministic iterates. Others use random numbers and have stochastic iterates.

• Gradients — Some solvers use estimated or user-supplied derivatives in calculating the iterates. Other solvers do not use or estimate derivatives, but use only objective and constraint function values.

• Start points — Most solvers require you to provide a starting point for the optimization in order to obtain the dimension of the decision variables. `ga` and `surrogateopt` do not require any starting points, because they take the dimension of the decision variables as an input or infer dimensions from bounds. These solvers generate a start point or population automatically, or they accept a point or points that you supply.

Compare the characteristics of Global Optimization Toolbox solvers to Optimization Toolbox solvers.

SolverConvergenceCharacteristics
`fmincon`, `fminunc`, `fseminf`, `lsqcurvefit`, `lsqnonlin`Proven quadratic convergence to local optima for smooth problemsDeterministic iterates
Gradient-based
User-supplied starting point
`fminsearch`No convergence proof — counterexamples exist.Deterministic iterates
No gradients
User-supplied start point
No constraints
`fminbnd`Proven convergence to local optima for smooth problems, slower than quadratic.Deterministic iterates
No gradients
User-supplied start interval
Only one-dimensional problems

All these Optimization Toolbox solvers:

• Have deterministic iterates

• Require a start point or interval

• Search just one basin of attraction

### Why Are Some Solvers Objects?

`GlobalSearch` and `MultiStart` are objects. What does this mean for you?

• You create a `GlobalSearch` or `MultiStart` object before running your problem.

• You can reuse the object for running multiple problems.

• `GlobalSearch` and `MultiStart` objects are containers for algorithms and global options. You use these objects to run a local solver multiple times. The local solver has its own options.

For more information, see the Classes documentation.