fmincon

Find minimum of constrained nonlinear multivariable function

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Syntax

x = fmincon(fun,x0,A,b)

x = fmincon(fun,x0,A,b,Aeq,beq)

x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub)

x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)

x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)

x = fmincon(problem)

[x,fval]
= fmincon(___)

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

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

Description

Nonlinear programming solver.

Finds the minimum of a problem specified by

$\min_{x} f (x) such that {\begin{matrix} c (x) \leq 0 \\ c e q (x) = 0 \\ A \cdot x \leq b \\ A e q \cdot x = b e q \\ l b \leq x \leq u b, \end{matrix}$

b and beq are vectors, A and Aeq are matrices, c(x) and ceq(x) are functions that return vectors, and f(x) is a function that returns a scalar. f(x), c(x), and ceq(x) can be nonlinear functions.

x, lb, and ub can be passed as vectors or matrices; see Matrix Arguments.

example

x = fmincon(fun,x0,A,b) starts at x0 and attempts to find a minimizer x of the function described in fun subject to the linear inequalities A*x ≤ b. 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.

example

x = fmincon(fun,x0,A,b,Aeq,beq) minimizes fun subject to the linear equalities Aeq*x = beq and A*x ≤ b. If no inequalities exist, set A = [] and b = [].

example

x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub) defines a set of lower and upper bounds on the design variables in x, so that the solution is always in the range lb ≤ x ≤ ub. If no equalities exist, set Aeq = [] and beq = []. If x(i) is unbounded below, set lb(i) = -Inf, and if x(i) is unbounded above, set ub(i) = Inf.

Note

If the specified input bounds for a problem are inconsistent, fmincon throws an error. In this case, output x is x0 and fval is [].

For the default 'interior-point' algorithm, fmincon sets components of x0 that violate the bounds lb ≤ x ≤ ub, or are equal to a bound, to the interior of the bound region. For the 'trust-region-reflective' algorithm, fmincon sets violating components to the interior of the bound region. For other algorithms, fmincon sets violating components to the closest bound. Components that respect the bounds are not changed. See Iterations Can Violate Constraints.

example

x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon) subjects the minimization to the nonlinear inequalities c(x) or equalities ceq(x) defined in nonlcon. fmincon optimizes such that c(x) ≤ 0 and ceq(x) = 0. If no bounds exist, set lb = [] and/or ub = [].

example

x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options) minimizes with the optimization options specified in options. Use optimoptions to set these options. If there are no nonlinear inequality or equality constraints, set nonlcon = [].

example

x = fmincon(problem) finds the minimum for problem, a structure described in problem.

example

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

example

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

example

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

lambda — Structure with fields containing the Lagrange multipliers at the solution x.
grad — Gradient of fun at the solution x.
hessian — Hessian of fun at the solution x. See fmincon Hessian.

Examples

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Linear Inequality Constraint

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Find the minimum value of Rosenbrock's function when there is a linear inequality constraint.

Set the objective function fun to be Rosenbrock's function. Rosenbrock's function is well-known to be difficult to minimize. It has its minimum objective value of 0 at the point (1,1). For more information, see Solve a Constrained Nonlinear Problem, Solver-Based.

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

Find the minimum value starting from the point [-1,2], constrained to have $x (1) + 2 x (2) \leq 1$ . Express this constraint in the form Ax <= b by taking A = [1,2] and b = 1. Notice that this constraint means that the solution will not be at the unconstrained solution (1,1), because at that point $x (1) + 2 x (2) = 3 > 1$ .

x0 = [-1,2];
A = [1,2];
b = 1;
x = fmincon(fun,x0,A,b)

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.

x = 1×2

    0.5022    0.2489

Linear Inequality and Equality Constraint

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Find the minimum value of Rosenbrock's function when there are both a linear inequality constraint and a linear equality constraint.

Set the objective function fun to be Rosenbrock's function.

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

Find the minimum value starting from the point [0.5,0], constrained to have $x (1) + 2 x (2) \leq 1$ and $2 x (1) + x (2) = 1$ .

Express the linear inequality constraint in the form A*x <= b by taking A = [1,2] and b = 1.
Express the linear equality constraint in the form Aeq*x = beq by taking Aeq = [2,1] and beq = 1.

x0 = [0.5,0];
A = [1,2];
b = 1;
Aeq = [2,1];
beq = 1;
x = fmincon(fun,x0,A,b,Aeq,beq)

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.

x = 1×2

    0.4149    0.1701

Minimize with Bound Constraints

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Find the minimum of an objective function in the presence of bound constraints.

The objective function is a simple algebraic function of two variables.

fun = @(x)1+x(1)/(1+x(2)) - 3*x(1)*x(2) + x(2)*(1+x(1));

Look in the region where $x$ has positive values, $x (1) \leq 1$ , and $x (2) \leq 2$ .

lb = [0,0];
ub = [1,2];

There are no linear constraints, so set those arguments to [].

A = [];
b = [];
Aeq = [];
beq = [];

Try an initial point in the middle of the region.

x0 = (lb + ub)/2;

Solve the problem.

x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub)

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.

x = 1×2

    1.0000    2.0000

A different initial point can lead to a different solution.

x0 = x0/5;
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub)

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.

x = 1×2
10^-6 ×

    0.4000    0.4000

To determine which solution is better, see Obtain the Objective Function Value.

Nonlinear Constraints

Open Script

Find the minimum of a function subject to nonlinear constraints

Find the point where Rosenbrock's function is minimized within a circle, also subject to bound constraints.

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

Look within the region $0 \le x(1) \le 0.5$ , $0.2 \le x(2) \le 0.8$ .

lb = [0,0.2];
ub = [0.5,0.8];

Also look within the circle centered at [1/3,1/3] with radius 1/3. Copy the following code to a file on your MATLAB® path named circlecon.m.


% Copyright 2015 The MathWorks, Inc.

function [c,ceq] = circlecon(x)
c = (x(1)-1/3)^2 + (x(2)-1/3)^2 - (1/3)^2;
ceq = [];

There are no linear constraints, so set those arguments to [].

A = [];
b = [];
Aeq = [];
beq = [];

Choose an initial point satisfying all the constraints.

x0 = [1/4,1/4];

Solve the problem.

nonlcon = @circlecon;
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.


x =

    0.5000    0.2500

Nondefault Options

Open Script

Set options to view iterations as they occur and to use a different algorithm.

To observe the fmincon solution process, set the Display option to 'iter'. Also, try the 'sqp' algorithm, which is sometimes faster or more accurate than the default 'interior-point' algorithm.

options = optimoptions('fmincon','Display','iter','Algorithm','sqp');

Find the minimum of Rosenbrock's function on the unit disk, $||x||^2 \le 1$ . First create a function that represents the nonlinear constraint. Save this as a file named unitdisk.m on your MATLAB® path.

function [c,ceq] = unitdisk(x)
c = x(1)^2 + x(2)^2 - 1;
ceq = [];

Create the remaining problem specifications. Then run fmincon.

fun = @(x)100*(x(2)-x(1)^2)^2 + (1-x(1))^2;
A = [];
b = [];
Aeq = [];
beq = [];
lb = [];
ub = [];
nonlcon = @unitdisk;
x0 = [0,0];
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)

 Iter  Func-count            Fval   Feasibility   Step Length       Norm of   First-order  
                                                                       step    optimality
    0           3    1.000000e+00     0.000e+00     1.000e+00     0.000e+00     2.000e+00  
    1          12    8.913011e-01     0.000e+00     1.176e-01     2.353e-01     1.107e+01  
    2          22    8.047847e-01     0.000e+00     8.235e-02     1.900e-01     1.330e+01  
    3          28    4.197517e-01     0.000e+00     3.430e-01     1.217e-01     6.172e+00  
    4          31    2.733703e-01     0.000e+00     1.000e+00     5.254e-02     5.705e-01  
    5          34    2.397111e-01     0.000e+00     1.000e+00     7.498e-02     3.164e+00  
    6          37    2.036002e-01     0.000e+00     1.000e+00     5.960e-02     3.106e+00  
    7          40    1.164353e-01     0.000e+00     1.000e+00     1.459e-01     1.059e+00  
    8          43    1.161753e-01     0.000e+00     1.000e+00     1.754e-01     7.383e+00  
    9          46    5.901601e-02     0.000e+00     1.000e+00     1.547e-02     7.278e-01  
   10          49    4.533081e-02     2.898e-03     1.000e+00     5.393e-02     1.252e-01  
   11          52    4.567454e-02     2.225e-06     1.000e+00     1.492e-03     1.679e-03  
   12          55    4.567481e-02     4.406e-12     1.000e+00     2.095e-06     1.502e-05  
   13          58    4.567481e-02     0.000e+00     1.000e+00     2.203e-12     1.406e-05  

Local minimum possible. Constraints satisfied.

fmincon stopped because the size of the current step is less than
the value of the step size tolerance and constraints are 
satisfied to within the value of the constraint tolerance.


x =

    0.7864    0.6177

Include Gradient

Open Script

Include gradient evaluation in the objective function for faster or more reliable computations.

Include the gradient evaluation as a conditionalized output in the objective function file. For details, see 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

$\nabla f(x) = \left[ {\begin{array}{*{20}{c}}
{ - 400\left( {{x_2} - x_1^2} \right){x_1} - 2\left( {1 - {x_1}} \right)}\\
{200\left( {{x_2} - x_1^2} \right)}
\end{array}} \right].$

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 named rosenbrockwithgrad.m on your MATLAB® path.

Create options to use the objective function gradient.

options = optimoptions('fmincon','SpecifyObjectiveGradient',true);

Create the other inputs for the problem. Then call fmincon.

fun = @rosenbrockwithgrad;
x0 = [-1,2];
A = [];
b = [];
Aeq = [];
beq = [];
lb = [-2,-2];
ub = [2,2];
nonlcon = [];
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.


x =

    1.0000    1.0000

Use a Problem Structure

Open Live Script

Solve the same problem as in Nondefault Options using a problem structure instead of separate arguments.

Create the options and a problem structure. See problem for the field names and required fields.

options = optimoptions('fmincon','Display','iter','Algorithm','sqp');
problem.options = options;
problem.solver = 'fmincon';
problem.objective = @(x)100*(x(2)-x(1)^2)^2 + (1-x(1))^2;
problem.x0 = [0,0];

The nonlinear constraint function unitdisk appears at the end of this example. Include the nonlinear constraint function in problem.

problem.nonlcon = @unitdisk;

Solve the problem.

x = fmincon(problem)

 Iter  Func-count            Fval   Feasibility   Step Length       Norm of   First-order  
                                                                       step    optimality
    0           3    1.000000e+00     0.000e+00     1.000e+00     0.000e+00     2.000e+00  
    1          12    8.913011e-01     0.000e+00     1.176e-01     2.353e-01     1.107e+01  
    2          22    8.047847e-01     0.000e+00     8.235e-02     1.900e-01     1.330e+01  
    3          28    4.197517e-01     0.000e+00     3.430e-01     1.217e-01     6.172e+00  
    4          31    2.733703e-01     0.000e+00     1.000e+00     5.254e-02     5.705e-01  
    5          34    2.397111e-01     0.000e+00     1.000e+00     7.498e-02     3.164e+00  
    6          37    2.036002e-01     0.000e+00     1.000e+00     5.960e-02     3.106e+00  
    7          40    1.164353e-01     0.000e+00     1.000e+00     1.459e-01     1.059e+00  
    8          43    1.161753e-01     0.000e+00     1.000e+00     1.754e-01     7.383e+00  
    9          46    5.901601e-02     0.000e+00     1.000e+00     1.547e-02     7.278e-01  
   10          49    4.533081e-02     2.898e-03     1.000e+00     5.393e-02     1.252e-01  
   11          52    4.567454e-02     2.225e-06     1.000e+00     1.492e-03     1.679e-03  
   12          55    4.567481e-02     4.406e-12     1.000e+00     2.095e-06     1.502e-05  
   13          58    4.567481e-02     0.000e+00     1.000e+00     2.203e-12     1.406e-05  

Local minimum possible. Constraints satisfied.

fmincon stopped because the size of the current step is less than
the value of the step size tolerance and constraints are 
satisfied to within the value of the constraint tolerance.

x = 1×2

    0.7864    0.6177

The iterative display and solution are the same as in Nondefault Options.

The following code creates the unitdisk function.

function [c,ceq] = unitdisk(x)
c = x(1)^2 + x(2)^2 - 1;
ceq = [];
end

Obtain the Objective Function Value

Open Live Script

Call fmincon with the fval output to obtain the value of the objective function at the solution.

The Minimize with Bound Constraints example shows two solutions. Which is better? Run the example requesting the fval output as well as the solution.

fun = @(x)1+x(1)./(1+x(2)) - 3*x(1).*x(2) + x(2).*(1+x(1));
lb = [0,0];
ub = [1,2];
A = [];
b = [];
Aeq = [];
beq = [];
x0 = (lb + ub)/2;
[x,fval] = fmincon(fun,x0,A,b,Aeq,beq,lb,ub)

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.

x = 1×2

    1.0000    2.0000

fval = -0.6667

Run the problem using a different starting point x0.

x0 = x0/5;
[x2,fval2] = fmincon(fun,x0,A,b,Aeq,beq,lb,ub)

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.

x2 = 1×2
10^-6 ×

    0.4000    0.4000

fval2 = 1.0000

This solution has an objective function value fval2 = 1, which is higher than the first value fval = –0.6667. The first solution x has a lower local minimum objective function value.

Examine Solution Using Extra Outputs

Open Script

To easily examine the quality of a solution, request the exitflag and output outputs.

Set up the problem of minimizing Rosenbrock's function on the unit disk, $||x||^2 \le 1$ . First create a function that represents the nonlinear constraint. Save this as a file named unitdisk.m on your MATLAB® path.

function [c,ceq] = unitdisk(x)
c = x(1)^2 + x(2)^2 - 1;
ceq = [];

Create the remaining problem specifications.

fun = @(x)100*(x(2)-x(1)^2)^2 + (1-x(1))^2;
nonlcon = @unitdisk;
A = [];
b = [];
Aeq = [];
beq = [];
lb = [];
ub = [];
x0 = [0,0];

Call fmincon using the fval, exitflag, and output outputs.

[x,fval,exitflag,output] = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.


x =

    0.7864    0.6177


fval =

    0.0457


exitflag =

     1


output = 

  struct with fields:

         iterations: 24
          funcCount: 84
    constrviolation: 0
           stepsize: 6.9162e-06
          algorithm: 'interior-point'
      firstorderopt: 2.4373e-08
       cgiterations: 4
            message: '...'
       bestfeasible: [1x1 struct]

The exitflag value 1 indicates that the solution is a local minimum.
The output structure reports several statistics about the solution process. In particular, it gives the number of iterations in output.iterations, number of function evaluations in output.funcCount, and the feasibility in output.constrviolation.

Obtain All Outputs

Open Script

fmincon optionally returns several outputs that you can use for analyzing the reported solution.

Set up the problem of minimizing Rosenbrock's function on the unit disk. First create a function that represents the nonlinear constraint. Save this as a file named unitdisk.m on your MATLAB® path.

function [c,ceq] = unitdisk(x)
c = x(1)^2 + x(2)^2 - 1;
ceq = [];

Create the remaining problem specifications.

fun = @(x)100*(x(2)-x(1)^2)^2 + (1-x(1))^2;
nonlcon = @unitdisk;
A = [];
b = [];
Aeq = [];
beq = [];
lb = [];
ub = [];
x0 = [0,0];

Request all fmincon outputs.

[x,fval,exitflag,output,lambda,grad,hessian] = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.


x =

    0.7864    0.6177


fval =

    0.0457


exitflag =

     1


output = 

  struct with fields:

         iterations: 24
          funcCount: 84
    constrviolation: 0
           stepsize: 6.9162e-06
          algorithm: 'interior-point'
      firstorderopt: 2.4373e-08
       cgiterations: 4
            message: '...'
       bestfeasible: [1x1 struct]


lambda = 

  struct with fields:

         eqlin: [0x1 double]
      eqnonlin: [0x1 double]
       ineqlin: [0x1 double]
         lower: [2x1 double]
         upper: [2x1 double]
    ineqnonlin: 0.1215


grad =

   -0.1911
   -0.1501


hessian =

  497.2903 -314.5589
 -314.5589  200.2392

The lambda.ineqnonlin output shows that the nonlinear constraint is active at the solution, and gives the value of the associated Lagrange multiplier.
The grad output gives the value of the gradient of the objective function at the solution x.
The hessian output is described in fmincon Hessian.

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 = fmincon(@myfun,x0,A,b)

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 = fmincon(@(x)norm(x)^2,x0,A,b);

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

options = optimoptions('fmincon','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 optimoptions and the Algorithm option is 'trust-region-reflective', 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.

If you can also compute the Hessian matrix and the Algorithm option is set to 'interior-point', there is a different way to pass the Hessian to fmincon. For more information, see Hessian for fmincon interior-point algorithm. For an example using Symbolic Math Toolbox™ to compute the gradient and Hessian, see Calculate Gradients and Hessians Using Symbolic Math Toolbox™.

The 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. 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.

'interior-point' algorithm — If the HonorBounds option is true (default), fmincon resets x0 components that are on or outside bounds lb or ub to values strictly between the bounds.
'trust-region-reflective' algorithm — fmincon resets infeasible x0 components to be feasible with respect to bounds or linear equalities.
'sqp', 'sqp-legacy', or 'active-set' algorithm — fmincon resets x0 components that are outside bounds to the values of the corresponding bounds.

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

Data Types: double

`A` — Linear inequality constraints
real matrix

Linear inequality constraints, specified as a real matrix. A is an M-by-N matrix, where M is the number of inequalities, and N is the number of variables (number of elements in x0). For large problems, pass A as a sparse matrix.

A encodes the M linear inequalities

A*x <= b,

where x is the column vector of N variables x(:), and b is a column vector with M elements.

For example, to specify

x₁ + 2x₂ ≤ 10
3x₁ + 4x₂ ≤ 20
5x₁ + 6x₂ ≤ 30,

enter these constraints:

A = [1,2;3,4;5,6];
b = [10;20;30];

Example: To specify that the x components sum to 1 or less, use A = ones(1,N) and b = 1.

Data Types: double

`b` — Linear inequality constraints
real vector

Linear inequality constraints, specified as a real vector. b is an M-element vector related to the A matrix. If you pass b as a row vector, solvers internally convert b to the column vector b(:). For large problems, pass b as a sparse vector.

b encodes the M linear inequalities

A*x <= b,

where x is the column vector of N variables x(:), and A is a matrix of size M-by-N.

For example, consider these inequalities:

x₁ + 2x₂ ≤ 10
3x₁ + 4x₂ ≤ 20
5x₁ + 6x₂ ≤ 30.

Specify the inequalities by entering the following constraints.

A = [1,2;3,4;5,6];
b = [10;20;30];

Example: To specify that the x components sum to 1 or less, use A = ones(1,N) and b = 1.

Data Types: double

`Aeq` — Linear equality constraints
real matrix

Linear equality constraints, specified as a real matrix. Aeq is an Me-by-N matrix, where Me is the number of equalities, and N is the number of variables (number of elements in x0). For large problems, pass Aeq as a sparse matrix.

Aeq encodes the Me linear equalities

Aeq*x = beq,

where x is the column vector of N variables x(:), and beq is a column vector with Me elements.

For example, to specify

x₁ + 2x₂ + 3x₃ = 10
2x₁ + 4x₂ + x₃ = 20,

enter these constraints:

Aeq = [1,2,3;2,4,1];
beq = [10;20];

Example: To specify that the x components sum to 1, use Aeq = ones(1,N) and beq = 1.

Data Types: double

`beq` — Linear equality constraints
real vector

Linear equality constraints, specified as a real vector. beq is an Me-element vector related to the Aeq matrix. If you pass beq as a row vector, solvers internally convert beq to the column vector beq(:). For large problems, pass beq as a sparse vector.

beq encodes the Me linear equalities

Aeq*x = beq,

where x is the column vector of N variables x(:), and Aeq is a matrix of size Me-by-N.

For example, consider these equalities:

x₁ + 2x₂ + 3x₃ = 10
2x₁ + 4x₂ + x₃ = 20.

Specify the equalities by entering the following constraints.

Aeq = [1,2,3;2,4,1];
beq = [10;20];

Example: To specify that the x components sum to 1, use Aeq = ones(1,N) and beq = 1.

Data Types: double

`lb` — Lower bounds
real vector | real array

Lower bounds, specified as a real vector or real array. If the number of elements in x0 is equal to the number of elements in lb, then lb specifies that

x(i) >= lb(i) for all i.

If numel(lb) < numel(x0), then lb specifies that

x(i) >= lb(i) for 1 <= i <= numel(lb).

If there are fewer elements in lb than in x0, solvers issue a warning.

Example: To specify that all x components are positive, use lb = zeros(size(x0)).

Data Types: double

`ub` — Upper bounds
real vector | real array

Upper bounds, specified as a real vector or real array. If the number of elements in x0 is equal to the number of elements in ub, then ub specifies that

x(i) <= ub(i) for all i.

If numel(ub) < numel(x0), then ub specifies that

x(i) <= ub(i) for 1 <= i <= numel(ub).

If there are fewer elements in ub than in x0, solvers issue a warning.

Example: To specify that all x components are less than 1, use ub = ones(size(x0)).

Data Types: double

`nonlcon` — Nonlinear constraints
function handle | function name

Nonlinear constraints, specified as a function handle or function name. nonlcon is a function that accepts a vector or array x and returns two arrays, c(x) and ceq(x).

c(x) is the array of nonlinear inequality constraints at x. fmincon attempts to satisfy
c(x) <= 0 for all entries of c.
ceq(x) is the array of nonlinear equality constraints at x. fmincon attempts to satisfy
ceq(x) = 0 for all entries of ceq.

For example,

x = fmincon(@myfun,x0,A,b,Aeq,beq,lb,ub,@mycon)

where mycon is a MATLAB function such as

function [c,ceq] = mycon(x)
c = ...     % Compute nonlinear inequalities at x.
ceq = ...   % Compute nonlinear equalities at x.

If the gradients of the constraints can also be computed and the SpecifyConstraintGradient option is true, as set by

options = optimoptions('fmincon','SpecifyConstraintGradient',true)

then nonlcon must also return, in the third and fourth output arguments, GC, the gradient of c(x), and GCeq, the gradient of ceq(x). GC and GCeq can be sparse or dense. If GC or GCeq is large, with relatively few nonzero entries, save running time and memory in the interior-point algorithm by representing them as sparse matrices. For more information, see Nonlinear Constraints.

Data Types: char | function_handle | string

`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 optimization algorithm: `'interior-point'` (default) `'trust-region-reflective'` `'sqp'` `'sqp-legacy'` (`optimoptions` only) `'active-set'` For information on choosing the algorithm, see Choosing the Algorithm. The `trust-region-reflective` algorithm requires: A gradient to be supplied in the objective function `SpecifyObjectiveGradient` to be set to `true` Either bound constraints or linear equality constraints, but not both If you select the `'trust-region-reflective'` algorithm and these conditions are not all satisfied, `fmincon` throws an error. The `'active-set'`, `'sqp-legacy'`, and `'sqp'` algorithms are not large-scale. See Large-Scale vs. Medium-Scale Algorithms.
`CheckGradients`	Compare user-supplied derivatives (gradients of objective or constraints) 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 Names.
`ConstraintTolerance`	Tolerance on the constraint violation, a positive scalar. The default is `1e-6`. See Tolerances and Stopping Criteria. For `optimset`, the name is `TolCon`. See Current and Legacy Option Names.
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. For `optimset`, the name is `FinDiffRelStep`. See Current and Legacy Option Names.
`FiniteDifferenceType`	Finite differences, used to estimate gradients, are either `'forward'` (default), or `'central'` (centered). `'central'` takes twice as many function evaluations but should be more accurate. The trust-region-reflective algorithm uses `FiniteDifferenceType` only when `CheckGradients` is set to `true`. `fmincon` is careful to obey bounds when estimating both types of finite differences. So, for example, it could take a backward, rather than a forward, difference to avoid evaluating at a point outside bounds. However, for the `interior-point` algorithm, `'central'` differences might violate bounds during their evaluation if the `HonorBounds` option is set to `false`. For `optimset`, the name is `FinDiffType`. See Current and Legacy Option Names.
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 for all algorithms except `interior-point` is `100*numberOfVariables`; for the `interior-point` algorithm the default is `3000`. See Tolerances and Stopping Criteria and Iterations and Function Counts. For `optimset`, the name is `MaxFunEvals`. See Current and Legacy Option Names.
`MaxIterations`	Maximum number of iterations allowed, a positive integer. The default value for all algorithms except `interior-point` is `400`; for the `interior-point` algorithm the default is `1000`. See Tolerances and Stopping Criteria and Iterations and Function Counts. For `optimset`, the name is `MaxIter`. See Current and Legacy Option Names.
`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 Names.
`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 and Plot 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 `'optimplotfvalconstr'` plots the best feasible objective function value found as a line plot. The plot shows infeasible points as red, and feasible points as blue, using a feasibility tolerance of `1e-6`. `'optimplotconstrviolation'` plots the maximum constraint violation `'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 for Optimization Toolbox™ and Output Function and Plot Function Syntax. For `optimset`, the name is `PlotFcns`. See Current and Legacy Option Names.
`SpecifyConstraintGradient`	Gradient for nonlinear constraint functions defined by the user. When set to the default, `false`, `fmincon` estimates gradients of the nonlinear constraints by finite differences. When set to `true`, `fmincon` expects the constraint function to have four outputs, as described in `nonlcon`. The `trust-region-reflective` algorithm does not accept nonlinear constraints. For `optimset`, the name is `GradConstr` and the values are `'on'` or `'off'`. See Current and Legacy Option Names.
`SpecifyObjectiveGradient`	Gradient for the objective function defined by the user. See the description of `fun` to see how to define the gradient in `fun`. The default, `false`, causes `fmincon` to estimate gradients using finite differences. Set to `true` to have `fmincon` use a user-defined gradient of the objective function. To use the `'trust-region-reflective'` algorithm, you must provide the gradient, and set `SpecifyObjectiveGradient` to `true`. For `optimset`, the name is `GradObj` and the values are `'on'` or `'off'`. See Current and Legacy Option Names.
`StepTolerance`	Termination tolerance on `x`, a positive scalar. The default value for all algorithms except `'interior-point'` is `1e-6`; for the `'interior-point'` algorithm, the default is `1e-10`. See Tolerances and Stopping Criteria. For `optimset`, the name is `TolX`. See Current and Legacy Option Names.
`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)`. `fmincon` uses `TypicalX` for scaling finite differences for gradient estimation. The `'trust-region-reflective'` algorithm uses `TypicalX` only for the `CheckGradients` option.
`UseParallel`	When `true`, `fmincon` estimates gradients in parallel. Disable by setting to the default, `false`. `trust-region-reflective` requires a gradient in the objective, so `UseParallel` does not apply. See Parallel Computing.
Trust-Region-Reflective 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 Names.
`HessianFcn`	If `[]` (default), `fmincon` approximates the Hessian using finite differences, or uses a Hessian multiply function (with option `HessianMultiplyFcn`). If `'objective'`, `fmincon` uses a user-defined Hessian (defined in `fun`). See Hessian as an Input. For `optimset`, the name is `HessFcn`. See Current and Legacy Option Names.
`HessianMultiplyFcn`	Hessian multiply function, specified as a function handle. 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 a 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 = HY`, although `H` is not formed explicitly. `fmincon` 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 `[]`, and `SubproblemAlgorithm` must be `'cg'` (default). See Hessian Multiply Function. See Minimization with Dense Structured Hessian, Linear Equalities for an example. For `optimset`, the name is `HessMult`. See Current and Legacy Option Names.
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)`. `fmincon` 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 `fmincon` computes a full finite-difference approximation in each iteration. This computation can be very 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 bound-constrained problems, and is `numberOfVariables` for equality-constrained problems. For more information, see Preconditioned Conjugate Gradient Method.
PrecondBandWidth	Upper bandwidth of preconditioner for PCG, a nonnegative integer. By default, diagonal preconditioning is used (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 fmincon Trust Region Reflective Algorithm.
TolPCG	Termination tolerance on the PCG iteration, a positive scalar. The default is `0.1`.
Active-Set 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 Names.
MaxSQPIter	Maximum number of SQP iterations allowed, a positive integer. The default is `10*max(numberOfVariables, numberOfInequalities + numberOfBounds)`.
RelLineSrchBnd	Relative bound (a real nonnegative scalar value) on the line search step length. The total displacement in x satisfies \|Δx(i)\| ≤ relLineSrchBnd· max(\|x(i)\|,\|typicalx(i)\|). This option provides control over the magnitude of the displacements in x for cases in which the solver takes steps that are considered too large. The default is no bounds (`[]`).
RelLineSrchBndDuration	Number of iterations for which the bound specified in `RelLineSrchBnd` should be active (default is `1`).
TolConSQP	Termination tolerance on inner iteration SQP constraint violation, a positive scalar. The default is `1e-6`.
Interior-Point Algorithm
`HessianApproximation`	Chooses how `fmincon` calculates the Hessian (see Hessian as an Input). The choices are: `'bfgs'` (default) `'finite-difference'` `'lbfgs'` `{'lbfgs',Positive Integer}` Note To use `HessianApproximation`, both `HessianFcn` and `HessianMultiplyFcn` must be empty entries (`[]`). For `optimset`, the name is `Hessian` and the values are `'user-supplied'`, `'bfgs'`, `'lbfgs'`, `'fin-diff-grads'`, `'on'`, or `'off'`. See Current and Legacy Option Names.
`HessianFcn`	If `[]` (default), `fmincon` approximates the Hessian using finite differences, or uses a supplied `HessianMultiplyFcn`. If a function handle, `fmincon` uses `HessianFcn` to calculate the Hessian. See Hessian as an Input. For `optimset`, the name is `HessFcn`. See Current and Legacy Option Names.
`HessianMultiplyFcn`	User-supplied function that gives a Hessian-times-vector product (see Hessian Multiply Function). Pass a function handle. Note To use the `HessianMultiplyFcn` option, `HessianFcn` must be set to `[]`, and `SubproblemAlgorithm` must be `'cg'`. For `optimset`, the name is `HessMult`. See Current and Legacy Option Names.
`HonorBounds`	The default `true` ensures that bound constraints are satisfied at every iteration. Disable by setting to `false`. For `optimset`, the name is `AlwaysHonorConstraints` and the values are `'bounds'` or `'none'`. See Current and Legacy Option Names.
InitBarrierParam	Initial barrier value, a positive scalar. Sometimes it might help to try a value above the default `0.1`, especially if the objective or constraint functions are large.
InitTrustRegionRadius	Initial radius of the trust region, a positive scalar. On badly scaled problems it might help to choose a value smaller than the default $\sqrt{n}$ , where n is the number of variables.
MaxProjCGIter	A tolerance (stopping criterion) for the number of projected conjugate gradient iterations; this is an inner iteration, not the number of iterations of the algorithm. This positive integer has a default value of `2*(numberOfVariables - numberOfEqualities)`.
`ObjectiveLimit`	A tolerance (stopping criterion) that is a scalar. If the objective function value goes below `ObjectiveLimit` and the iterate is feasible, the iterations halt, because the problem is presumably unbounded. The default value is `-1e20`.
`ScaleProblem`	`true` causes the algorithm to normalize all constraints and the objective function. Disable by setting to the default `false`. For `optimset`, the values are `'obj-and-constr'` or `'none'`. See Current and Legacy Option Names.
`SubproblemAlgorithm`	Determines how the iteration step is calculated. The default, `'factorization'`, is usually faster than `'cg'` (conjugate gradient), though `'cg'` might be faster for large problems with dense Hessians. See fmincon Interior Point Algorithm.
TolProjCG	A relative tolerance (stopping criterion) for projected conjugate gradient algorithm; this is for an inner iteration, not the algorithm iteration. This positive scalar has a default of `0.01`.
TolProjCGAbs	Absolute tolerance (stopping criterion) for projected conjugate gradient algorithm; this is for an inner iteration, not the algorithm iteration. This positive scalar has a default of `1e-10`.
SQP and SQP Legacy Algorithms
`ObjectiveLimit`	A tolerance (stopping criterion) that is a scalar. If the objective function value goes below `ObjectiveLimit` and the iterate is feasible, the iterations halt, because the problem is presumably unbounded. The default value is `-1e20`.
`ScaleProblem`	`true` causes the algorithm to normalize all constraints and the objective function. Disable by setting to the default `false`. For `optimset`, the values are `'obj-and-constr'` or `'none'`. See Current and Legacy Option Names.

Example: options = optimoptions('fmincon','SpecifyObjectiveGradient',true,'SpecifyConstraintGradient',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`
`Aineq`	Matrix for linear inequality constraints
`bineq`	Vector for linear inequality constraints
`Aeq`	Matrix for linear equality constraints
`beq`	Vector for linear equality constraints
`lb`	Vector of lower bounds
`ub`	Vector of upper bounds
`nonlcon`	Nonlinear constraint function
`solver`	`'fmincon'`
`options`	Options created with `optimoptions`

You must supply at least the objective, x0, solver, and options fields in the problem structure.

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 `fmincon` stopped
integer

Reason fmincon stopped, returned as an integer.

All Algorithms:
`1`	First-order optimality measure was less than `options.OptimalityTolerance`, and maximum constraint violation was less than `options.ConstraintTolerance`.
`0`	Number of iterations exceeded `options.MaxIterations` or number of function evaluations exceeded `options.MaxFunctionEvaluations`.
`-1`	Stopped by an output function or plot function.
`-2`	No feasible point was found.
All algorithms except `active-set`:
`2`	Change in `x` was less than `options.StepTolerance` and maximum constraint violation was less than `options.ConstraintTolerance`.
`trust-region-reflective` algorithm only:
`3`	Change in the objective function value was less than `options.FunctionTolerance` and maximum constraint violation was less than `options.ConstraintTolerance`.
`active-set` algorithm only:
`4`	Magnitude of the search direction was less than 2*`options.StepTolerance` and maximum constraint violation was less than `options.ConstraintTolerance`.
`5`	Magnitude of directional derivative in search direction was less than 2*`options.OptimalityTolerance` and maximum constraint violation was less than `options.ConstraintTolerance`.
`interior-point`, `sqp-legacy`, and `sqp` algorithms:
`-3`	Objective function at current iteration went below `options.ObjectiveLimit` and maximum constraint violation was less than `options.ConstraintTolerance`.

`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
`lssteplength`	Size of line search step relative to search direction (`active-set` and `sqp` algorithms only)
`constrviolation`	Maximum of constraint functions
`stepsize`	Length of last displacement in `x` (not in `active-set` algorithm)
`algorithm`	Optimization algorithm used
`cgiterations`	Total number of PCG iterations (`trust-region-reflective` and `interior-point` algorithms)
`firstorderopt`	Measure of first-order optimality
`bestfeasible`	Best (lowest objective function) feasible point encountered. A structure with these fields: `x` `fval` `firstorderopt` `constrviolation` If no feasible point is found, the `bestfeasible` field is empty. For this purpose, a point is feasible when the maximum of the constraint functions does not exceed `options.ConstraintTolerance`. The `bestfeasible` point can differ from the returned solution point `x` for a variety of reasons. For an example, see Obtain Best Feasible Point.
`message`	Exit message

`lambda` — Lagrange multipliers at the solution
structure

Lagrange multipliers at the solution, returned as a structure with fields:

`lower`	Lower bounds corresponding to `lb`
`upper`	Upper bounds corresponding to `ub`
`ineqlin`	Linear inequalities corresponding to `A` and `b`
`eqlin`	Linear equalities corresponding to `Aeq` and `beq`
`ineqnonlin`	Nonlinear inequalities corresponding to the `c` in `nonlcon`
`eqnonlin`	Nonlinear equalities corresponding to the `ceq` in `nonlcon`

`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.

Limitations

fmincon is a gradient-based method that is designed to work on problems where the objective and constraint functions are both continuous and have continuous first derivatives.
For the 'trust-region-reflective' algorithm, you must provide the gradient in fun and set the 'SpecifyObjectiveGradient' option to true.
The 'trust-region-reflective' algorithm does not allow equal upper and lower bounds. For example, if lb(2)==ub(2), fmincon gives this error:
```
Equal upper and lower bounds not permitted in trust-region-reflective algorithm. Use
either interior-point or SQP algorithms instead.
```
There are two different syntaxes for passing a Hessian, and there are two different syntaxes for passing a HessianMultiplyFcn function; one for trust-region-reflective, and another for interior-point. See Including Hessians.
- For trust-region-reflective, the Hessian of the Lagrangian is the same as the Hessian of the objective function. You pass that Hessian as the third output of the objective function.
- For interior-point, the Hessian of the Lagrangian involves the Lagrange multipliers and the Hessians of the nonlinear constraint functions. You pass the Hessian as a separate function that takes into account both the current point x and the Lagrange multiplier structure lambda.
When the problem is infeasible, fmincon attempts to minimize the maximum constraint value.

More About

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Hessian as an Input

fmincon uses a Hessian as an optional input. This Hessian is the matrix of second derivatives of the Lagrangian (see Equation 1), namely,

\nabla_{x x}^{2} L (x, λ) = \nabla^{2} f (x) + \sum λ_{i} \nabla^{2} c_{i} (x) + \sum λ_{i} \nabla^{2} c e q_{i} (x) .

(1)

For details of how to supply a Hessian to the trust-region-reflective or interior-point algorithms, see Including Hessians.

The active-set and sqp algorithms do not accept an input Hessian. They compute a quasi-Newton approximation to the Hessian of the Lagrangian.

The interior-point algorithm has several choices for the 'HessianApproximation' option; see Choose Input Hessian Approximation for interior-point fmincon:

'bfgs' — fmincon calculates the Hessian by a dense quasi-Newton approximation. This is the default Hessian approximation.
'lbfgs' — fmincon calculates the Hessian by a limited-memory, large-scale quasi-Newton approximation. The default memory, 10 iterations, is used.
{'lbfgs',positive integer} — fmincon calculates the Hessian by a limited-memory, large-scale quasi-Newton approximation. The positive integer specifies how many past iterations should be remembered.
'finite-difference' — fmincon calculates a Hessian-times-vector product by finite differences of the gradient(s). You must supply the gradient of the objective function, and also gradients of nonlinear constraints (if they exist). Set the 'SpecifyObjectiveGradient' option to true and, if applicable, the 'SpecifyConstraintGradient' option to true. You must set the 'SubproblemAlgorithm' to 'cg'.

Hessian Multiply Function

The 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. For details, see Hessian Multiply Function.

Algorithms

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Choosing the Algorithm

For help choosing the algorithm, see fmincon Algorithms. To set the algorithm, use optimoptions to create options, and use the 'Algorithm' name-value pair.

The rest of this section gives brief summaries or pointers to information about each algorithm.

Interior-Point Optimization

This algorithm is described in fmincon Interior Point Algorithm. There is more extensive description in [1], [41], and [9].

SQP and SQP-Legacy Optimization

The fmincon 'sqp' and 'sqp-legacy' algorithms are similar to the 'active-set' algorithm described in Active-Set Optimization. fmincon SQP Algorithm describes the main differences. In summary, these differences are:

Active-Set Optimization

fmincon uses a sequential quadratic programming (SQP) method. In this method, the function solves a quadratic programming (QP) subproblem at each iteration. fmincon updates an estimate of the Hessian of the Lagrangian at each iteration using the BFGS formula (see fminunc and references [7] and [8]).

fmincon performs a line search using a merit function similar to that proposed by [6], [7], and [8]. The QP subproblem is solved using an active set strategy similar to that described in [5]. fmincon Active Set Algorithm describes this algorithm in detail.

See also SQP Implementation for more details on the algorithm used.

Trust-Region-Reflective Optimization

The 'trust-region-reflective' algorithm is a subspace trust-region method and is based on the interior-reflective Newton method described in [3] and [4]. Each iteration involves the approximate solution of a large linear system using the method of preconditioned conjugate gradients (PCG). See the trust-region and preconditioned conjugate gradient method descriptions in fmincon Trust Region Reflective Algorithm.

Alternative Functionality

App

The Optimize Live Editor task provides a visual interface for fmincon.

References

[1] Byrd, R. H., J. C. Gilbert, and J. Nocedal. “A Trust Region Method Based on Interior Point Techniques for Nonlinear Programming.” Mathematical Programming, Vol 89, No. 1, 2000, pp. 149–185.

[2] Byrd, R. H., Mary E. Hribar, and Jorge Nocedal. “An Interior Point Algorithm for Large-Scale Nonlinear Programming.” SIAM Journal on Optimization, Vol 9, No. 4, 1999, pp. 877–900.

[3] 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.

[4] 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.

[5] Gill, P. E., W. Murray, and M. H. Wright. Practical Optimization, London, Academic Press, 1981.

[6] Han, S. P. “A Globally Convergent Method for Nonlinear Programming.” Journal of Optimization Theory and Applications, Vol. 22, 1977, pp. 297.

[7] Powell, M. J. D. “A Fast Algorithm for Nonlinearly Constrained Optimization Calculations.” Numerical Analysis, ed. G. A. Watson, Lecture Notes in Mathematics, Springer-Verlag, Vol. 630, 1978.

[8] Powell, M. J. D. “The Convergence of Variable Metric Methods For Nonlinearly Constrained Optimization Calculations.” Nonlinear Programming 3 (O. L. Mangasarian, R. R. Meyer, and S. M. Robinson, eds.), Academic Press, 1978.

[9] Waltz, R. A., J. L. Morales, J. Nocedal, and D. Orban. “An interior algorithm for nonlinear optimization that combines line search and trust region steps.” Mathematical Programming, Vol 107, No. 3, 2006, pp. 391–408.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

Generated code has limited error checking for options. The recommended way to update an option is to use optimoptions, not dot notation.

opts = optimoptions('fmincon','Algorithm','sqp');
opts = optimoptions(opts,'MaxIterations',1e4); % Recommended
opts.MaxIterations = 1e4; % Not recommended

Do not load options from a file. Doing so can cause code generation to fail. Instead, create options in your code.
Usually, if you specify an option that is not supported, the option is silently ignored during code generation. However, if you specify a plot function or output function by using dot notation, code generation can issue an error. For reliability, specify only supported options.
Because output functions and plot functions are not supported, fmincon does not return the exit flag –1.
Code generated from fmincon does not contain the bestfeasible field in a returned output structure.

For an example, see Code Generation for Optimization Basics.

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.

Documentation

fmincon

Syntax

Description

Examples

Linear Inequality Constraint

Linear Inequality and Equality Constraint

Minimize with Bound Constraints

Nonlinear Constraints

Nondefault Options

Include Gradient

Use a Problem Structure

Obtain the Objective Function Value

Examine Solution Using Extra Outputs

Obtain All Outputs

Input Arguments

fun — Function to minimize function handle | function name

x0 — Initial point real vector | real array

A — Linear inequality constraints real matrix

b — Linear inequality constraints real vector

Aeq — Linear equality constraints real matrix

beq — Linear equality constraints real vector

lb — Lower bounds real vector | real array

ub — Upper bounds real vector | real array

nonlcon — Nonlinear constraints function handle | function name

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

problem — Problem structure structure

Output Arguments

x — Solution real vector | real array

fval — Objective function value at solution real number

exitflag — Reason fmincon stopped integer

output — Information about the optimization process structure

lambda — Lagrange multipliers at the solution structure

grad — Gradient at the solution real vector

hessian — Approximate Hessian real matrix

Limitations

More About

Hessian as an Input

Hessian Multiply Function

Algorithms

Choosing the Algorithm

Interior-Point Optimization

SQP and SQP-Legacy Optimization

Active-Set Optimization

Trust-Region-Reflective Optimization

Alternative Functionality

App

References

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using MATLAB® Coder™.

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

See Also

Topics

Optimization Toolbox Documentation

Support

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

`x0` — Initial point
real vector | real array

`A` — Linear inequality constraints
real matrix

`b` — Linear inequality constraints
real vector

`Aeq` — Linear equality constraints
real matrix

`beq` — Linear equality constraints
real vector

`lb` — Lower bounds
real vector | real array

`ub` — Upper bounds
real vector | real array

`nonlcon` — Nonlinear constraints
function handle | function name

`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 `fmincon` stopped
integer

`output` — Information about the optimization process
structure

`lambda` — Lagrange multipliers at the solution
structure

`grad` — Gradient at the solution
real vector

`hessian` — Approximate Hessian
real matrix

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

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