Optimization Toolbox

fmincon

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Find minimum of constrained nonlinear multivariable function

Equation

Finds the minimum of a problem specified by

subject to

where x, b, beq, lb, and ub 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.

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

Description

fmincon attempts to find a constrained minimum of a scalar function of several variables starting at an initial estimate. This is generally referred to as constrained nonlinear optimization or nonlinear programming.

x = fmincon(fun,x0,A,b) starts at x0 and attempts to find a minimum x to the function described in fun subject to the linear inequalities A*x <= b. x0 can be a scalar, vector, or matrix.

x = fmincon(fun,x0,A,b,Aeq,beq) minimizes fun subject to the linear equalities Aeq*x = beq as well as A*x <= b. Set A=[] and b=[] if no inequalities exist.

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. Set Aeq=[] and beq=[] if no equalities exist. Set lb(i)=-Inf if x(i) is unbounded below, and set ub(i)=Inf if x(i) is unbounded above.

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. Set lb=[] and/or ub=[] if no bounds exist.

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

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

[x,fval,exitflag] = fmincon(...) returns a value exitflag that describes the exit condition of fmincon.

[x,fval,exitflag,output] = fmincon(...) returns a structure output with information about the optimization.

[x,fval,exitflag,output,lambda] = fmincon(...) returns a structure lambda whose fields contain the Lagrange multipliers at the solution x.

[x,fval,exitflag,output,lambda,grad] = fmincon(...) returns the value of the gradient of fun at the solution x.

[x,fval,exitflag,output,lambda,grad,hessian] = fmincon(...) returns the value of the Hessian at the solution x. See Hessian.

Avoiding Global Variables via Anonymous and Nested Functions explains how to parameterize the objective function fun, if necessary.

Note   If the specified input bounds for a problem are inconsistent, the output x is x0 and the output fval is []. Components of x0 that violate the bounds lb <= x <=ub are reset to the interior of the box defined by the bounds. Components that respect the bounds are not changed.

Input Arguments

Function Arguments contains general descriptions of arguments passed into fmincon. This section provides function-specific details for fun, nonlcon, and options:

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

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

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

options = optimset('GradObj','on')

function [f,g] = myfun(x)
f = ...         % Compute the function value at x
if nargout > 1  % fun called with two output arguments
   g = ...      % Compute the gradient evaluated at x
end

function [f,g,H] = myfun(x)
f = ...     % Compute the objective function value at x
if nargout > 1   % fun called with two output arguments
   g = ...  % Gradient of the function evaluated at x
   if nargout > 2
      H = ...  % Hessian evaluated at x
   end
end

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

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

options = optimset('GradConstr','on')

function [c,ceq,GC,GCeq] = mycon(x)
c = ...          % Nonlinear inequalities at x
ceq = ...        % Nonlinear equalities at x
if nargout > 2   % nonlcon called with 4 outputs
   GC = ...      % Gradients of the inequalities
   GCeq = ...    % Gradients of the equalities
end

Output Arguments

Function Arguments contains general descriptions of arguments returned by fmincon. This section provides function-specific details for exitflag, lambda, and output:

Hessian

fmincon computes the output argument hessian as follows:

• When using the medium-scale algorithm, the function computes a quasi-Newton approximation to the Hessian of the Lagrangian at x.

• When using the large-scale algorithm, the function uses

  • options.Hessian, if you supply it, to compute the Hessian at x.

  • A finite-difference approximation to the Hessian at x, if you supply only the gradient. Note that because the large-scale algorithm does not take nonlinear constraints, the Hessian of the Lagrangian is the same as the Hessian of the objective function.

Options

Optimization options used by fmincon. Some options apply to all algorithms, some are only relevant when you are using the large-scale algorithm, and others are only relevant when you are using the medium-scale algorithm. You can use optimset to set or change the values of these fields in the options structure options. See Optimization Options for detailed information.

The LargeScale option specifies a preference for which algorithm to use. It is only a preference because certain conditions must be met to use the large-scale algorithm. For fmincon, you must provide the gradient (see the preceding description of fun to see how) or else the medium-scale algorithm is used:

Medium-Scale and Large-Scale Algorithms

These options are used by both the medium-scale and large-scale algorithms:

Large-Scale Algorithm Only

These options are used only by the large-scale algorithm:

W = hmfun(Hinfo,Y,p1,p2,...)

[f,g,Hinfo] = fun(x)

Medium-Scale Algorithm Only

These options are used only by the medium-scale algorithm:

Examples

Find values of x that minimize , starting at the point x = [10; 10; 10] and subject to the constraints:

First, write an M-file that returns a scalar value f of the function evaluated at x.

function f = myfun(x)
f = -x(1) * x(2) * x(3);

Then rewrite the constraints as both less than or equal to a constant,

Since both constraints are linear, formulate them as the matrix inequality where

Next, supply a starting point and invoke an optimization routine.

x0 = [10; 10; 10];    % Starting guess at the solution
[x,fval] = fmincon(@myfun,x0,A,b)

After 66 function evaluations, the solution is

x =
    24.0000
    12.0000
    12.0000

where the function value is

fval =
    -3.4560e+03

and linear inequality constraints evaluate to be less than or equal to 0.

A*x-b= 
    -72
      0

Notes

Large-Scale Optimization

To use the large-scale method, you must

• Supply the gradient in fun

• Set GradObj to 'on' in options

• Specify the feasible region using one, but not both, of the following types of constraints:

  • Upper and lower bounds constraints.

  • Linear equality constraints, in which the equality constraint matrix Aeq cannot have more rows than columns. Aeq is typically sparse.

You cannot use inequality constraints with the large-scale algorithm. If the preceding conditions are not met, fmincon reverts to the medium-scale algorithm.

The function fmincon returns a warning if no gradient is provided and the LargeScale option is not 'off'. fmincon permits g(x) to be an approximate gradient but this option is not recommended; the numerical behavior of most optimization methods is considerably more robust when the true gradient is used. See Large-Scale Problem Coverage and Requirements for more information on what problem formulations are covered and what information you must be provide.

The large-scale method in fmincon is most effective when the matrix of second derivatives, i.e., the Hessian matrix H(x), is also computed. However, evaluation of the true Hessian matrix is not required. For example, if you can supply the Hessian sparsity structure (using the HessPattern option in options), fmincon computes a sparse finite-difference approximation to H(x).

If x0 is not strictly feasible, fmincon chooses a new strictly feasible (centered) starting point.

If components of x have no upper (or lower) bounds, then fmincon prefers that the corresponding components of ub (or lb) be set to Inf (or -Inf for lb) as opposed to an arbitrary but very large positive (or negative in the case of lower bounds) number.

Several aspects of linearly constrained minimization should be noted:

• A dense (or fairly dense) column of matrix Aeq can result in considerable fill and computational cost.

• fmincon removes (numerically) linearly dependent rows in Aeq; however, this process involves repeated matrix factorizations and therefore can be costly if there are many dependencies.

• Each iteration involves a sparse least-squares solution with matrix

  

  where R^T is the Cholesky factor of the preconditioner. Therefore, there is a potential conflict between choosing an effective preconditioner and minimizing fill in .

Medium-Scale Optimization

Better numerical results are likely if you specify equalities explicitly, using Aeq and beq, instead of implicitly, using lb and ub.

If equality constraints are present and dependent equalities are detected and removed in the quadratic subproblem, 'dependent' is displayed under the Procedures heading (when you ask for output by setting the Display option to'iter'). The dependent equalities are only removed when the equalities are consistent. If the system of equalities is not consistent, the subproblem is infeasible and 'infeasible' is displayed under the Procedures heading.

Algorithm

Large-Scale Optimization

The large-scale algorithm is a subspace trust-region method and is based on the interior-reflective Newton method described in [1] and [2]. 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 Large-Scale Algorithms.

Medium-Scale Optimization

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

A line search is performed using a merit function similar to that proposed by [4], [5], and [6]. The QP subproblem is solved using an active set strategy similar to that described in [3]. A full description of this algorithm is found in Constrained Optimization in Standard Algorithms.

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

Limitations

fmincon only handles real variables.

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.

fmincon might only give local solutions.

When the problem is infeasible, fmincon attempts to minimize the maximum constraint value.

The objective function and constraint function must be real-valued; that is, they cannot return complex values.

The large-scale method does not allow equal upper and lower bounds. For example, if lb(2)==ub(2), then fmincon gives this error:

Equal upper and lower bounds not permitted in this
large-scale method.
Use equality constraints and the medium-scale
method instead.

If you have only equality constraints, you can still use the large-scale method. But if you have both equalities and bounds, you must use the medium-scale method.

References

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

[2] 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, pp. 189-224, 1994.

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

[4] Han, S.P., "A Globally Convergent Method for Nonlinear Programming," Vol. 22, Journal of Optimization Theory and Applications, p. 297, 1977.

[5] 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.

[6] 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.

See Also

@ (function_handle), fminbnd, fminsearch, fminunc, optimsetoptimtool

fminbnd

fminimax

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