fminunc

Find minimum of unconstrained multivariable function

Syntax

x = fminunc(fun,x0)

x = fminunc(fun,x0,options)

x = fminunc(problem)

[x,fval] = fminunc(___)

[x,fval,exitflag,output] = fminunc(___)

[x,fval,exitflag,output,grad,hessian] = fminunc(___)

Description

Nonlinear programming solver.

Finds the minimum of a problem specified by

$$\underset{x}{\min }f(x)$$

where f(x) is a function that returns a scalar.

x is a vector or a matrix; see Matrix Arguments.

x = fminunc(fun,x0) starts at the point x0 and attempts to find a local minimum x of the function described in fun. The point x0 can be a scalar, vector, or matrix.

Note

Passing Extra Parameters explains how to pass extra parameters to the objective function and nonlinear constraint functions, if necessary.

fminunc is for nonlinear problems without constraints. If your problem has constraints, generally use fmincon. See Optimization Decision Table.

x = fminunc(fun,x0,options) minimizes fun with the optimization options specified in options. Use optimoptions to set these options.

x = fminunc(problem) finds the minimum for problem, where problem is a structure described in Input Arguments. Create the problem structure by exporting a problem from Optimization app, as described in Exporting Your Work.

[x,fval] = fminunc(___), for any syntax, returns the value of the objective function fun at the solution x.

[x,fval,exitflag,output] = fminunc(___) additionally returns a value exitflag that describes the exit condition of fminunc, and a structure output with information about the optimization process.

[x,fval,exitflag,output,grad,hessian] = fminunc(___) additionally returns:

• grad — Gradient of fun at the solution x.

• hessian — Hessian of fun at the solution x. See fminunc Hessian.

Examples

Minimize a Polynomial

Minimize the function $$f(x)=3x_1^2+2x_1{x}_2+x_2^2-4x_1+5x_2$$.

To do so, write an anonymous function fun that calculates the objective.

fun = @(x)3*x(1)^2 + 2*x(1)*x(2) + x(2)^2 - 4*x(1) + 5*x(2);

Call fminunc to find a minimum of fun near [1,1].

x0 = [1,1];
[x,fval] = fminunc(fun,x0)

Local minimum found.

Optimization completed because the size of the gradient is less than
the value of the optimality tolerance.

x = 1×2

    2.2500   -4.7500

fval = -16.3750

Supply Gradient

fminunc can be faster and more reliable when you provide derivatives.

Write an objective function that returns the gradient as well as the function value. Use the conditionalized form described in Including Gradients and Hessians. The objective function is Rosenbrock's function,

$$f(x)=100{\left(x_2-x_1^2\right)}^2+(1-x_1{)}^2,$$

which has gradient

$$\nabla f(x)=\left[\begin{array}{c}-400\left(x_2-x_1^2\right)x_1-2\left(1-x_1\right)\\ 200\left(x_2-x_1^2\right)\end{array}\right]$$.

The code for the objective function with gradient appears at the end of this example.

Create options to use the objective function’s gradient. Also, set the algorithm to 'trust-region'.

options = optimoptions('fminunc','Algorithm','trust-region','SpecifyObjectiveGradient',true);

Set the initial point to [-1,2]. Then call fminunc.

x0 = [-1,2];
fun = @rosenbrockwithgrad;
x = fminunc(fun,x0,options)

Local minimum found.

Optimization completed because the size of the gradient is less than
the value of the optimality tolerance.

x = 1×2

    1.0000    1.0000

The following code creates the rosenbrockwithgrad function, which includes the gradient as the second output.

function [f,g] = rosenbrockwithgrad(x)
% Calculate objective f
f = 100*(x(2) - x(1)^2)^2 + (1-x(1))^2;

if nargout > 1 % gradient required
    g = [-400*(x(2)-x(1)^2)*x(1) - 2*(1-x(1));
        200*(x(2)-x(1)^2)];
end
end

Use Problem Structure

Solve the same problem as in Supply Gradient using a problem structure instead of separate arguments.

Write an objective function that returns the gradient as well as the function value. Use the conditionalized form described in Including Gradients and Hessians. The objective function is Rosenbrock's function,

$$f(x)=100{\left(x_2-x_1^2\right)}^2+(1-x_1{)}^2$$,

which has gradient

$$\nabla f(x)=\left[\begin{array}{c}-400\left(x_2-x_1^2\right)x_1-2\left(1-x_1\right)\\ 200\left(x_2-x_1^2\right)\end{array}\right]$$.

The code for the objective function with gradient appears at the end of this example.

Create options to use the objective function’s gradient. Also, set the algorithm to 'trust-region'.

options = optimoptions('fminunc','Algorithm','trust-region','SpecifyObjectiveGradient',true);

Create a problem structure including the initial point x0 = [-1,2]. For the required fields in this structure, see problem.

problem.options = options;
problem.x0 = [-1,2];
problem.objective = @rosenbrockwithgrad;
problem.solver = 'fminunc';

Solve the problem.

x = fminunc(problem)

Local minimum found.

Optimization completed because the size of the gradient is less than
the value of the optimality tolerance.

x = 1×2

    1.0000    1.0000

The following code creates the rosenbrockwithgrad function, which includes the gradient as the second output.

function [f,g] = rosenbrockwithgrad(x)
% Calculate objective f
f = 100*(x(2) - x(1)^2)^2 + (1-x(1))^2;

if nargout > 1 % gradient required
    g = [-400*(x(2)-x(1)^2)*x(1)-2*(1-x(1));
        200*(x(2)-x(1)^2)];
end
end

Obtain Optimal Objective Function Value

Find both the location of the minimum of a nonlinear function and the value of the function at that minimum. The objective function is

$$f(x)=x(1)e^{-{\Vert x\Vert }_2^2}+{\Vert x\Vert }_2^2/20$$.

fun = @(x)x(1)*exp(-(x(1)^2 + x(2)^2)) + (x(1)^2 + x(2)^2)/20;

Find the location and objective function value of the minimizer starting at x0 = [1,2].

x0 = [1,2];
[x,fval] = fminunc(fun,x0)

Local minimum found.

Optimization completed because the size of the gradient is less than
the value of the optimality tolerance.

x = 1×2

   -0.6691    0.0000

fval = -0.4052

Examine the Solution Process

Choose fminunc options and outputs to examine the solution process.

Set options to obtain iterative display and use the 'quasi-newton' algorithm.

options = optimoptions(@fminunc,'Display','iter','Algorithm','quasi-newton');

The objective function is

$$f(x)=x(1)e^{-{\Vert x\Vert }_2^2}+{\Vert x\Vert }_2^2/20.$$

fun = @(x)x(1)*exp(-(x(1)^2 + x(2)^2)) + (x(1)^2 + x(2)^2)/20;

Start the minimization at x0 = [1,2], and obtain outputs that enable you to examine the solution quality and process.

x0 = [1,2];
[x,fval,exitflag,output] = fminunc(fun,x0,options)

                                                        First-order 
 Iteration  Func-count       f(x)        Step-size       optimality
     0           3         0.256738                         0.173
     1           6         0.222149              1          0.131  
     2           9          0.15717              1          0.158  
     3          18        -0.227902       0.438133          0.386  
     4          21        -0.299271              1           0.46  
     5          30        -0.404028       0.102071         0.0458  
     6          33        -0.404868              1         0.0296  
     7          36        -0.405236              1        0.00119  
     8          39        -0.405237              1       0.000252  
     9          42        -0.405237              1       7.97e-07  

Local minimum found.

Optimization completed because the size of the gradient is less than
the value of the optimality tolerance.

x = 1×2

   -0.6691    0.0000

fval = -0.4052

exitflag = 1

output = struct with fields:
       iterations: 9
        funcCount: 42
         stepsize: 2.9343e-04
     lssteplength: 1
    firstorderopt: 7.9721e-07
        algorithm: 'quasi-newton'
          message: '...'

• The exit flag 1 shows that the solution is a local optimum.

• The output structure shows the number of iterations, number of function evaluations, and other information.

• The iterative display also shows the number of iterations and function evaluations.

Input Arguments

fun — Function to minimize

function handle | function name

Function to minimize, specified as a function handle or function name. fun is a function that accepts a vector or array x and returns a real scalar f, the objective function evaluated at x.

Specify fun as a function handle for a file:

x = fminunc(@myfun,x0)

where myfun is a MATLAB^® function such as

function f = myfun(x)
f = ...            % Compute function value at x

You can also specify fun as a function handle for an anonymous function:

x = fminunc(@(x)norm(x)^2,x0);

If you can compute the gradient of fun and the SpecifyObjectiveGradient option is set to true, as set by

options = optimoptions('fminunc','SpecifyObjectiveGradient',true)

then fun must return the gradient vector g(x) in the second output argument.

If you can also compute the Hessian matrix and the HessianFcn option is set to 'objective' via options = optimoptions('fminunc','HessianFcn','objective') and the Algorithm option is set to 'trust-region', fun must return the Hessian value H(x), a symmetric matrix, in a third output argument. fun can give a sparse Hessian. See Hessian for fminunc trust-region or fmincon trust-region-reflective algorithms for details.

The trust-region algorithm allows you to supply a Hessian multiply function. This function gives the result of a Hessian-times-vector product without computing the Hessian directly. This can save memory. See Hessian Multiply Function.

Example: fun = @(x)sin(x(1))*cos(x(2))

Data Types: char | function_handle | string

x0 — Initial point

real vector | real array

Initial point, specified as a real vector or real array. Solvers use the number of elements in x0 and the size of x0 to determine the number and size of variables that fun accepts.

Example: x0 = [1,2,3,4]

Data Types: double

options — Optimization options

output of optimoptions | structure such as optimset returns

Optimization options, specified as the output of optimoptions or a structure such as optimset returns.

Some options apply to all algorithms, and others are relevant for particular algorithms. See Optimization Options Reference for detailed information.

Some options are absent from the optimoptions display. These options appear in italics in the following table. For details, see View Options.

All Algorithms
Algorithm	Choose the fminunc algorithm. Choices are 'quasi-newton' (default) or 'trust-region'. The 'trust-region' algorithm requires you to provide the gradient (see the description of fun), or else fminunc uses the 'quasi-newton' algorithm. For information on choosing the algorithm, see Choosing the Algorithm.
CheckGradients	Compare user-supplied derivatives (gradient of objective) to finite-differencing derivatives. Choices are false (default) or true. For optimset, the name is DerivativeCheck and the values are 'on' or 'off'. See Current and Legacy Option Name Tables.
Diagnostics	Display diagnostic information about the function to be minimized or solved. Choices are 'off' (default) or 'on'.
DiffMaxChange	Maximum change in variables for finite-difference gradients (a positive scalar). The default is Inf.
DiffMinChange	Minimum change in variables for finite-difference gradients (a positive scalar). The default is 0.
Display	Level of display (see Iterative Display): 'off' or 'none' displays no output. 'iter' displays output at each iteration, and gives the default exit message. 'iter-detailed' displays output at each iteration, and gives the technical exit message. 'notify' displays output only if the function does not converge, and gives the default exit message. 'notify-detailed' displays output only if the function does not converge, and gives the technical exit message. 'final' (default) displays only the final output, and gives the default exit message. 'final-detailed' displays only the final output, and gives the technical exit message.
FiniteDifferenceStepSize	Scalar or vector step size factor for finite differences. When you set FiniteDifferenceStepSize to a vector v, the forward finite differences delta are delta = v.*sign′(x).*max(abs(x),TypicalX); where sign′(x) = sign(x) except sign′(0) = 1. Central finite differences are delta = v.*max(abs(x),TypicalX); Scalar FiniteDifferenceStepSize expands to a vector. The default is sqrt(eps) for forward finite differences, and eps^(1/3) for central finite differences. The trust-region algorithm uses FiniteDifferenceStepSize only when CheckGradients is set to true. For optimset, the name is FinDiffRelStep. See Current and Legacy Option Name Tables.
FiniteDifferenceType	Finite differences, used to estimate gradients, are either 'forward' (the default), or 'central' (centered). 'central' takes twice as many function evaluations, but should be more accurate. The trust-region algorithm uses FiniteDifferenceType only when CheckGradients is set to true. For optimset, the name is FinDiffType. See Current and Legacy Option Name Tables.
FunValCheck	Check whether objective function values are valid. The default setting, 'off', does not perform a check. The 'on' setting displays an error when the objective function returns a value that is complex, Inf, or NaN.
MaxFunctionEvaluations	Maximum number of function evaluations allowed, a positive integer. The default value is 100*numberOfVariables. See Tolerances and Stopping Criteria and Iterations and Function Counts. For optimset, the name is MaxFunEvals. See Current and Legacy Option Name Tables.
MaxIterations	Maximum number of iterations allowed, a positive integer. The default value is 400. See Tolerances and Stopping Criteria and Iterations and Function Counts. For optimset, the name is MaxIter. See Current and Legacy Option Name Tables.
OptimalityTolerance	Termination tolerance on the first-order optimality (a positive scalar). The default is 1e-6. See First-Order Optimality Measure. For optimset, the name is TolFun. See Current and Legacy Option Name Tables.
OutputFcn	Specify one or more user-defined functions that an optimization function calls at each iteration. Pass a function handle or a cell array of function handles. The default is none ([]). See Output Function Syntax.
PlotFcn	Plots various measures of progress while the algorithm executes; select from predefined plots or write your own. Pass a built-in plot function name, a function handle, or a cell array of built-in plot function names or function handles. For custom plot functions, pass function handles. The default is none ([]): 'optimplotx' plots the current point. 'optimplotfunccount' plots the function count. 'optimplotfval' plots the function value. 'optimplotstepsize' plots the step size. 'optimplotfirstorderopt' plots the first-order optimality measure. Custom plot functions use the same syntax as output functions. See Output Functions and Output Function Syntax. For optimset, the name is PlotFcns. See Current and Legacy Option Name Tables.
SpecifyObjectiveGradient	Gradient for the objective function defined by the user. See the description of fun to see how to define the gradient in fun. Set to true to have fminunc use a user-defined gradient of the objective function. The default false causes fminunc to estimate gradients using finite differences. You must provide the gradient, and set SpecifyObjectiveGradient to true, to use the trust-region algorithm. This option is not required for the quasi-Newton algorithm. For optimset, the name is GradObj and the values are 'on' or 'off'. See Current and Legacy Option Name Tables.
StepTolerance	Termination tolerance on x, a positive scalar. The default value is 1e-6. See Tolerances and Stopping Criteria. For optimset, the name is TolX. See Current and Legacy Option Name Tables.
TypicalX	Typical x values. The number of elements in TypicalX is equal to the number of elements in x0, the starting point. The default value is ones(numberofvariables,1). fminunc uses TypicalX for scaling finite differences for gradient estimation. The trust-region algorithm uses TypicalX only for the CheckGradients option.
trust-region Algorithm
FunctionTolerance	Termination tolerance on the function value, a positive scalar. The default is 1e-6. See Tolerances and Stopping Criteria. For optimset, the name is TolFun. See Current and Legacy Option Name Tables.
HessianFcn	If set to [] (default), fminunc approximates the Hessian using finite differences. If set to 'objective', fminunc uses a user-defined Hessian for the objective function. The Hessian is the third output of the objective function (see fun). For optimset, the name is HessFcn. See Current and Legacy Option Name Tables.
HessianMultiplyFcn	Hessian multiply function, specified as a function handle. For large-scale structured problems, this function computes the Hessian matrix product H*Y without actually forming H. The function is of the form W = hmfun(Hinfo,Y) where Hinfo contains the matrix used to compute H*Y. The first argument is the same as the third argument returned by the objective function fun, for example [f,g,Hinfo] = fun(x) Y is a matrix that has the same number of rows as there are dimensions in the problem. The matrix W = H*Y, although H is not formed explicitly. fminunc uses Hinfo to compute the preconditioner. For information on how to supply values for any additional parameters hmfun needs, see Passing Extra Parameters. Note To use the HessianMultiplyFcn option, HessianFcn must be set to []. For an example, see Minimization with Dense Structured Hessian, Linear Equalities. For optimset, the name is HessMult. See Current and Legacy Option Name Tables.
HessPattern	Sparsity pattern of the Hessian for finite differencing. Set HessPattern(i,j) = 1 when you can have ∂2fun/∂x(i)∂x(j) ≠ 0. Otherwise, set HessPattern(i,j) = 0. Use HessPattern when it is inconvenient to compute the Hessian matrix H in fun, but you can determine (say, by inspection) when the ith component of the gradient of fun depends on x(j). fminunc can approximate H via sparse finite differences (of the gradient) if you provide the sparsity structure of H as the value for HessPattern. In other words, provide the locations of the nonzeros. When the structure is unknown, do not set HessPattern. The default behavior is as if HessPattern is a dense matrix of ones. Then fminunc computes a full finite-difference approximation in each iteration. This computation can be expensive for large problems, so it is usually better to determine the sparsity structure.
MaxPCGIter	Maximum number of preconditioned conjugate gradient (PCG) iterations, a positive scalar. The default is max(1,floor(numberOfVariables/2)). For more information, see Trust Region Algorithm.
PrecondBandWidth	Upper bandwidth of preconditioner for PCG, a nonnegative integer. By default, fminunc uses diagonal preconditioning (upper bandwidth of 0). For some problems, increasing the bandwidth reduces the number of PCG iterations. Setting PrecondBandWidth to Inf uses a direct factorization (Cholesky) rather than the conjugate gradients (CG). The direct factorization is computationally more expensive than CG, but produces a better quality step towards the solution.
SubproblemAlgorithm	Determines how the iteration step is calculated. The default, 'cg', takes a faster but less accurate step than 'factorization'. See fminunc trust-region Algorithm.
TolPCG	Termination tolerance on the PCG iteration, a positive scalar. The default is 0.1.
quasi-newton Algorithm
HessUpdate	Method for choosing the search direction in the Quasi-Newton algorithm. The choices are: 'bfgs', the default 'dfp' 'steepdesc' See Quasi-Newton Algorithm and Hessian Update for a description of these methods.
ObjectiveLimit	A tolerance (stopping criterion) that is a scalar. If the objective function value at an iteration is less than or equal to ObjectiveLimit, the iterations halt because the problem is presumably unbounded. The default value is -1e20.
UseParallel	When true, fminunc estimates gradients in parallel. Disable by setting to the default, false. trust-region requires a gradient in the objective, so UseParallel does not apply. See Parallel Computing.

Example: options = optimoptions('fminunc','SpecifyObjectiveGradient',true)

problem — Problem structure

structure

Problem structure, specified as a structure with the following fields:

Field Name	Entry
objective	Objective function
x0	Initial point for x
solver	'fminunc'
options	Options created with optimoptions

The simplest way to obtain a problem structure is to export the problem from the Optimization app.

Data Types: struct

Output Arguments

x — Solution

real vector | real array

Solution, returned as a real vector or real array. The size of x is the same as the size of x0. Typically, x is a local solution to the problem when exitflag is positive. For information on the quality of the solution, see When the Solver Succeeds.

fval — Objective function value at solution

real number

Objective function value at the solution, returned as a real number. Generally, fval = fun(x).

exitflag — Reason fminunc stopped

integer

Reason fminunc stopped, returned as an integer.

1	Magnitude of gradient is smaller than the OptimalityTolerance tolerance.
2	Change in x was smaller than the StepTolerance tolerance.
3	Change in the objective function value was less than the FunctionTolerance tolerance.
5	Predicted decrease in the objective function was less than the FunctionTolerance tolerance.
0	Number of iterations exceeded MaxIterations or number of function evaluations exceeded MaxFunctionEvaluations.
-1	Algorithm was terminated by the output function.
-3	Objective function at current iteration went below ObjectiveLimit.

output — Information about the optimization process

structure

Information about the optimization process, returned as a structure with fields:

iterations	Number of iterations taken
funcCount	Number of function evaluations
firstorderopt	Measure of first-order optimality
algorithm	Optimization algorithm used
cgiterations	Total number of PCG iterations ('trust-region' algorithm only)
lssteplength	Size of line search step relative to search direction ('quasi-newton' algorithm only)
stepsize	Final displacement in x
message	Exit message

grad — Gradient at the solution

real vector

Gradient at the solution, returned as a real vector. grad gives the gradient of fun at the point x(:).

hessian — Approximate Hessian

real matrix

Approximate Hessian, returned as a real matrix. For the meaning of hessian, see Hessian Output.

Algorithms

Quasi-Newton Algorithm

The quasi-newton algorithm uses the BFGS Quasi-Newton method with a cubic line search procedure. This quasi-Newton method uses the BFGS ([1],[5],[8], and [9]) formula for updating the approximation of the Hessian matrix. You can select the DFP ([4],[6], and [7]) formula, which approximates the inverse Hessian matrix, by setting the HessUpdate option to 'dfp' (and the Algorithm option to 'quasi-newton'). You can select a steepest descent method by setting HessUpdate to 'steepdesc' (and Algorithm to 'quasi-newton'), although this setting is usually inefficient. See fminunc quasi-newton Algorithm.

Trust Region Algorithm

The trust-region algorithm requires that you supply the gradient in fun and set SpecifyObjectiveGradient to true using optimoptions. This algorithm is a subspace trust-region method and is based on the interior-reflective Newton method described in [2] and [3]. Each iteration involves the approximate solution of a large linear system using the method of preconditioned conjugate gradients (PCG). See fminunc trust-region Algorithm, Trust-Region Methods for Nonlinear Minimization and Preconditioned Conjugate Gradient Method.

References

[1] Broyden, C. G. “The Convergence of a Class of Double-Rank Minimization Algorithms.” Journal Inst. Math. Applic., Vol. 6, 1970, pp. 76–90.

[2] Coleman, T. F. and Y. Li. “An Interior, Trust Region Approach for Nonlinear Minimization Subject to Bounds.” SIAM Journal on Optimization, Vol. 6, 1996, pp. 418–445.

[3] Coleman, T. F. and Y. Li. “On the Convergence of Reflective Newton Methods for Large-Scale Nonlinear Minimization Subject to Bounds.” Mathematical Programming, Vol. 67, Number 2, 1994, pp. 189–224.

[4] Davidon, W. C. “Variable Metric Method for Minimization.” A.E.C. Research and Development Report, ANL-5990, 1959.

[5] Fletcher, R. “A New Approach to Variable Metric Algorithms.” Computer Journal, Vol. 13, 1970, pp. 317–322.

[6] Fletcher, R. “Practical Methods of Optimization.” Vol. 1, Unconstrained Optimization, John Wiley and Sons, 1980.

[7] Fletcher, R. and M. J. D. Powell. “A Rapidly Convergent Descent Method for Minimization.” Computer Journal, Vol. 6, 1963, pp. 163–168.

[8] Goldfarb, D. “A Family of Variable Metric Updates Derived by Variational Means.” Mathematics of Computing, Vol. 24, 1970, pp. 23–26.

[9] Shanno, D. F. “Conditioning of Quasi-Newton Methods for Function Minimization.” Mathematics of Computing, Vol. 24, 1970, pp. 647–656.

Extended Capabilities

Automatic Parallel Support

Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.

To run in parallel, set the 'UseParallel' option to true.

options = optimoptions('solvername','UseParallel',true)

For more information, see Using Parallel Computing in Optimization Toolbox.