lsqnonlin (Optimization Toolbox)

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

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

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 parameters specified in the structure options. Use optimset to set these parameters. Pass empty matrices for lb and ub if no bounds exist.

x = lsqnonlin(fun,x0,lb,ub,options,P1,P2,...)

passes the problem-dependent parameters P1, P2, etc. directly to the function fun. Pass an empty matrix for options to use the default values for options.

[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.

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


`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. 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 an inline object. x = lsqnonlin(inline('sin(x.x)'),x0); If the Jacobian can also be computed and* the `Jacobian` parameter 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` parameters.

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


`exitflag`	Describes the exit condition:
	`> 0`	The function converged to a solution `x`.
	`0`	The maximum number of function evaluations or iterations was exceeded.
	`< 0`	The function did not converge to a solution.
`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 are
	`iterations`	Number of iterations taken
	`funcCount`	The number of function evaluations
	`algorithm`	Algorithm used
	`cgiterations`	Number of PCG iterations (large-scale algorithm only)
	`stepsize`	The final step size taken (medium-scale algorithm only)
	`firstorderopt`	Measure of first-order optimality (large-scale algorithm only) For large-scale bound constrained problems, the first-order optimality is the infinity norm of `v.*g`, where `v` is defined as in Box Constraints, and `g` is the gradient g = J^TF (see Nonlinear Least-Squares).

Optimization parameter options. You can set or change the values of these parameters using the optimset function. Some parameters 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 Parameters 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'`.

Medium-Scale and Large-Scale Algorithms. These parameters 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.
`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 a user-defined function that an opimization function calls at each iteration. See Output Function.
`TolFun`	Termination tolerance on the function value.
`TolX`	Termination tolerance on `x`.

Large-Scale Algorithm Only. These parameters 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 products 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.

```
[F,Jinfo] = fun(x,p1,p2,...)
```

The parameters p1,p2,... are the same additional parameters that are passed to lsqnonlin (and to fun).

```
lsqnonlin(fun,...,options,p1,p2,...)
```

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.

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 Upper bandwidth of preconditioner for PCG. By default, diagonal preconditioning is used (upper bandwidth of 0). For some problems, increasing the bandwidth reduces the number of PCG iterations.

TolPCG Termination tolerance on the PCG iteration.

TypicalX Typical x values.

Medium-Scale Algorithm Only. These parameters are used only by the medium-scale algorithm:


`DiffMaxChange`	Maximum change in variables for finite differencing.
`DiffMinChange`	Minimum change in variables for finite differencing.
`LevenbergMarquardt`	Choose Levenberg-Marquardt over Gauss-Newton algorithm.
`LineSearchType`	Line search algorithm choice.

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

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], [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 parameter set to 'off' with optimset, lsqnonlin uses the Levenberg-Marquardt method with line search [4], [5], [6]. Alternatively, you can select a Gauss-Newton method [3] with line search by setting the LevenbergMarquardt parameter. Setting LevenbergMarquardt to 'off' (and LargeScale to 'off') selects the Gauss-Newton method, which is generally faster when the residual

is small.

The default line search algorithm, i.e., the LineSearchType parameter set to 'quadcubic', is a safeguarded mixed quadratic and cubic polynomial interpolation and extrapolation method. You can select a safeguarded cubic polynomial method by setting the LineSearchType parameter 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 the Standard Algorithms chapter.

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

(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.)

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 Table 2-4, 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.

[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.

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