lsqnonneg

Solve nonnegative linear least-squares problem

Syntax

x = lsqnonneg(C,d)

x = lsqnonneg(C,d,options)

x = lsqnonneg(problem)

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

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

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

Description

Solve nonnegative least-squares curve fitting problems of the form

$$\underset{x}{\min }{\Vert C\cdot x-d\Vert }_2^2,\text{ where }x\ge 0.$$

x = lsqnonneg(C,d) returns the vector x that minimizes norm(C*x-d) subject to x ≥ 0. Arguments C and d must be real.

x = lsqnonneg(C,d,options) minimizes with the optimization options specified in the structure options. Use optimset (Optimization Toolbox) to set these options.

x = lsqnonneg(problem) finds the minimum for problem, where problem is a structure.

[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 residual d-C*x.

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

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

Examples

Nonnegative Linear Least Squares

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

Prepare a C matrix and d vector for the problem $$\min ||Cx-d||$$.

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 = 2×1

         0
    0.6929

xunc = C\d

xunc = 2×1

   -2.5627
    3.1108

All entries in x are nonnegative, but some entries in xunc are 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

Set the Display option to 'final' to see output when lsqnonneg finishes.

Create the options.

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

Prepare a C matrix and d vector for the problem $$\min ||Cx-d||$$.

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];

Call lsqnonneg with the options structure.

x = lsqnonneg(C,d,options);

Optimization terminated.

Obtain Residuals from Nonnegative Least Squares

Call lsqnonneg with outputs to obtain the solution, residual norm, and residual vector.

Prepare a C matrix and d vector for the problem $$\min ||Cx-d||$$.

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,residual] = lsqnonneg(C,d)

x = 2×1

         0
    0.6929

resnorm = 0.8315

residual = 4×1

    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

Request all output arguments to examine the solution and solution process after lsqnonneg finishes.

Prepare a C matrix and d vector for the problem $$\min ||Cx-d||$$.

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 = 2×1

         0
    0.6929

resnorm = 0.8315

residual = 4×1

    0.6599
   -0.3119
   -0.3580
    0.4130

exitflag = 1

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

lambda = 2×1

   -0.1506
   -0.0000

exitflag is 1, indicating a correct solution.

x(1) = 0, and the corresponding lambda(1) $$\ne$$ 0, showing the correct duality. Similarly, x(2) > 0, and the corresponding lambda(2) = 0.

Input Arguments

C — Linear multiplier

real matrix

Linear multiplier, specified as a real matrix. Represents the variable C in the problem

$$\underset{x}{\min }{\Vert C\cdot x-d\Vert }_2^2,\text{ where }x\ge 0.$$

For compatibility, the number of rows of C must equal the length of d.

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 variable d in the problem

$$\underset{x}{\min }{\Vert C\cdot x-d\Vert }_2^2,\text{ where }x\ge 0.$$

For compatibility, the length of d must equal the number of rows of C.

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

Data Types: double

options — Optimization options

structure such as optimset returns

Optimization options, specified as a structure such as optimset returns. You can use optimset to set or change the values of these fields in the options structure. See Set Optimization Options for 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 on x, a positive scalar. The default is 10*eps*norm(C,1)*length(C). See Tolerances 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 a problem structure is to export the problem from the Optimization app.

Data Types: struct

Output Arguments

x — Solution

real vector

Solution, returned as a real vector. The length of x is the same as the length of d.

resnorm — Squared residual norm

nonnegative scalar

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

residual — Residual

real vector

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

exitflag — Reason lsqnonneg stopped

integer

Reason lsqnonneg stopped, 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 condition x'*lambda = 0. This means lambda(i) < 0 when x(i) is approximately 0, and lambda(i) is approximately 0 when x(i) > 0.

Algorithms

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

Alternative Functionality

App

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

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.

• Code generation does not support sparse matrix inputs for this function.