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).)

Note Passing Extra Parameters explains how to pass extra parameters to the vector function f, if necessary.

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.

Note If the specified input bounds for a problem are inconsistent, the output x is x0 and the outputs resnorm and residual are [].

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 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.

Note The sum of squares should not be formed explicitly. Instead, your function should return a vector of function values. See Examples.

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

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

the function fun must return, in a second output argument, the Jacobian value J, a matrix, at x. 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, 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). (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

lb Vector of lower bounds

ub Vector 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`	Output function terminated the algorithm.
	`-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 (trust-region-reflective algorithm, [ ] for others)
	`iterations`	Number of iterations taken
	`funcCount`	The number of function evaluations
	`cgiterations`	Total number of PCG iterations (trust-region-reflective algorithm, [ ] for others)
	`stepsize`	Final displacement in `x` (Levenberg-Marquardt algorithm)
	`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 trust-region-reflective algorithm, and others are only relevant when you are using the Levenberg-Marquardt algorithm. See Optimization Options for detailed information.

Algorithm Options

Both algorithms use the following options:

`Algorithm`	Choose between `'trust-region-reflective'` (default) and `'levenberg-marquardt'`. Set the initial Levenberg-Marquardt parameter λ by setting `Algorithm` to a cell array such as `{'levenberg-marquardt',.005}`. The default λ = `0.01`. The `Algorithm` option specifies a preference for which algorithm to use. It is only a preference, because certain conditions must be met to use each algorithm. For the trust-region-reflective 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`. The Levenberg-Marquardt algorithm does not handle bound constraints.
`DerivativeCheck`	Compare user-supplied derivatives (gradients of objective or constraints) to finite-differencing derivatives. The choices are `'on'` or the default `'off'`.
`Diagnostics`	Display diagnostic information about the function to be minimized or solved. The choices are `'on'` or the default `'off'`.
`DiffMaxChange`	Maximum change in variables for finite-difference gradients (a positive scalar). The default is `0.1`.
`DiffMinChange`	Minimum change in variables for finite-difference gradients (a positive scalar). The default is `1e-8`.
`Display`	Level of display. `'off'` displays no output, and `'final'` (default) displays just the final output.
`FunValCheck`	Check whether function values are valid. `'on'` displays an error when the function returns a value that is `complex`, `Inf`, or `NaN`. The default `'off'` displays no error.
`Jacobian`	If `'on'`, `lsqnonlin` uses a user-defined Jacobian (defined in `fun`), or Jacobian information (when using `JacobMult`), for the objective function. If `'off'` (default), `lsqnonlin` approximates the Jacobian using finite differences.
`MaxFunEvals`	Maximum number of function evaluations allowed, a positive integer. The default is `100*numberOfVariables`.
`MaxIter`	Maximum number of iterations allowed, a positive integer. The default is `400`.
`OutputFcn`	Specify one or more user-defined functions that an optimization function calls at each iteration, either as a function handle or as a cell array of function handles. The default is none (`[]`). See Output Function.
`PlotFcns`	Plots various measures of progress while the algorithm executes, select from predefined plots or write your own. Pass a function handle or a cell array of function handles. The default is none (`[]`): `@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 optimality measure.
`TolFun`	Termination tolerance on the function value, a positive scalar. The default is `1e-6`.
`TolX`	Termination tolerance on `x`, a positive scalar. The default is `1e-6`.
`TypicalX`	Typical `x` values. The number of elements in `TypicalX` is equal to the number of elements in `x0`, the starting point. The default value is `ones(numberofvariables,1)`. `lsqnonlin` uses `TypicalX` for scaling finite differences for gradient estimation.

Trust-Region-Reflective Algorithm Only

The trust-region-reflective algorithm uses the following options:

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)

where Jinfo contains the matrix 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. See Passing Extra Parameters for information on how to supply values for any additional parameters jmfun needs.

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

See Example: Nonlinear Minimization with a Dense but Structured Hessian and Equality Constraints and Example: Jacobian Multiply Function with Linear Least Squares for similar examples.

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 you supply the structure of J (locations of the nonzeros) 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, a positive scalar. The default is max(1, numberOfVariables/2)). For more information, see Algorithm.

PrecondBandWidth

Upper bandwidth of preconditioner for PCG, a nonnegative integer. 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, a positive scalar. The default is 0.1.

Levenberg-Marquardt Algorithm Only

The Levenberg-Marquardt algorithm uses the following option:

ScaleProblem

'Jacobian' can sometimes improve the convergence of a poorly scaled problem; the default is 'none'.

Gauss-Newton Algorithm Only

The Gauss-Newton algorithm uses the following option:

`LargeScale` and `LevenbergMarquardt`	Specify `LargeScale` as `'off'` and `LevenbergMarquardt` as `'off'` to choose the Gauss-Newton algorithm. These options are being obsoleted, and the other algorithms do not use them.
`LineSearchType`	The choices are `'cubicpoly'` or the default, `'quadcubic'`.

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

Trust-Region-Reflective Optimization

By default, lsqnonlin chooses the trust-region-reflective 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 Large-Scale Least Squares, and in particular Large Scale Nonlinear Least Squares.

Levenberg-Marquardt Optimization

If you set the Algorithm option to 'levenberg-marquardt' using optimset, lsqnonlin uses the Levenberg-Marquardt method [4], [5], and [6]. See Levenberg-Marquardt Method.

Gauss-Newton

The Gauss-Newton method [6] is going to be removed in a future version of MATLAB. Currently, you get a warning when using it. It is not a large-scale method.

Select the Gauss-Newton method [3] with line search by setting the options LevenbergMarquardt to 'off' and LargeScale to 'off' with optimset. The Gauss-Newton method can be faster than the Levenberg-Marquardt method when the residual is small. See Gauss-Newton Method.

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.

Diagnostics

Trust-Region-Reflective Optimization

The trust-region-reflective 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 (the user-defined function must only return real values). When x has complex variables, the variables must be split into real and imaginary parts.

Trust-Region-Reflective Optimization

The trust-region-reflective algorithm for lsqnonlin does not solve underdetermined systems; it requires that the number of equations, i.e., the row dimension of F, be at least as great as the number of variables. In the underdetermined case, the Levenberg-Marquardt algorithm is used instead.

The preconditioner computation used in the preconditioned conjugate gradient part of the trust-region-reflective method forms J^TJ (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 J^TJ, can lead to a costly solution process for large problems.

If components of x have no upper (or lower) bounds, 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.

Trust-Region-Reflective Problem Coverage and Requirements

For Large Problems

For Large Problems
Provide sparsity structure of the Jacobian or compute the Jacobian in `fun`. The Jacobian should be sparse.

Provide sparsity structure of the Jacobian or compute the Jacobian in fun.
The Jacobian should be sparse.

Levenberg-Marquardt Optimization

The Levenberg-Marquardt algorithm does not handle bound constraints.

Since the trust-region-reflective algorithm does not handle underdetermined systems and the Levenberg-Marquardt 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.

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

Equation

Syntax

Description

Input Arguments

Output Arguments

Options

Algorithm Options

Trust-Region-Reflective Algorithm Only

Levenberg-Marquardt Algorithm Only

Gauss-Newton Algorithm Only

Examples

Algorithm

Trust-Region-Reflective Optimization

Levenberg-Marquardt Optimization

Gauss-Newton

Diagnostics

Trust-Region-Reflective Optimization

Limitations

Trust-Region-Reflective Optimization

Levenberg-Marquardt Optimization

References

See Also

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