lsqnonneg

Solve nonnegative linear least-squares problem

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Solve nonnegative least-squares curve fitting problems of the form

$\min_{x} {‖ C \cdot x - d ‖}_{2}^{2}, where x \geq 0.$

Syntax

x = lsqnonneg(C,d)

x = lsqnonneg(C,d,options)

x = lsqnonneg(problem)

[x, resnorm残留]= lsqnonneg(___)

[x,resnorm,residual,exitflag,output] = lsqnonneg(___)

[x,resnorm,residual,exitflag,output,lambda] = lsqnonneg(___)

Description

example

x= lsqnonneg(C,d)returns the vectorxthat minimizesnorm(C*x-d)subject tox ≥ 0. ArgumentsCanddmust be real.

example

x= lsqnonneg(C,d,options)minimizes with the optimization options specified in the structureoptions. Useoptimsetto set these options.

x= lsqnonneg(problem)finds the minimum forproblem, whereproblemis a structure. Create theproblemargument by exporting a problem from Optimization app, as described inExporting Your Work.

example

[x,resnorm,residual] = lsqnonneg(___), for any previous syntax, additionally returns the value of the squared 2-norm of the residual,norm(C*x-d)^2, and returns the residuald-C*x.

[x,resnorm,residual,exitflag,output] = lsqnonneg(___)additionally returns a valueexitflagthat describes the exit condition oflsqnonneg, and a structureoutput与信息;n about the optimization process.

example

[x,resnorm,residual,exitflag,output,lambda] = lsqnonneg(___)additionally returns the Lagrange multiplier vectorlambda.

Examples

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Nonnegative Linear Least Squares

Open Live Script

Compute a nonnegative solution to a linear least-squares problem, and how to compare the result to the solution of an unconstrained problem.

Prepare aCmatrix anddvector for the problem.

C = [0.0372 0.2869 0.6861 0.7071 0.6233 0.6245 0.6344 0.6170]; d = [0.8587 0.1781 0.0747 0.8405];

Compute the constrained and unconstrained solutions.

x = lsqnonneg(C,d)

x =0 0.6929

xunc = C\d

xunc =-2.5627 3.1108

All entries inxare nonnegative, but some entries inxuncare negative.

Compute the norms of the residuals for the two solutions.

constrained_norm = norm(C*x - d)

constrained_norm = 0.9118

unconstrained_norm = norm(C*xunc - d)

unconstrained_norm = 0.6674

The unconstrained solution has a smaller residual norm because constraints can only increase a residual norm.

Nonnegative Least Squares with Nondefault Options

Open Live Script

Set theDisplayoption to'final'to see output whenlsqnonnegfinishes.

Create the options.

options = optimset('Display','final');

Prepare aCmatrix anddvector for the problem.

C = [0.0372 0.2869 0.6861 0.7071 0.6233 0.6245 0.6344 0.6170]; d = [0.8587 0.1781 0.0747 0.8405];

Calllsqnonnegwith the options structure.

x = lsqnonneg(C,d,options);

Optimization terminated.

Obtain Residuals from Nonnegative Least Squares

Open Live Script

Calllsqnonnegwith outputs to obtain the solution, residual norm, and residual vector.

Prepare aCmatrix anddvector for the problem.

C = [0.0372 0.2869 0.6861 0.7071 0.6233 0.6245 0.6344 0.6170]; d = [0.8587 0.1781 0.0747 0.8405];

Obtain the solution and residual information.

[x, resnorm残留]= lsqnonneg(C,d)

x =0 0.6929

resnorm = 0.8315

residual =0.6599 -0.3119 -0.3580 0.4130

Verify that the returned residual norm is the square of the norm of the returned residual vector.

norm(residual)^2

ans = 0.8315

Inspect the Result of Nonnegative Least Squares

Open Live Script

Request all output arguments to examine the solution and solution process afterlsqnonnegfinishes.

Prepare aCmatrix anddvector for the problem.

C = [0.0372 0.2869 0.6861 0.7071 0.6233 0.6245 0.6344 0.6170]; d = [0.8587 0.1781 0.0747 0.8405];

Solve the problem, requesting all output arguments.

[x,resnorm,residual,exitflag,output,lambda] = lsqnonneg(C,d)

x =0 0.6929

resnorm = 0.8315

residual =0.6599 -0.3119 -0.3580 0.4130

exitflag = 1

output =struct with fields:iterations: 1 algorithm: 'active-set' message: 'Optimization terminated.'

lambda =-0.1506 -0.0000

exitflagis1, indicating a correct solution.

x(1) = 0, and the correspondinglambda(1)0, showing the correct duality. Similarly,x(2) > 0, and the correspondinglambda(2) = 0.

Input Arguments

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`C`— Linear multiplier
real matrix

Linear multiplier, specified as a real matrix. Represents the variableCin the problem

$\min_{x} {‖ C x - d ‖}_{2}^{2} .$

For compatibility, the number of rows ofCmust equal the length ofd.

Example:C = [1,2;3,-1;-4,4]

Data Types:double

`d`— Additive term
real vector

Additive term, specified as a real vector. Represents the variabledin the problem

$\min_{x} {‖ C x - d ‖}_{2}^{2} .$

For compatibility, the length ofdmust equal the number of rows ofC.

Example:d = [1;-6;5]

Data Types:double

`options`— Optimization options
structure such as`optimset`returns

Optimization options, specified as a structure such asoptimsetreturns. You can useoptimsetto set or change the values of these fields in the options structure. SeeOptimization Options Referencefor detailed information.

Display

Level of display:

'notify'(default) displays output only if the function does not converge.
'off'or'none'displays no output.
'final'displays just the final output.

TolX

Termination tolerance onx, a positive scalar. The default is10*eps*norm(C,1)*length(C). SeeTolerances and Stopping Criteria.

Example:options = optimset('Display','final')

Data Types:struct

`problem`— Problem structure
structure

Problem structure, specified as a structure with the following fields.

Field Name	Entry
`C`	Real matrix
`d`	Real vector
`solver`	`'lsqnonneg'`
`options`	Options structure such as returned by`optimset`

The simplest way to obtain aproblemstructure is to export the problem from the Optimization app.

Data Types:struct

Output Arguments

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`x`— Solution
real vector

Solution, returned as a real vector. The length ofxis the same as the length ofd.

`resnorm`— Squared residual norm
nonnegative scalar

Squared residual norm, returned as a nonnegative scalar. Equal tonorm(C*x-d)^2.

`residual`— Residual
real vector

Residual, returned as a real vector. The residual isd - C*x.

`exitflag`— Reason`lsqnonneg`stopped
integer

Reasonlsqnonnegstopped, returned as an integer.

`1`	Function converged to a solution`x`.
`0`	Number of iterations exceeded`options.MaxIter`.

`output`— Information about the optimization process
structure

Information about the optimization process, returned as a structure with fields:

`iterations`	Number of iterations taken
`algorithm`	`'active-set'`
`message`	Exit message

`lambda`— Lagrange multipliers
real vector

Lagrange multipliers, returned as a real vector. The entries satisfy the complementarity conditionx'*lambda = 0. This meanslambda(i) < 0whenx(i)is approximately0, andlambda(i)is approximately0whenx(i) > 0.

Tips

For problems wheredhas length over 20,lsqlinmight be faster thanlsqnonneg. Whendhas length under 20,lsqnonnegis generally more efficient.
To convert between the solvers whenChas more rows than columns (meaning the system is overdetermined),
```
[x,resnorm,residual,exitflag,output,lambda] = lsqnonneg(C,d)
```
相当于
```
[m,n] = size(C); [x,resnorm,residual,exitflag,output,lambda_lsqlin] = ... lsqlin(C,d,-eye(n,n),zeros(n,1));
```
The only difference is that the corresponding Lagrange multipliers have opposite signs:lambda = -lambda_lsqlin.ineqlin.

Algorithms

lsqnonneguses the algorithm described in[1]. The algorithm starts with a set of possible basis vectors and computes the associated dual vectorlambda. It then selects the basis vector corresponding to the maximum value inlambdato swap it out of the basis in exchange for another possible candidate. This continues untillambda ≤ 0.

References

[1] Lawson, C. L. and R. J. Hanson.Solving Least-Squares Problems. Upper Saddle River, NJ: Prentice Hall. 1974. Chapter 23, p. 161.

Extended Capabilities

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

For C/C++ code generation:

You must enable support for variable-size arrays.
The exit message in the output structure is not translated.

Documentation

lsqnonneg

Syntax

Description

Examples

Nonnegative Linear Least Squares

Nonnegative Least Squares with Nondefault Options

Obtain Residuals from Nonnegative Least Squares

Inspect the Result of Nonnegative Least Squares

Input Arguments

`C`— Linear multiplier
real matrix

`d`— Additive term
real vector

`options`— Optimization options
structure such as`optimset`returns

`problem`— Problem structure
structure

Output Arguments

`x`— Solution
real vector

`resnorm`— Squared residual norm
nonnegative scalar

`residual`— Residual
real vector

`exitflag`— Reason`lsqnonneg`stopped
integer

`output`— Information about the optimization process
structure

`lambda`— Lagrange multipliers
real vector

Tips

Algorithms

References

Extended Capabilities

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

See Also

Introduced before R2006a

Optimization Toolbox Documentation

Other Documentation

金宝app

Try MATLAB, Simulink, and Other Products

Documentation

lsqnonneg

Syntax

Description

Examples

Nonnegative Linear Least Squares

Nonnegative Least Squares with Nondefault Options

Obtain Residuals from Nonnegative Least Squares

Inspect the Result of Nonnegative Least Squares

Input Arguments

C— Linear multiplierreal matrix

d— Additive termreal vector

options— Optimization optionsstructure such asoptimsetreturns

problem— Problem structurestructure

Output Arguments

x— Solutionreal vector

resnorm— Squared residual normnonnegative scalar

residual— Residualreal vector

exitflag— Reasonlsqnonnegstoppedinteger

output— Information about the optimization processstructure

lambda— Lagrange multipliersreal vector

Tips

Algorithms

References

Extended Capabilities

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

See Also

Introduced before R2006a

Optimization Toolbox Documentation

Other Documentation

金宝app

Try MATLAB, Simulink, and Other Products

`C`— Linear multiplier
real matrix

`d`— Additive term
real vector

`options`— Optimization options
structure such as`optimset`returns

`problem`— Problem structure
structure

`x`— Solution
real vector

`resnorm`— Squared residual norm
nonnegative scalar

`residual`— Residual
real vector

`exitflag`— Reason`lsqnonneg`stopped
integer

`output`— Information about the optimization process
structure

`lambda`— Lagrange multipliers
real vector

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