Choosing the Algorithm
fmincon Algorithms
fmincon
has five algorithm options:
'interiorpoint'
(default)'trustregionreflective'
'sqp'
'sqplegacy'
'activeset'
Use optimoptions
to set the
Algorithm
option at the command line.
Recommendations 


Reasoning Behind the Recommendations
'interiorpoint'
handles large, sparse problems, as well as small dense problems. The algorithm satisfies bounds at all iterations, and can recover fromNaN
orInf
results. It is a largescale algorithm; see LargeScale vs. MediumScale Algorithms. The algorithm can use special techniques for largescale problems. For details, see InteriorPoint Algorithm infmincon
options
.'sqp'
satisfies bounds at all iterations. The algorithm can recover fromNaN
orInf
results. It is not a largescale algorithm; see LargeScale vs. MediumScale Algorithms.'sqplegacy'
is similar to'sqp'
, but usually is slower and uses more memory.'activeset'
can take large steps, which adds speed. The algorithm is effective on some problems with nonsmooth constraints. It is not a largescale algorithm; see LargeScale vs. MediumScale Algorithms.'trustregionreflective'
requires you to provide a gradient, and allows only bounds or linear equality constraints, but not both. Within these limitations, the algorithm handles both large sparse problems and small dense problems efficiently. It is a largescale algorithm; see LargeScale vs. MediumScale Algorithms. The algorithm can use special techniques to save memory usage, such as a Hessian multiply function. For details, see TrustRegionReflective Algorithm infmincon
options
.
For descriptions of the algorithms, see Constrained Nonlinear Optimization Algorithms.
fsolve Algorithms
fsolve
has three algorithms:
'trustregiondogleg'
(default)'trustregion'
'levenbergmarquardt'
Use optimoptions
to set the
Algorithm
option at the command line.
Recommendations 


Reasoning Behind the Recommendations
'trustregiondogleg'
is the only algorithm that is specially designed to solve nonlinear equations. The others attempt to minimize the sum of squares of the function.The
'trustregion'
algorithm is effective on sparse problems. It can use special techniques such as a Jacobian multiply function for largescale problems.
For descriptions of the algorithms, see Equation Solving Algorithms.
fminunc Algorithms
fminunc
has two algorithms:
'quasinewton'
(default)'trustregion'
Use optimoptions
to set the
Algorithm
option at the command line.
Recommendations 

For help if the minimization fails, see When the Solver Fails or When the Solver Might Have Succeeded. 
For descriptions of the algorithms, see Unconstrained Nonlinear Optimization Algorithms.
Least Squares Algorithms
lsqlin
lsqlin
has three algorithms:
'interiorpoint'
, the default'trustregionreflective'
'activeset'
Use optimoptions
to set the
Algorithm
option at the command line.
Recommendations 

For help if the minimization fails, see When the Solver Fails or When the Solver Might Have Succeeded. 
For descriptions of the algorithms, see LeastSquares (Model Fitting) Algorithms.
lsqcurvefit and lsqnonlin
lsqcurvefit
and lsqnonlin
have three
algorithms:
'trustregionreflective'
(default for unconstrained or boundconstrained problems)'levenbergmarquardt'
'interiorpoint'
(default for problems with linear or nonlinear constraints)
Use optimoptions
to set the
Algorithm
option at the command line.
Recommendations 

For help if the minimization fails, see When the Solver Fails or When the Solver Might Have Succeeded. 
For descriptions of the algorithms, see LeastSquares (Model Fitting) Algorithms.
Linear Programming Algorithms
linprog
has four algorithms:
'dualsimplexhighs'
, the default'dualsimplexlegacy'
'interiorpoint'
'interiorpointlegacy'
Use optimoptions
to set the
Algorithm
option at the command line.
Recommendations 

Use the For help if the minimization fails, see When the Solver Fails or When the Solver Might Have Succeeded. 
Reasoning Behind the Recommendations
Often, the
'dualsimplexhighs'
,'dualsimplexlegacy'
, and'interiorpoint'
algorithms are fast, and use relatively little memory.The
'interiorpointlegacy'
algorithm is similar to'interiorpoint'
, but'interiorpointlegacy'
can be slower, less robust, or use more memory.
For descriptions of the algorithms, see Linear Programming Algorithms.
MixedInteger Linear Programming Algorithms
intlinprog
has two algorithms:
'highs'
, the default'legacy'
Use optimoptions
to set the
Algorithm
option at the command line.
Recommendations 

Use the For help if the minimization fails, see When the Solver Fails or When the Solver Might Have Succeeded. 
Reasoning Behind the Recommendations
Often, the
'highs'
algorithm works faster or more successfully than the'legacy'
algorithm.The
'highs'
algorithm has many fewer tuning options. Therefore, you have fewer choices to make when solving a problem.The
'legacy'
algorithm will be removed in a future release.
For descriptions of the algorithms, see MixedInteger Linear Programming (MILP) Algorithms.
Quadratic Programming Algorithms
quadprog
has three algorithms:
'interiorpointconvex'
(default)'trustregionreflective'
'activeset'
Use optimoptions
to set the
Algorithm
option at the command line.
Recommendations 

For help if the minimization fails, see When the Solver Fails or When the Solver Might Have Succeeded. 
For descriptions of the algorithms, see Quadratic Programming Algorithms.
LargeScale vs. MediumScale Algorithms
An optimization algorithm is large scale when it uses linear algebra that does not need to store, nor operate on, full matrices. This may be done internally by storing sparse matrices, and by using sparse linear algebra for computations whenever possible. Furthermore, the internal algorithms either preserve sparsity, such as a sparse Cholesky decomposition, or do not generate matrices, such as a conjugate gradient method.
In contrast, mediumscale methods internally create full matrices and use dense linear algebra. If a problem is sufficiently large, full matrices take up a significant amount of memory, and the dense linear algebra may require a long time to execute.
Don't let the name “large scale” mislead you; you can use a largescale algorithm on a small problem. Furthermore, you do not need to specify any sparse matrices to use a largescale algorithm. Choose a mediumscale algorithm to access extra functionality, such as additional constraint types, or possibly for better performance.
Potential Inaccuracy with InteriorPoint Algorithms
Interiorpoint algorithms in fmincon
,
quadprog
, lsqlin
, and
linprog
have many good characteristics, such as low memory
usage and the ability to solve large problems quickly. However, their solutions can
be slightly less accurate than those from other algorithms. The reason for this
potential inaccuracy is that the (internally calculated) barrier function keeps
iterates away from inequality constraint boundaries.
For most practical purposes, this inaccuracy is usually quite small.
To reduce the inaccuracy, try to:
Rerun the solver with smaller
StepTolerance
,OptimalityTolerance
, and possiblyConstraintTolerance
tolerances (but keep the tolerances sensible.) See Tolerances and Stopping Criteria).Run a different algorithm, starting from the interiorpoint solution. This can fail, because some algorithms can use excessive memory or time, and all
linprog
and somequadprog
algorithms do not accept an initial point.
For example, try to minimize the function x when bounded below
by 0. Using the fmincon
default
interiorpoint
algorithm:
options = optimoptions(@fmincon,'Algorithm','interiorpoint','Display','off'); x = fmincon(@(x)x,1,[],[],[],[],0,[],[],options)
x = 2.0000e08
Using the fmincon
sqp
algorithm:
options.Algorithm = 'sqp';
x2 = fmincon(@(x)x,1,[],[],[],[],0,[],[],options)
x2 = 0
Similarly, solve the same problem using the linprog
interiorpointlegacy
algorithm:
opts = optimoptions(@linprog,'Display','off','Algorithm','interiorpointlegacy'); x = linprog(1,[],[],[],[],0,[],1,opts)
x = 2.0833e13
Using the linprog
dualsimplex
algorithm:
opts.Algorithm = 'dualsimplex';
x2 = linprog(1,[],[],[],[],0,[],1,opts)
x2 = 0
In these cases, the interiorpoint algorithms are less accurate, but the answers are quite close to the correct answer.