## Writing Scalar Objective Functions

### Function Files

A scalar objective function file accepts one input, say `x`

, and
returns one real scalar output, say `f`

. The input
`x`

can be a scalar, vector, or matrix. A function file can return more outputs (see Including Gradients and Hessians).

For example, suppose your objective is a function of three variables,
*x*, *y*, and *z*:

*f*(*x*) = 3*(*x* –
*y*)^{4} + 4*(*x*
+ *z*)^{2} / (1 +
*x*^{2} +
*y*^{2} +
*z*^{2}) +
cosh(*x* – 1) + tanh(*y* +
*z*).

Write this function as a file that accepts the vector

`xin`

= [*x*;*y*;*z*] and returns*f*:function f = myObjective(xin) f = 3*(xin(1)-xin(2))^4 + 4*(xin(1)+xin(3))^2/(1+norm(xin)^2) ... + cosh(xin(1)-1) + tanh(xin(2)+xin(3));

Save it as a file named

`myObjective.m`

to a folder on your MATLAB^{®}path.Check that the function evaluates correctly:

myObjective([1;2;3]) ans = 9.2666

For information on how to include extra parameters, see Passing Extra Parameters. For more complex examples of function files, see Minimization with Gradient and Hessian Sparsity Pattern or Minimization with Bound Constraints and Banded Preconditioner.

#### Local Functions and Nested Functions

Functions can exist inside other files as local functions or nested functions. Using local functions or nested functions can lower the number of distinct files you save. Using nested functions also lets you access extra parameters, as shown in Nested Functions.

For example, suppose you want to minimize the `myObjective.m`

objective function, described in Function Files, subject to the `ellipseparabola.m`

constraint, described in Nonlinear Constraints. Instead of writing
two files, `myObjective.m`

and
`ellipseparabola.m`

, write one file that contains both
functions as local functions:

function [x fval] = callObjConstr(x0,options) % Using a local function for just one file if nargin < 2 options = optimoptions('fmincon','Algorithm','interior-point'); end [x fval] = fmincon(@myObjective,x0,[],[],[],[],[],[], ... @ellipseparabola,options); function f = myObjective(xin) f = 3*(xin(1)-xin(2))^4 + 4*(xin(1)+xin(3))^2/(1+sum(xin.^2)) ... + cosh(xin(1)-1) + tanh(xin(2)+xin(3)); function [c,ceq] = ellipseparabola(x) c(1) = (x(1)^2)/9 + (x(2)^2)/4 - 1; c(2) = x(1)^2 - x(2) - 1; ceq = [];

Solve the constrained minimization starting from the point
`[1;1;1]`

:

[x fval] = callObjConstr(ones(3,1)) Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the default value of the function tolerance, and constraints are satisfied to within the default value of the constraint tolerance. x = 1.1835 0.8345 -1.6439 fval = 0.5383

### Anonymous Function Objectives

Use anonymous functions to write simple objective functions. For more information about anonymous functions, see What Are Anonymous Functions?. Rosenbrock's function is simple enough to write as an anonymous function:

anonrosen = @(x)(100*(x(2) - x(1)^2)^2 + (1-x(1))^2);

`anonrosen`

evaluates correctly at ```
[-1
2]
```

:anonrosen([-1 2]) ans = 104

`anonrosen`

with `fminunc`

yields the
following
results:options = optimoptions(@fminunc,'Algorithm','quasi-newton'); [x fval] = fminunc(anonrosen,[-1;2],options) Local minimum found. Optimization completed because the size of the gradient is less than the default value of the function tolerance. x = 1.0000 1.0000 fval = 1.2266e-10

### Including Gradients and Hessians

#### Provide Derivatives For Solvers

For `fmincon`

and `fminunc`

, you can
include gradients in the objective function. Generally, solvers are more robust,
and can be slightly faster when you include gradients. See Benefits of Including Derivatives. To also include second
derivatives (Hessians), see Including Hessians.

The following table shows which algorithms can use gradients and Hessians.

Solver | Algorithm | Gradient | Hessian |
---|---|---|---|

`fmincon` | `active-set` | Optional | No |

`interior-point` | Optional | Optional (see Hessian for fmincon interior-point algorithm) | |

`sqp` | Optional | No | |

`trust-region-reflective` | Required | Optional (see Hessian for fminunc trust-region or fmincon trust-region-reflective algorithms) | |

`fminunc` | `quasi-newton` | Optional | No |

`trust-region` | Required | Optional (see Hessian for fminunc trust-region or fmincon trust-region-reflective algorithms) |

#### How to Include Gradients

Write code that returns:

The objective function (scalar) as the first output

The gradient (vector) as the second output

Set the

`SpecifyObjectiveGradient`

option to`true`

using`optimoptions`

. If appropriate, also set the`SpecifyConstraintGradient`

option to`true`

.Optionally, check if your gradient function matches a finite-difference approximation. See Checking Validity of Gradients or Jacobians.

**Tip**

For most flexibility, write *conditionalized* code.
Conditionalized means that the number of function outputs can vary, as shown
in the following example. Conditionalized code does not error depending on
the value of the `SpecifyObjectiveGradient`

option.
Unconditionalized code requires you to set options appropriately.

For example, consider Rosenbrock's function

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

which is described and plotted in Solve a Constrained Nonlinear Problem, Solver-Based. The gradient of *f*(*x*) is

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

`rosentwo`

is a conditionalized function that returns
whatever the solver requires:

function [f,g] = rosentwo(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

`nargout`

checks the number of arguments that a calling
function specifies. See Find Number of Function Arguments.

The `fminunc`

solver, designed for unconstrained
optimization, allows you to minimize Rosenbrock's function. Tell
`fminunc`

to use the gradient and Hessian by setting
`options`

:

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

Run `fminunc`

starting at `[-1;2]`

:

[x fval] = fminunc(@rosentwo,[-1;2],options) Local minimum found. Optimization completed because the size of the gradient is less than the default value of the function tolerance. x = 1.0000 1.0000 fval = 1.9886e-17

If you have a Symbolic Math Toolbox™ license, you can calculate gradients and Hessians automatically, as described in Calculate Gradients and Hessians Using Symbolic Math Toolbox.

#### Including Hessians

You can include second derivatives with the `fmincon`

`'trust-region-reflective'`

and
`'interior-point'`

algorithms, and with the
`fminunc`

`'trust-region'`

algorithm. There are several ways to include
Hessian information, depending on the type of information and on the
algorithm.

You must also include gradients (set
`SpecifyObjectiveGradient`

to `true`

and,
if applicable, `SpecifyConstraintGradient`

to
`true`

) in order to include Hessians.

**Hessian for fminunc trust-region or
fmincon trust-region-reflective algorithms. **These algorithms either have no constraints, or have only bound or linear
equality constraints. Therefore the Hessian is the matrix of second
derivatives of the objective function.

Include the Hessian matrix as the third output of the objective function.
For example, the Hessian *H*(*x*) of Rosenbrock’s function is (see How to Include Gradients)

$$H(x)=\left[\begin{array}{cc}1200{x}_{1}^{2}-400{x}_{2}+2& -400{x}_{1}\\ -400{x}_{1}& 200\end{array}\right].$$

Include this Hessian in the objective:

function [f, g, H] = rosenboth(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)]; if nargout > 2 % Hessian required H = [1200*x(1)^2-400*x(2)+2, -400*x(1); -400*x(1), 200]; end end

Set `HessianFcn`

to `'objective'`

. For
example,

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

**Hessian for fmincon interior-point algorithm. **The Hessian is the Hessian of the Lagrangian, where the Lagrangian

*L*(

*x*,

*λ*) is

$$L(x,\lambda )=f(x)+{\displaystyle \sum {\lambda}_{g,i}{g}_{i}(x)}+{\displaystyle \sum {\lambda}_{h,i}{h}_{i}(x)}.$$

*g* and *h* are vector functions
representing all inequality and equality constraints respectively (meaning
bound, linear, and nonlinear constraints), so the minimization problem
is

$$\underset{x}{\mathrm{min}}f(x)\text{subjectto}g(x)\le 0,\text{}h(x)=0.$$

For details, see Constrained Optimality Theory. The Hessian of the Lagrangian is

$${\nabla}_{xx}^{2}L(x,\lambda )={\nabla}^{2}f(x)+{\displaystyle \sum {\lambda}_{g,i}{\nabla}^{2}{g}_{i}(x)}+{\displaystyle \sum {\lambda}_{h,i}{\nabla}^{2}{h}_{i}(x)}.$$ | (1) |

To include a Hessian, write a function with the syntax

hessian = hessianfcn(x,lambda)

`hessian`

is an
*n*-by-*n* matrix, sparse or dense,
where *n* is the number of variables. If
`hessian`

is large and has relatively few nonzero
entries, save running time and memory by representing
`hessian`

as a sparse matrix. `lambda`

is a structure with the Lagrange multiplier vectors associated with the
nonlinear constraints:

lambda.ineqnonlin lambda.eqnonlin

`fmincon`

computes the structure
`lambda`

and passes it to your Hessian function.
`hessianfcn`

must calculate the sums in Equation 1. Indicate that you are supplying a
Hessian by setting these options:

options = optimoptions('fmincon','Algorithm','interior-point',... 'SpecifyObjectiveGradient',true,'SpecifyConstraintGradient',true,... 'HessianFcn',@hessianfcn);

For example, to include a Hessian for Rosenbrock’s function constrained to the unit disk $${x}_{1}^{2}+{x}_{2}^{2}\le 1$$, notice that the constraint function $$g(x)={x}_{1}^{2}+{x}_{2}^{2}-1\le 0$$ has gradient and second derivative matrix

$$\begin{array}{c}\nabla g(x)=\left[\begin{array}{c}2{x}_{1}\\ 2{x}_{2}\end{array}\right]\\ {H}_{g}(x)=\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right].\end{array}$$

Write the Hessian function as

function Hout = hessianfcn(x,lambda) % Hessian of objective H = [1200*x(1)^2-400*x(2)+2, -400*x(1); -400*x(1), 200]; % Hessian of nonlinear inequality constraint Hg = 2*eye(2); Hout = H + lambda.ineqnonlin*Hg;

Save `hessianfcn`

on your MATLAB path. To complete the example, the constraint function
including gradients is

function [c,ceq,gc,gceq] = unitdisk2(x) c = x(1)^2 + x(2)^2 - 1; ceq = [ ]; if nargout > 2 gc = [2*x(1);2*x(2)]; gceq = []; end

Solve the problem including gradients and Hessian.

fun = @rosenboth; nonlcon = @unitdisk2; x0 = [-1;2]; options = optimoptions('fmincon','Algorithm','interior-point',... 'SpecifyObjectiveGradient',true,'SpecifyConstraintGradient',true,... 'HessianFcn',@hessianfcn); [x,fval,exitflag,output] = fmincon(fun,x0,[],[],[],[],[],[],@unitdisk2,options);

For other examples using an interior-point Hessian, see fmincon Interior-Point Algorithm with Analytic Hessian and Calculate Gradients and Hessians Using Symbolic Math Toolbox.

**Hessian Multiply Function. **Instead of a complete Hessian function, both the
`fmincon`

`interior-point`

and
`trust-region-reflective`

algorithms allow 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. The `SubproblemAlgorithm`

option must be
`'cg'`

for a Hessian multiply function to work; this is
the `trust-region-reflective`

default.

The syntaxes for the two algorithms differ.

For the

`interior-point`

algorithm, the syntax isW = HessMultFcn(x,lambda,v);

The result

`W`

should be the product`H*v`

, where`H`

is the Hessian of the Lagrangian at`x`

(see Equation 1),`lambda`

is the Lagrange multiplier (computed by`fmincon`

), and`v`

is a vector of size*n*-by-1. Set options as follows:options = optimoptions('fmincon','Algorithm','interior-point','SpecifyObjectiveGradient',true,... 'SpecifyConstraintGradient',true,'SubproblemAlgorithm','cg','HessianMultiplyFcn',@HessMultFcn);

Supply the function

`HessMultFcn`

, which returns an*n*-by-1 vector, where*n*is the number of dimensions of*x*. The`HessianMultiplyFcn`

option enables you to pass the result of multiplying the Hessian by a vector without calculating the Hessian.The

`trust-region-reflective`

algorithm does not involve`lambda`

:W = HessMultFcn(H,v);

The result

`W = H*v`

.`fmincon`

passes`H`

as the value returned in the third output of the objective function (see Hessian for fminunc trust-region or fmincon trust-region-reflective algorithms).`fmincon`

also passes`v`

, a vector or matrix with*n*rows. The number of columns in`v`

can vary, so write`HessMultFcn`

to accept an arbitrary number of columns.`H`

does not have to be the Hessian; rather, it can be anything that enables you to calculate`W = H*v`

.Set options as follows:

options = optimoptions('fmincon','Algorithm','trust-region-reflective',... 'SpecifyObjectiveGradient',true,'HessianMultiplyFcn',@HessMultFcn);

For an example using a Hessian multiply function with the

`trust-region-reflective`

algorithm, see Minimization with Dense Structured Hessian, Linear Equalities.

#### Benefits of Including Derivatives

If you do not provide gradients, solvers estimate gradients via finite differences. If you provide gradients, your solver need not perform this finite difference estimation, so can save time and be more accurate, although a finite-difference estimate can be faster for complicated derivatives. Furthermore, solvers use an approximate Hessian, which can be far from the true Hessian. Providing a Hessian can yield a solution in fewer iterations. For example, see the end of Calculate Gradients and Hessians Using Symbolic Math Toolbox.

For constrained problems, providing a gradient has another advantage. A solver
can reach a point `x`

such that `x`

is
feasible, but, for this `x`

, finite differences around
`x`

always lead to an infeasible point. Suppose further
that the objective function at an infeasible point returns a complex output,
`Inf`

, `NaN`

, or error. In this case, a
solver can fail or halt prematurely. Providing a gradient allows a solver to
proceed. To obtain this benefit, you might also need to include the gradient of
a nonlinear constraint function, and set the
`SpecifyConstraintGradient`

option to
`true`

. See Nonlinear Constraints.

#### Choose Input Hessian Approximation for interior-point `fmincon`

The `fmincon`

`interior-point`

algorithm has many options for selecting an
input Hessian approximation. For syntax details, see Hessian as an Input. Here are the options, along with estimates of their
relative characteristics.

Hessian | Relative Memory Usage | Relative Efficiency |
---|---|---|

`'bfgs'` (default) | High (for large problems) | High |

`'lbfgs'` | Low to Moderate | Moderate |

`'fin-diff-grads'` | Low | Moderate |

`'HessianMultiplyFcn'` | Low (can depend on your code) | Moderate |

`'HessianFcn'` | ? (depends on your code) | High (depends on your code) |

Use the default `'bfgs'`

Hessian unless you

Run out of memory — Try

`'lbfgs'`

instead of`'bfgs'`

. If you can provide your own gradients, try`'fin-diff-grads'`

, and set the`SpecifyObjectiveGradient`

and`SpecifyConstraintGradient`

options to`true`

.Want more efficiency — Provide your own gradients and Hessian. See Including Hessians, fmincon Interior-Point Algorithm with Analytic Hessian, and Calculate Gradients and Hessians Using Symbolic Math Toolbox.

The reason `'lbfgs'`

has only moderate efficiency is twofold.
It has relatively expensive Sherman-Morrison updates. And the resulting
iteration step can be somewhat inaccurate due to the `'lbfgs'`

limited memory.

The reason `'fin-diff-grads'`

and
`HessianMultiplyFcn`

have only moderate efficiency is that
they use a conjugate gradient approach. They accurately estimate the Hessian of
the objective function, but they do not generate the most accurate iteration
step. For more information, see fmincon Interior Point Algorithm, and its discussion of the LDL
approach and the conjugate gradient approach to solving Equation 38.