lsqnonlin

Solve nonlinear least-squares (nonlinear data-fitting) problems

Equation

Solves nonlinear least-squares curve fitting problems of the form

Syntax

x = lsqnonlin(fun,x0)
x = lsqnonlin(fun,x0,lb,ub)
x = lsqnonlin(fun,x0,lb,ub,options)
x = lsqnonlin(problem)
[x,resnorm] = lsqnonlin(...)
[x,resnorm,residual] = lsqnonlin(...)
[x,resnorm,residual,exitflag] = lsqnonlin(...)
[x,resnorm,residual,exitflag,output] = lsqnonlin(...)
[x,resnorm,residual,exitflag,output,lambda] = lsqnonlin(...)
[x,resnorm,residual,exitflag,output,lambda,jacobian] = lsqnonlin(...)

Description

lsqnonlin solves nonlinear least-squares problems, including nonlinear data-fitting problems.

Rather than compute the value (the sum of squares), lsqnonlin requires the user-defined function to compute the vector-valued function

Then, in vector terms, you can restate this optimization problem as

where x is a vector and f(x) is a function that returns a vector value.

x = lsqnonlin(fun,x0) starts at the point x0 and finds a minimum of the sum of squares of the functions described in fun. fun should return a vector of values and not the sum of squares of the values. (The algorithm implicitly sums and squares fun(x).)

x = lsqnonlin(fun,x0,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.

x = lsqnonlin(fun,x0,lb,ub,options) minimizes with the optimization options specified in the structure options. Use optimset to set these options. Pass empty matrices for lb and ub if no bounds exist.

x = lsqnonlin(problem) finds the minimum for problem, where problem is a structure described in Input Arguments.

Create the structure problem by exporting a problem from Optimization Tool, as described in Exporting to the MATLAB® Workspace.

[x,resnorm] = lsqnonlin(...) returns the value of the squared 2-norm of the residual at x: sum(fun(x).^2).

[x,resnorm,residual] = lsqnonlin(...) returns the value of the residual fun(x) at the solution x.

[x,resnorm,residual,exitflag] = lsqnonlin(...) returns a value exitflag that describes the exit condition.

[x,resnorm,residual,exitflag,output] = lsqnonlin(...) returns a structure output that contains information about the optimization.

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

[x,resnorm,residual,exitflag,output,lambda,jacobian] = lsqnonlin(...) returns the Jacobian of fun at the solution x.

Input Arguments

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

fun

The function whose sum of squares is minimized. fun is a function that accepts a vector x and returns a vector F, the objective functions evaluated at x. The function fun can be specified as a function handle for an M-file function

x = lsqnonlin(@myfun,x0)

where myfun is a MATLAB® function such as

function F = myfun(x)
F = ...            % Compute function values at x

fun can also be a function handle for an anonymous function.

x = lsqnonlin(@(x)sin(x.*x),x0);

If the user-defined values for x and F are matrices, they are converted to a vector using linear indexing.

If the Jacobian can also be computed and the Jacobian option is 'on', set by

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

then the function fun must return, in a second output argument, the Jacobian value J, a matrix, at x. Note that by checking the value of nargout the function can avoid computing J when fun is called with only one output argument (in the case where the optimization algorithm only needs the value of F but not J).

function [F,J] = myfun(x)
F = ...          % Objective function values at x
if nargout > 1   % Two output arguments
   J = ...   % Jacobian of the function evaluated at x
end

If fun returns a vector (matrix) of m components and x has length n, where n is the length of x0, then the Jacobian J is an m-by-n matrix where J(i,j) is the partial derivative of F(i) with respect to x(j). (Note that the Jacobian J is the transpose of the gradient of F.)

options

Options provides the function-specific details for the options values.

problem

objective

Objective function

x0

Initial point for x
lbVector of lower bounds
ubVector of upper bounds

solver

'lsqnonlin'

options

Options structure created with optimset

Output Arguments

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

exitflag

Integer identifying the reason the algorithm terminated. The following lists the values of exitflag and the corresponding reasons the algorithm terminated:

 

1

Function converged to a solution x.

 

2

Change in x was less than the specified tolerance.

 

3

Change in the residual was less than the specified tolerance.

 

4

Magnitude of search direction was smaller than the specified tolerance.

 

0

Number of iterations exceeded options.MaxIter or number of function evaluations exceeded options.FunEvals.

 

-1

Algorithm was terminated by the output function.

 

-2

Problem is infeasible: the bounds lb and ub are inconsistent.

 

-4

Line search could not sufficiently decrease the residual along the current search direction.

lambda

Structure containing the Lagrange multipliers at the solution x (separated by constraint type). The fields are

 

lower

Lower bounds lb

 

upper

Upper bounds ub

output

Structure containing information about the optimization. The fields of the structure are

 firstorderopt

Measure of first-order optimality (large-scale algorithm, [ ] for others)

 iterations

Number of iterations taken

 funcCount

The number of function evaluations

 cgiterations

Total number of PCG iterations (large-scale algorithm, [ ] for others))

 stepsize

Final displacement in x (medium-scale algorithm only)

 algorithm

Optimization algorithm used

 message

Exit message

Options

Optimization options. You can set or change the values of these options using the optimset function. 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. 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 or medium-scale algorithm. For the large-scale algorithm, the nonlinear system of equations cannot be underdetermined; that is, the number of equations (the number of elements of F returned by fun) must be at least as many as the length of x. Furthermore, only the large-scale algorithm handles bound constraints:

LargeScale

Use large-scale algorithm if possible when set to 'on'. Use medium-scale algorithm when set to 'off'.

The large-scale algorithm is a more modern algorithm than the medium-scale algorithms. The large-scale algorithm handles both large-scale and medium-scale problems effectively.

Medium-Scale and Large-Scale Algorithms

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

DerivativeCheck

Compare user-supplied derivatives (Jacobian) to finite-differencing derivatives.

Diagnostics

Display diagnostic information about the function to be minimized.

DiffMaxChange

Maximum change in variables for finite differencing.

DiffMinChange

Minimum change in variables for finite differencing.

Display

Level of display. 'off' displays no output; 'iter' displays output at each iteration; 'final' (default) displays just the final output.

Jacobian

If 'on', lsqnonlin uses a user-defined Jacobian (defined in fun), or Jacobian information (when using JacobMult), for the objective function. If 'off', lsqnonlin approximates the Jacobian using finite differences.

MaxFunEvals

Maximum number of function evaluations allowed.

MaxIter

Maximum number of iterations allowed.

OutputFcn

Specify one or more user-defined functions that an optimization function calls at each iteration. See Output Function.

PlotFcns

Plots various measures of progress while the algorithm executes, select from predefined plots or write your own. Specifying @optimplotx plots the current point; @optimplotfunccount plots the function count; @optimplotfval plots the function value; @optimplotresnorm plots the norm of the residuals; @optimplotstepsize plots the step size; @optimplotfirstorderopt plots the first-order of optimality.

TolFun

Termination tolerance on the function value.

TolX

Termination tolerance on x.

TypicalX

Typical x values.

Large-Scale Algorithm Only

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

JacobMult

Function handle for Jacobian multiply function. For large-scale structured problems, this function computes the Jacobian matrix product J*Y, J'*Y, or J'*(J*Y) without actually forming J. The function is of the form

W = jmfun(Jinfo,Y,flag,p1,p2,...)
 

where Jinfo and the additional parameters p1,p2,... contain the matrices used to compute J*Y (or J'*Y, or J'*(J*Y)). The first argument Jinfo must be the same as the second argument returned by the objective function fun, for example by

[F,Jinfo] = fun(x)

Y is a matrix that has the same number of rows as there are dimensions in the problem. flag determines which product to compute:

  • If flag == 0 then W = J'*(J*Y).

  • If flag > 0 then W = J*Y.

  • If flag < 0 then W = J'*Y.

    In each case, J is not formed explicitly. lsqnonlin uses Jinfo to compute the preconditioner. The optional parameters p1, p2, ... can be any additional parameters needed by jmfun. See Passing Extra Parameters for information on how to supply values for these parameters.

 

    Note   'Jacobian' must be set to 'on' for Jinfo to be passed from fun to jmfun.

See Nonlinear Minimization with a Dense but Structured Hessian and Equality Constraints for a similar example.

JacobPattern

Sparsity pattern of the Jacobian for finite differencing. If it is not convenient to compute the Jacobian matrix J in fun, lsqnonlin can approximate J via sparse finite differences, provided the structure of J, i.e., locations of the nonzeros, is supplied as the value for JacobPattern. In the worst case, if the structure is unknown, you can set JacobPattern to be a dense matrix and a full finite-difference approximation is computed in each iteration (this is the default if JacobPattern is not set). This can be very expensive for large problems, so it is usually worth the effort to determine the sparsity structure.

MaxPCGIter

Maximum number of PCG (preconditioned conjugate gradient) iterations (see Algorithm).

PrecondBandWidth

The default PrecondBandWidth is 'Inf', which means a direct factorization (Cholesky) is used rather than the conjugate gradients (CG). The direct factorization is computationally more expensive than CG, but produces a better quality step towards the solution. Set PrecondBandWidth to 0 for diagonal preconditioning (upper bandwidth of 0). For some problems, an intermediate bandwidth reduces the number of PCG iterations.

TolPCG

Termination tolerance on the PCG iteration.

Medium-Scale Algorithm Only

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

LevenbergMarquardt

Choose Levenberg-Marquardt over Gauss-Newton algorithm.

LineSearchType

Line search algorithm choice.

Examples

Find x that minimizes

starting at the point x = [0.3, 0.4].

Because lsqnonlin assumes that the sum of squares is not explicitly formed in the user-defined function, the function passed to lsqnonlin should instead compute the vector-valued function

for k = 1 to 10 (that is, F should have k components).

First, write an M-file to compute the k-component vector F.

function F = myfun(x)
k = 1:10;
F = 2 + 2*k-exp(k*x(1))-exp(k*x(2));

Next, invoke an optimization routine.

x0 = [0.3 0.4]                        % Starting guess
[x,resnorm] = lsqnonlin(@myfun,x0)    % Invoke optimizer

After about 24 function evaluations, this example gives the solution

x = 
     0.2578   0.2578
resnorm       % Residual or sum of squares
resnorm = 
     124.3622

Algorithm

Large-Scale Optimization

By default lsqnonlin chooses the large-scale algorithm. This 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 Trust-Region Methods for Nonlinear Minimization and Preconditioned Conjugate Gradients.

Medium-Scale Optimization

If you set the LargeScale option to 'off' with optimset, lsqnonlin uses the Levenberg-Marquardt method with line search [4], [5], and [6]. Alternatively, you can select a Gauss-Newton method [6] with line search by setting the LevenbergMarquardt option to 'off' (and LargeScale to 'off') with optimset. The Gauss-Newton method is generally faster when the residual is small.

The default line search algorithm, i.e., the LineSearchType option, is 'quadcubic'. This is a safeguarded mixed quadratic and cubic polynomial interpolation and extrapolation method. You can select a safeguarded cubic polynomial method by setting the LineSearchType option to 'cubicpoly'. This method generally requires fewer function evaluations but more gradient evaluations. Thus, if gradients are being supplied and can be calculated inexpensively, the cubic polynomial line search method is preferable. The algorithms used are described fully in Standard Algorithms.

Diagnostics

Large-Scale Optimization

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

Equal upper and lower bounds not permitted.

(lsqnonlin does not handle equality constraints, which is another way to formulate equal bounds. If equality constraints are present, use fmincon, fminimax, or fgoalattain for alternative formulations where equality constraints can be included.)

Limitations

The function to be minimized must be continuous. lsqnonlin might only give local solutions.

lsqnonlin only handles real variables. When x has complex variables, the variables must be split into real and imaginary parts.

Large-Scale Optimization

The large-scale method for lsqnonlin does not solve underdetermined systems; it requires that the number of equations (i.e., the number of elements of F) be at least as great as the number of variables. In the underdetermined case, the medium-scale algorithm is used instead. (If bound constraints exist, a warning is issued and the problem is solved with the bounds ignored.) See Large-Scale Problem Coverage and Requirements for more information on what problem formulations are covered and what information must be provided.

The preconditioner computation used in the preconditioned conjugate gradient part of the large-scale method forms JTJ (where J is the Jacobian matrix) before computing the preconditioner; therefore, a row of J with many nonzeros, which results in a nearly dense product JTJ, can lead to a costly solution process for large problems.

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

Medium-Scale Optimization

The medium-scale algorithm does not handle bound constraints.

Because the large-scale algorithm does not handle underdetermined systems and the medium-scale algorithm does not handle bound constraints, problems with both these characteristics cannot be solved by lsqnonlin.

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] Dennis, J.E., Jr., "Nonlinear Least-Squares," State of the Art in Numerical Analysis, ed. D. Jacobs, Academic Press, pp. 269–312, 1977.

[4] Levenberg, K., "A Method for the Solution of Certain Problems in Least-Squares," Quarterly Applied Math. 2, pp. 164–168, 1944.

[5] Marquardt, D., "An Algorithm for Least-Squares Estimation of Nonlinear Parameters," SIAM Journal Applied Math., Vol. 11, pp. 431–441, 1963.

[6] Moré, J.J., "The Levenberg-Marquardt Algorithm: Implementation and Theory," Numerical Analysis, ed. G. A. Watson, Lecture Notes in Mathematics 630, Springer Verlag, pp. 105–116, 1977.

See Also

@ (function_handle), lsqcurvefit, lsqlin, optimset, optimtool

  


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