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fmincon

Find minimum of constrained nonlinear multivariable function

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Equation

Finds the minimum of a problem specified by

min

f(x)suchthat



c(x)ceq(x)A⋅xAeq⋅xlb≤0=0≤b=beq≤x≤ub,

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.

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,exitag] = fmincon(...)

[x,fval,exitag,output] = fmincon(...)

[x,fval,exitag,output,lambda] = fmincon(...)

[x,fval,exitag,output,lambda,grad] = fmincon(...)

[x,fval,exitag,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 optimizationornonlinear

programming.

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

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.

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 = [].

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 speci+ed 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 de+ned by the bounds. Components that respect the bounds are not

changed.

See!Iterations Can Violate Constraints.

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 = [].

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 = [].

x = fmincon(problem)finds the minimum forproblem, whereproblemis a structure described

inInput Arguments. Create theproblemstructure by exporting a problem from Optimization app, as

described inExporting Your Work.

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

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

[x,fval,exitag,output] = fmincon(...)returns a structureoutputwith information about the

optimization.

[x,fval,exitag,output,lambda] = fmincon(...)returns a structurelambdawhose fields contain the

Lagrange multipliers at the solutionx.

[x,fval,exitag,output,lambda,grad] = fmincon(...)returns the value of the gradient offunat the

solutionx.

[x,fval,exitag,output,lambda,grad,hessian] = fmincon(...)returns the value of the Hessian at

the solutionx. Seefmincon Hessian.

Input Arguments

Function Argumentsdescribes the arguments passed tofmincon.Optionsprovides the function-

specific details for theoptionsvalues. This section provides function-specific details forfun,nonlcon,

andproblem.

fun The function to be minimized.!fun!is a function that accepts a vector!x!and

returns a scalar!f, the objective function evaluated at!x.!fun!can be speci+ed

as a function handle for a +le:

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

where!myfun!is a MATLAB

!function such as

function f = myfun(x)

f = ... % Compute function value at x

fun!can also be a function handle for an anonymous function:

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

If the gradient of!fun!can also be computed!and!the!GradObj!option is!'on',

as set by

options = optimoptions('fmincon','GradObj','on')

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

argument.

If the Hessian matrix can also be computed!and!the!Hessian!option

is!'on'!via!options!=!optimoptions('fmincon','Hessian','user-supplied')and!the!

Algorithm!option is!trust-region-reective,!fun!must return the Hessian

value!H(x), a symmetric matrix, in a third output argument.!fun!can give a

sparse Hessian. See!Writing Objective Functions!for details.

If the Hessian matrix can be computed and the!Algorithm!option is!interior-

point, there are several ways to pass the Hessian to!fmincon. For more

information, see!Hessian.

A,!b,!Ae

q,!beq

Linear constraint matrices!A!and!Aeq, and their corresponding

vectors!b!and!beq, can be sparse or dense. The!trust-region-

reective!andinterior-point!algorithms use sparse linear algebra.

If!A!or!Aeq!is large, with relatively few nonzero entries, save running time

and memory in the!trust-region-reective!or!interior-point!algorithms by

using sparse matrices.

nonlcon The function that computes the nonlinear inequality constraints!c(x)≤ 0!and

the nonlinear equality constraints!ceq(x) = 0.!nonlcon!accepts a vector!xand

returns the two vectors!c!and!ceq.!c!is a vector that contains the nonlinear

inequalities evaluated at!x, and!ceq!is a vector that contains the nonlinear

equalities evaluated at!x.!nonlcon!should be speci+ed as a function handle

to a +le or to an anonymous function, such as!mycon:

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!GradConstr!option is!'on', as set by

options = optimoptions('fmincon','GradConstr','on')

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.

Note!! Because Optimization Toolbox™ functions only accept inputs of type!double

functions must return outputs of type!double.

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C202_2021

2018-09-09

就是MATLAB软件对这个函数的解释而已

mikelang2015

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