Optimization Toolbox™

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.

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:

x = lsqnonlin(@myfun,x0)

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

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

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

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

Output Arguments

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

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

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:

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:

Large-Scale Algorithm Only

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

W = jmfun(Jinfo,Y,flag,p1,p2,...)

[F,Jinfo] = fun(x)

Medium-Scale Algorithm Only

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

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 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, 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

lsqlin

lsqnonneg