fminunc

Find minimum of unconstrained multivariable function

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Nonlinear programming solver.

Finds the minimum of a problem specified by

$\min_{x} f (x)$

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

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

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

example

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.

example

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

example

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.

example

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

example

[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

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Minimize a Polynomial

Minimize the function $f (x) = 3 x_{1}^{2} + 2 x_{1} x_{2} + x_{2}^{2} - 4 x_{1} + 5 x_{2}$ .

Write an anonymous function 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);

After a few iterations, fminunc returns the solution, x, and the value of the function at x, fval.

x,fval

x =

    2.2500   -4.7500


fval =

  -16.3750

Supply the 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,

$f (x) = 100 {(x_{2} - x_{1}^{2})}^{2} + {(1 - x_{1})}^{2},$

which has gradient

$\nabla f (x) = [\begin{matrix} - 400 (x_{2} - x_{1}^{2}) x_{1} - 2 (1 - x_{1}) \\ 200 (x_{2} - x_{1}^{2}) \end{matrix}] .$

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

Save this code as a file on your MATLAB^® path, named rosenbrockwithgrad.m.

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 default value of the function tolerance.

<stopping criteria details>


x =

    1.0000    1.0000

Use a Problem Structure

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

$f (x) = 100 {(x_{2} - x_{1}^{2})}^{2} + {(1 - x_{1})}^{2},$

which has gradient

$\nabla f (x) = [\begin{matrix} - 400 (x_{2} - x_{1}^{2}) x_{1} - 2 (1 - x_{1}) \\ 200 (x_{2} - x_{1}^{2}) \end{matrix}] .$

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

Save this code as a file on your MATLAB path, named rosenbrockwithgrad.m.

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].

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 default value of the function tolerance.

<stopping criteria details>


x =

    1.0000    1.0000

Obtain the 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

$f (x) = x (1) e^{- {‖ x ‖}_{2}^{2}} + {‖ x ‖}_{2}^{2} / 20.$

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 default value of the function tolerance.

<stopping criteria details>


x =

   -0.6691    0.0000


fval =

   -0.4052

Examine the Solution Process

Open Live Script

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

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 default value of the optimality tolerance.

x = 

   -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: 'Local minimum found....'

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

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`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, and 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 are listed in italics. 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`.
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`, forward finite differences steps `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`.
`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`.
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.
`MaxIterations`	Maximum number of iterations allowed, a positive integer. The default value is `400`. See Tolerances and Stopping Criteria and Iterations and Function Counts.
`OptimalityTolerance`	Termination tolerance on the first-order optimality, a positive scalar. The default is `1e-6`. See First-Order Optimality Measure.
`OutputFcn`	Specify one or more user-defined functions that an optimization function calls at each iteration, either as a function handle or as a cell array of function handles. The default is none (`[]`). See Output Function.
`PlotFcn`	Plot various measures of progress while the algorithm executes. Select from predefined plots or write your own. Pass a function handle or a cell array of 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. For information on writing a custom plot function, see Plot Functions.
`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.
`StepTolerance`	Termination tolerance on `x`, a positive scalar. The default value is `1e-6`. See Tolerances and Stopping Criteria.
`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.
`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`).
`HessianMultiplyFcn`	Function handle for Hessian multiply function. For large-scale structured problems, this function computes the Hessian matrix product `HY` without actually forming `H`. The function is of the form W = hmfun(Hinfo,Y) where `Hinfo` contains the matrix used to compute `HY`. 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.
HessPattern	Sparsity pattern of the Hessian for finite differencing. Set `HessPattern(i,j) = 1` when you can have ∂²`fun`/∂`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 `i`th 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

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`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.

Algorithms

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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.

Documentation

fminunc

Syntax

Description

Note

Examples

Minimize a Polynomial

Supply the Gradient

Use a Problem Structure

Obtain the Optimal Objective Function Value

Examine the Solution Process

Input Arguments

`fun` — Function to minimize
function handle | function name

`x0` — Initial point
real vector | real array

`options` — Optimization options
output of `optimoptions` | structure such as `optimset` returns

Note

`problem` — Problem structure
structure

Output Arguments

`x` — Solution
real vector | real array

`fval` — Objective function value at solution
real number

`exitflag` — Reason `fminunc` stopped
integer

`output` — Information about the optimization process
structure

`grad` — Gradient at the solution
real vector

`hessian` — Approximate Hessian
real matrix

Algorithms

Quasi-Newton Algorithm

Trust Region Algorithm

References

See Also

Topics

Introduced before R2006a

Optimization Toolbox Documentation

Other Documentation

Support

Documentation

fminunc

Syntax

Description

Note

Examples

Minimize a Polynomial

Supply the Gradient

Use a Problem Structure

Obtain the Optimal Objective Function Value

Examine the Solution Process

Input Arguments

fun — Function to minimize function handle | function name

x0 — Initial point real vector | real array

options — Optimization options output of optimoptions | structure such as optimset returns

Note

problem — Problem structure structure

Output Arguments

x — Solution real vector | real array

fval — Objective function value at solution real number

exitflag — Reason fminunc stopped integer

output — Information about the optimization process structure

grad — Gradient at the solution real vector

hessian — Approximate Hessian real matrix

Algorithms

Quasi-Newton Algorithm

Trust Region Algorithm

References

See Also

Topics

Introduced before R2006a

Optimization Toolbox Documentation

Other Documentation

Support

`fun` — Function to minimize
function handle | function name

`x0` — Initial point
real vector | real array

`options` — Optimization options
output of `optimoptions` | structure such as `optimset` returns

`problem` — Problem structure
structure

`x` — Solution
real vector | real array

`fval` — Objective function value at solution
real number

`exitflag` — Reason `fminunc` stopped
integer

`output` — Information about the optimization process
structure

`grad` — Gradient at the solution
real vector

`hessian` — Approximate Hessian
real matrix