Find global minimum
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Description
A GlobalSearch object contains properties (options) that affect how run repeatedly runs a local solver to generate a GlobalOptimSolution object. When run, the solver attempts to locate a solution that has the lowest objective function value.
Creation
Syntax
gs = GlobalSearch
gs = GlobalSearch(Name,Value)
gs = GlobalSearch(oldGS,Name,Value)
gs = GlobalSearch(ms)
Description
example
gs = GlobalSearch creates gs, a GlobalSearch solver with its properties set to the defaults.
example
gs = GlobalSearch(Name,Value) sets properties using name-value pairs.
example
gs = GlobalSearch(oldGS,Name,Value) creates a copy of the oldGS GlobalSearch solver, and sets properties using name-value pairs.
example
gs = GlobalSearch(ms) creates gs, a GlobalSearch solver, with common property values from the ms MultiStart solver.
Properties
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BasinRadiusFactor — Basin radius decrease factor
0.2 (default) | scalar from 0 through 1
Basin radius decrease factor, specified as a scalar from 0 through 1. A basin radius decreases after MaxWaitCycle consecutive start points are within the basin. The basin radius decreases by a factor of 1–BasinRadiusFactor.
Set BasinRadiusFactor to 0 to disable updates of the basin radius.
Example: 0.5
Data Types: double
Display — Level of display to the Command Window
'final' (default) | 'iter' | 'off'
Level of display to the Command Window, specified as one of the following character vectors or strings:
'final'– Report summary results afterrunfinishes.'iter'– Report results after the initialfminconrun, after Stage 1, after every 200 start points, and after every run offmincon, in addition to the final summary.'off'– No display.
Example: 'iter'
Data Types: char | string
DistanceThresholdFactor — Multiplier for determining trial point is in existing basin
0.75 (default) | nonnegative scalar
Multiplier for determining whether a trial point is in an existing basin of attraction, specified as a nonnegative scalar. For details, see Examine Stage 2 Trial Point to See if fmincon Runs.
Example: 0.5
Data Types: double
FunctionTolerance — Tolerance on function values for considering solutions equal
1e-6 (default) | nonnegative scalar
Tolerance on function values for considering solutions equal, specified as a nonnegative scalar. Solvers consider two solutions identical if they are within XTolerance relative distance of each other and have objective function values within FunctionTolerance relative difference of each other. If both conditions are not met, solvers report the solutions as distinct. Set FunctionTolerance to 0 to obtain the results of every local solver run. Set FunctionTolerance to a larger value to have fewer results. For GlobalSearch details, see Update Solution Set in When fmincon Runs. For MultiStart details, see Create GlobalOptimSolution Object.
Example: 1e-4
Data Types: double
MaxTime — Maximum time in seconds that GlobalSearch runs
Inf (default) | positive scalar
Maximum time in seconds that GlobalSearch runs, specified as a positive scalar. GlobalSearch and its local solvers halt when MaxTime seconds have passed since the beginning of the run, as measured by tic and toc.
MaxTime does not interrupt local solvers during a run, so the total time can exceed MaxTime.
Example: 180 stops the solver the first time a local solver call finishes after 180 seconds.
Data Types: double
MaxWaitCycle — Algorithm control parameter
20 (default) | positive integer
Algorithm control parameter, specified as a positive integer.
If the observed penalty function of
MaxWaitCycleconsecutive trial points is at least the penalty threshold, then raise the penalty threshold (see PenaltyThresholdFactor).If
MaxWaitCycleconsecutive trial points are in a basin, then update that basin's radius (see BasinRadiusFactor).
Example: 40
Data Types: double
NumStageOnePoints — Number of Stage 1 points
200 (default) | positive integer
Number of Stage 1 points, specified as a positive integer. For details, see Obtain Stage 1 Start Point, Run.
Example: 1000
Data Types: double
NumTrialPoints — Number of potential start points
1000 (default) | positive integer
Number of potential start points, specified as a positive integer.
Example: 3e4
Data Types: double
OutputFcn — Report on solver progress or halt solver
[] (default) | @savelocalsolutions | function handle | cell array of function handles
Report on solver progress or halt solver, specified as a function handle or cell array of function handles. Output functions run after each local solver call. They also run when the global solver starts and ends. Write output functions using the syntax described in OutputFcn. See Custom GlobalSearch Output Function.
The built-in @savelocalsolutions output function saves all local solutions to the localSolTable variable in the base workspace. This output function saves the infeasible local solutions as well as the feasible local solutions. For an example, see Collect All Local Solutions.
Data Types: cell | function_handle
PenaltyThresholdFactor — Increase in penalty threshold
0.2 (default) | positive scalar
Increase in the penalty threshold, specified as a positive scalar. For details, see React to Large Counter Values.
Example: 0.4
Data Types: double
PlotFcn — Plot solver progress
[] (default) | function handle | cell array of function handles
Plot solver progress, specified as a function handle or cell array of function handles. Plot functions run after each local solver call. They also run when the global solver starts and ends. Write plot functions using the syntax described in OutputFcn.
There are two built-in plot functions:
@gsplotbestfplots the best objective function value.@gsplotfunccountplots the number of function evaluations.
See MultiStart Plot Function.
Example: @gsplotbestf
Data Types: cell | function_handle
StartPointsToRun — Start points to run
'all' (default) | 'bounds' | 'bounds-ineqs'
Start points to run, specified as:
'all'— Run all start points.'bounds'— Run only start points that satisfy bounds.'bounds-ineqs'— Run only start points that satisfy bounds and inequality constraints.
GlobalSearch checks the StartPointsToRun property only during Stage 2 of the GlobalSearch algorithm (the main loop). For more information, see GlobalSearch Algorithm.
Example: 'bounds' runs only points that satisfy all bounds.
Data Types: char | string
XTolerance — Tolerance on distance for considering solutions equal
1e-6 (default) | nonnegative scalar
Tolerance on distance for considering solutions equal, specified as a nonnegative scalar. Solvers consider two solutions identical if they are within XTolerance relative distance of each other and have objective function values within FunctionTolerance relative difference of each other. If both conditions are not met, solvers report the solutions as distinct. Set XTolerance to 0 to obtain the results of every local solver run. Set XTolerance to a larger value to have fewer results. For GlobalSearch details, see Update Solution Set in When fmincon Runs. For MultiStart details, see Create GlobalOptimSolution Object.
Example: 2e-4
Data Types: double
Object Functions
| run | Run multiple-start solver |
Examples
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Run GlobalSearch on Multidimensional Problem
Open Live Script
Create an optimization problem that has several local minima, and try to find the global minimum using GlobalSearch. The objective is the six-hump camel back problem (see Run the Solver).
rng default % For reproducibilitygs = GlobalSearch;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]);x = run(gs,problem)
GlobalSearch stopped because it analyzed all the trial points.All 8 local solver runs converged with a positive local solver exit flag.
x = 1×2 -0.0898 0.7127You can request the objective function value at x when you call run by using the following syntax:
[x,fval] = run(gs,problem)
However, if you neglected to request fval, you can still compute the objective function value at x.
fval = sixmin(x)
fval = -1.0316
Run GlobalSearch on 1-D Problem
Open Live Script
Consider a function with several local minima.
fun = @(x) x.^2 + 4*sin(5*x);fplot(fun,[-5,5])
To search for the global minimum, run GlobalSearch using the fmincon 'sqp' algorithm.
rng default % For reproducibilityopts = optimoptions(@fmincon,'Algorithm','sqp');problem = createOptimProblem('fmincon','objective',... fun,'x0',3,'lb',-5,'ub',5,'options',opts);gs = GlobalSearch;[x,f] = run(gs,problem)
GlobalSearch stopped because it analyzed all the trial points.All 23 local solver runs converged with a positive local solver exit flag.
x = -0.3080
f = -3.9032
GlobalSearch Using Common Properties from MultiStart
Open Live Script
Create a nondefault MultiStart object.
ms = MultiStart('FunctionTolerance',2e-4,'UseParallel',true)
ms = MultiStart with properties: UseParallel: 1 Display: 'final' FunctionTolerance: 2.0000e-04 MaxTime: Inf OutputFcn: [] PlotFcn: [] StartPointsToRun: 'all' XTolerance: 1.0000e-06
Create a GlobalSearch object that uses the available properties from ms.
gs = GlobalSearch(ms)
gs = GlobalSearch with properties: NumTrialPoints: 1000 BasinRadiusFactor: 0.2000 DistanceThresholdFactor: 0.7500 MaxWaitCycle: 20 NumStageOnePoints: 200 PenaltyThresholdFactor: 0.2000 Display: 'final' FunctionTolerance: 2.0000e-04 MaxTime: Inf OutputFcn: [] PlotFcn: [] StartPointsToRun: 'all' XTolerance: 1.0000e-06
gs has the same nondefault value of FunctionTolerance as ms. But gs does not use the UseParallel property.
Update GlobalSearch Properties
Open Live Script
Create a GlobalSearch object with a FunctionTolerance of 1e-4.
gs = GlobalSearch('FunctionTolerance',1e-4)gs = GlobalSearch with properties: NumTrialPoints: 1000 BasinRadiusFactor: 0.2000 DistanceThresholdFactor: 0.7500 MaxWaitCycle: 20 NumStageOnePoints: 200 PenaltyThresholdFactor: 0.2000 Display: 'final' FunctionTolerance: 1.0000e-04 MaxTime: Inf OutputFcn: [] PlotFcn: [] StartPointsToRun: 'all' XTolerance: 1.0000e-06
Update the XTolerance property to 1e-3 and the StartPointsToRun property to 'bounds'.
gs = GlobalSearch(gs,'XTolerance',1e-3,'StartPointsToRun','bounds')
gs = GlobalSearch with properties: NumTrialPoints: 1000 BasinRadiusFactor: 0.2000 DistanceThresholdFactor: 0.7500 MaxWaitCycle: 20 NumStageOnePoints: 200 PenaltyThresholdFactor: 0.2000 Display: 'final' FunctionTolerance: 1.0000e-04 MaxTime: Inf OutputFcn: [] PlotFcn: [] StartPointsToRun: 'bounds' XTolerance: 1.0000e-03
You can also update properties one at a time by using dot notation.
gs.MaxTime = 1800
gs = GlobalSearch with properties: NumTrialPoints: 1000 BasinRadiusFactor: 0.2000 DistanceThresholdFactor: 0.7500 MaxWaitCycle: 20 NumStageOnePoints: 200 PenaltyThresholdFactor: 0.2000 Display: 'final' FunctionTolerance: 1.0000e-04 MaxTime: 1800 OutputFcn: [] PlotFcn: [] StartPointsToRun: 'bounds' XTolerance: 1.0000e-03
Algorithms
For a detailed description of the algorithm, see GlobalSearch Algorithm. Ugray et al. [1] describe both the algorithm and the scatter-search method of generating trial points.
References
[1] Ugray, Zsolt, Leon Lasdon, John Plummer, Fred Glover, James Kelly, and Rafael Martí. Scatter Search and Local NLP Solvers: A Multistart Framework for Global Optimization. INFORMS Journal on Computing, Vol. 19, No. 3, 2007, pp. 328–340.
Version History
Introduced in R2010a
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R2024a: Output Function: Built-in output function and updated optimValues structure
The built-in @savelocalsolutions output function saves all local solutions to the localSolTable variable in the base workspace. This output function saves the infeasible local solutions as well as the feasible local solutions.
The optimValues Structure in output functions and plot functions now contains the constrviolation field, which has the maximum constraint violation. This value is 0 for no constraint violations.
See Also
GlobalOptimSolution | MultiStart | run
Topics
- Example of Run with GlobalSearch
- Workflow for GlobalSearch and MultiStart
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