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There are seven Global Optimization Toolbox solvers:
ga (genetic algorithm)
patternsearch, also called direct search
simulannealbnd (simulated annealing)
gamultiobj, which is not a minimizer; see Multiobjective Optimization
Choose an optimizer based on problem characteristics and on the type of solution you want. Solver Characteristics contains more information to help you decide which solver is likely to be most suitable.
Desired Solution | Smooth Objective and Constraints | Nonsmooth Objective or Constraints |
---|---|---|
Explanation of "Desired Solution" | Choosing Between Solvers for Smooth Problems | Choosing Between Solvers for Nonsmooth Problems |
Single local solution | Optimization Toolbox™ functions; see Optimization Decision Table in the Optimization Toolbox documentation | fminbnd, patternsearch, fminsearch, ga, particleswarm, simulannealbnd |
Multiple local solutions | GlobalSearch, MultiStart | |
Single global solution | GlobalSearch, MultiStart, patternsearch, ga, simulannealbnd | patternsearch, ga, particleswarm, simulannealbnd |
Single local solution using parallel processing | MultiStart, Optimization Toolbox functions | patternsearch, ga, particleswarm |
Multiple local solutions using parallel processing | MultiStart | |
Single global solution using parallel processing | MultiStart | patternsearch, ga, particleswarm |
To understand the meaning of the terms in "Desired Solution," consider the example
f(x)=100x^{2}(1–x)^{2}–x,
which has local minima x1 near 0 and x2 near 1:
Code for generating the figure
The minima are located at:
x1 = fminsearch(@(x)(100*x^2*(x - 1)^2 - x),0) x1 = 0.0051 x2 = fminsearch(@(x)(100*x^2*(x - 1)^2 - x),1) x2 = 1.0049
Description of the Terms
Term | Meaning |
---|---|
Single local solution | Find 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 solutions | Find a set of local solutions. In the example, the complete set of local solutions is {x1,x2}. |
Single global solution | Find the point x where the objective function f(x) is a global minimum. In the example, the global solution is x2. |
Try GlobalSearch first. It is most focused on finding a global solution, and has an efficient local solver, fmincon.
Try MultiStart second. It has efficient local solvers, and can search a wide variety of start points.
Try patternsearch third. It is less efficient, since it does not use gradients. However, patternsearch is robust and is more efficient than the remaining local solvers.
Try particleswarm fourth, 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.
Try ga fifth. It can handle all types of constraints, and is usually more efficient than simulannealbnd.
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.
Choose the applicable solver with the lowest number. For problems with integer constraints, use ga.
Use fminbnd first on one-dimensional bounded problems only. fminbnd provably converges quickly in one dimension.
Use patternsearch on any other type of problem. patternsearch provably converges, and handles all types of constraints.
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.
Try particleswarm next on unbounded or bound-constrained problems. particleswarm has little supporting theory, but is often an efficient algorithm.
Try ga next. ga has little supporting theory and is often less efficient than patternsearch or particleswarm. It handles all types of constraints. ga is the only solver that handles integer constraints.
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 | Convergence | Characteristics |
---|---|---|
GlobalSearch | Fast convergence to local optima for smooth problems. | Deterministic iterates |
Gradient-based | ||
Automatic stochastic start points | ||
Removes many start points heuristically | ||
MultiStart | Fast convergence to local optima for smooth problems. | Deterministic iterates |
Can run in parallel; see How to Use Parallel Processing | ||
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 solvers. | Deterministic iterates |
Can run in parallel; see How to Use Parallel Processing | ||
No gradients | ||
User-supplied start point | ||
particleswarm | No convergence proof. | Stochastic iterates |
Can run in parallel; see How to Use Parallel Processing | ||
Population-based | ||
No gradients | ||
Automatic start population, or user-supplied population, or combination of both | ||
Only bound constraints | ||
ga | No convergence proof. | Stochastic iterates |
Can run in parallel; see How to Use Parallel Processing | ||
Population-based | ||
No gradients | ||
Allows integer constraints; see Mixed Integer Optimization | ||
Automatic start population, or user-supplied population, or combination of both | ||
simulannealbnd | Proven to converge to global optimum for bounded problems with very slow cooling schedule. | Stochastic 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. One reason they require a start point is to obtain the dimension of the decision variables. ga does not require any starting points, because it takes the dimension of the decision variables as an input. ga can generate its start population automatically.
Compare the characteristics of Global Optimization Toolbox solvers to Optimization Toolbox solvers.
Solver | Convergence | Characteristics |
---|---|---|
fmincon, fminunc, fseminf, lsqcurvefit, lsqnonlin | Proven quadratic convergence to local optima for smooth problems | Deterministic 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 point | ||
Only one-dimensional problems |
All these Optimization Toolbox solvers:
Have deterministic iterates
Start from one user-supplied point
Search just one basin of attraction
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 Object-Oriented Programming documentation.