# lsqnonlin

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

## Syntax

``x = lsqnonlin(fun,x0)``
``x = lsqnonlin(fun,x0,lb,ub)``
``x = lsqnonlin(fun,x0,lb,ub,A,b,Aeq,beq)``
``x = lsqnonlin(fun,x0,lb,ub,A,b,Aeq,beq,nonlcon)``
``x = lsqnonlin(fun,x0,lb,ub,options)``
``x = lsqnonlin(fun,x0,lb,ub,A,b,Aeq,beq,nonlcon,options)``
``x = lsqnonlin(problem)``
``````[x,resnorm] = lsqnonlin(___)``````
``````[x,resnorm,residual,exitflag,output] = lsqnonlin(___)``````
``````[x,resnorm,residual,exitflag,output,lambda,jacobian] = lsqnonlin(___)``````

## Description

Nonlinear least-squares solver

Solves nonlinear least-squares curve fitting problems of the form

`$\underset{x}{\mathrm{min}}{‖f\left(x\right)‖}_{2}^{2}=\underset{x}{\mathrm{min}}\left({f}_{1}{\left(x\right)}^{2}+{f}_{2}{\left(x\right)}^{2}+...+{f}_{n}{\left(x\right)}^{2}\right)$`

subject to the constraints

`$\begin{array}{c}\text{lb}\le x\\ x\le \text{ub}\\ Ax\le b\\ \text{Aeq}x=\text{beq}\\ c\left(x\right)\le 0\\ \text{ceq}\left(x\right)=0.\end{array}$`

x, `lb`, and `ub` can be vectors or matrices; see Matrix Arguments.

Do not specify the objective function as the scalar value ${‖f\left(x\right)‖}_{2}^{2}$ (the sum of squares). `lsqnonlin` requires the objective function to be the vector-valued function

`$f\left(x\right)=\left[\begin{array}{c}{f}_{1}\left(x\right)\\ {f}_{2}\left(x\right)\\ ⋮\\ {f}_{n}\left(x\right)\end{array}\right].$`

example

````x = lsqnonlin(fun,x0)` starts at the point `x0` and finds a minimum of the sum of squares of the functions described in `fun`. The function `fun` should return a vector (or array) of values and not the sum of squares of the values. (The algorithm implicitly computes the sum of squares of the components of `fun(x)`.) NotePassing Extra Parameters explains how to pass extra parameters to the vector function `fun(x)`, if necessary. ```

example

````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`. You can fix the solution component `x(i)` by specifying `lb(i) = ub(i)`. NoteIf 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. ```

example

````x = lsqnonlin(fun,x0,lb,ub,A,b,Aeq,beq)` constrains the solution to satisfy the linear constraintsA x ≤ bAeq x = beq.```

example

````x = lsqnonlin(fun,x0,lb,ub,A,b,Aeq,beq,nonlcon)` constrain the solution to satisfy the nonlinear constraints in the `nonlcon`(`x`) function. `nonlcon` returns two outputs, `c` and `ceq`. The solver attempts to satisfy the constraintsc ≤ 0`ceq` = 0.```

example

````x = lsqnonlin(fun,x0,lb,ub,options)` and `x = lsqnonlin(fun,x0,lb,ub,A,b,Aeq,beq,nonlcon,options)` minimizes with the optimization options specified in `options`. Use `optimoptions` to set these options. Pass empty matrices for `lb` and `ub` and for other input arguments if the arguments do not exist.```
````x = lsqnonlin(problem)` finds the minimum for `problem`, a structure described in `problem`.```

example

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

example

``````[x,resnorm,residual,exitflag,output] = lsqnonlin(___)``` additionally returns the value of the residual `fun(x)` at the solution `x`, a value `exitflag` that describes the exit condition, and a structure `output` that contains information about the optimization process.```
``````[x,resnorm,residual,exitflag,output,lambda,jacobian] = lsqnonlin(___)``` additionally returns a structure `lambda` whose fields contain the Lagrange multipliers at the solution `x`, and the Jacobian of `fun` at the solution `x`.```

## Examples

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Fit a simple exponential decay curve to data.

Generate data from an exponential decay model plus noise. The model is

`$y=\mathrm{exp}\left(-1.3t\right)+\epsilon ,$`

with $t$ ranging from 0 through 3, and $\epsilon$ normally distributed noise with mean 0 and standard deviation 0.05.

```rng default % for reproducibility d = linspace(0,3); y = exp(-1.3*d) + 0.05*randn(size(d));```

The problem is: given the data (`d`, `y`), find the exponential decay rate that best fits the data.

Create an anonymous function that takes a value of the exponential decay rate $r$ and returns a vector of differences from the model with that decay rate and the data.

`fun = @(r)exp(-d*r)-y;`

Find the value of the optimal decay rate. Arbitrarily choose an initial guess `x0` = 4.

```x0 = 4; x = lsqnonlin(fun,x0)```
```Local minimum possible. lsqnonlin stopped because the final change in the sum of squares relative to its initial value is less than the value of the function tolerance. ```
```x = 1.2645 ```

Plot the data and the best-fitting exponential curve.

```plot(d,y,'ko',d,exp(-x*d),'b-') legend('Data','Best fit') xlabel('t') ylabel('exp(-tx)')```

Find the best-fitting model when some of the fitting parameters have bounds.

Find a centering $b$ and scaling $a$ that best fit the function

`$a\mathrm{exp}\left(-t\right)\mathrm{exp}\left(-\mathrm{exp}\left(-\left(t-b\right)\right)\right)$`

to the standard normal density,

`$\frac{1}{\sqrt{2\pi }}\mathrm{exp}\left(-{t}^{2}/2\right).$`

Create a vector `t` of data points, and the corresponding normal density at those points.

```t = linspace(-4,4); y = 1/sqrt(2*pi)*exp(-t.^2/2);```

Create a function that evaluates the difference between the centered and scaled function from the normal `y`, with `x(1)` as the scaling $a$ and `x(2)` as the centering $b$.

`fun = @(x)x(1)*exp(-t).*exp(-exp(-(t-x(2)))) - y;`

Find the optimal fit starting from `x0` = `[1/2,0]`, with the scaling $a$ between 1/2 and 3/2, and the centering $b$ between -1 and 3.

```lb = [1/2,-1]; ub = [3/2,3]; x0 = [1/2,0]; x = lsqnonlin(fun,x0,lb,ub)```
```Local minimum possible. lsqnonlin stopped because the final change in the sum of squares relative to its initial value is less than the value of the function tolerance. ```
```x = 1×2 0.8231 -0.2444 ```

Plot the two functions to see the quality of the fit.

```plot(t,y,'r-',t,fun(x)+y,'b-') xlabel('t') legend('Normal density','Fitted function')```

Consider the following objective function, a sum of squares:

`$\sum _{k=1}^{10}{\left(2+2k+\mathrm{exp}\left(k\phantom{\rule{0.16666666666666666em}{0ex}}{x}_{1}\right)+2\mathrm{exp}\left(2k\phantom{\rule{0.16666666666666666em}{0ex}}{x}_{2}^{2}\right)\right)}^{2}.$`

The code for this objective function appears as the `myfun` function at the end of this example.

Minimize this function subject to the linear constraint ${x}_{1}\le \frac{{x}_{2}}{2}$. Write this constraint as ${x}_{1}-\frac{{x}_{2}}{2}\le 0$.

```A = [1 -1/2]; b = 0;```

Impose the bounds ${x}_{1}\ge 0$, ${x}_{2}\ge 0$, ${x}_{1}\le 2$, and ${x}_{2}\le 4$.

```lb = [0 0]; ub = [2 4];```

Start the optimization process from the point `x0 = [0.3 0.4]`.

`x0 = [0.3 0.4];`

The problem has no linear equality constraints.

```Aeq = []; beq = [];```

Run the optimization.

`x = lsqnonlin(@myfun,x0,lb,ub,A,b,Aeq,beq)`
```Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance. ```
```x = 1×2 0.1695 0.3389 ```
```function F = myfun(x) k = 1:10; F = 2 + 2*k - exp(k*x(1)) - 2*exp(2*k*(x(2)^2)); end```

Consider the following objective function, a sum of squares:

`$\sum _{k=1}^{10}{\left(2+2k+\mathrm{exp}\left(k\phantom{\rule{0.16666666666666666em}{0ex}}{x}_{1}\right)+2\mathrm{exp}\left(2k\phantom{\rule{0.16666666666666666em}{0ex}}{x}_{2}^{2}\right)\right)}^{2}.$`

The code for this objective function appears as the `myfun` function at the end of this example.

Minimize this function subject to the nonlinear constraint $\mathrm{sin}\left({x}_{1}\right)\le \mathrm{cos}\left({x}_{2}\right)$. The code for this nonlinear constraint function appears as the `nlcon` function at the end of this example.

Impose the bounds ${x}_{1}\ge 0$, ${x}_{2}\ge 0$, ${x}_{1}\le 2$, and ${x}_{2}\le 4$.

```lb = [0 0]; ub = [2 4];```

Start the optimization process from the point `x0 = [0.3 0.4]`.

`x0 = [0.3 0.4];`

The problem has no linear constraints.

```A = []; b = []; Aeq = []; beq = [];```

Run the optimization.

`x = lsqnonlin(@myfun,x0,lb,ub,A,b,Aeq,beq,@nlcon)`
```Local minimum possible. Constraints satisfied. fmincon stopped because the size of the current step is less than the value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance. ```
```x = 1×2 0.2133 0.3266 ```
```function F = myfun(x) k = 1:10; F = 2 + 2*k - exp(k*x(1)) - 2*exp(2*k*(x(2)^2)); end function [c,ceq] = nlcon(x) ceq = []; c = sin(x(1)) - cos(x(2)); end```

Compare the results of a data-fitting problem when using different `lsqnonlin` algorithms.

Suppose that you have observation time data `xdata` and observed response data `ydata`, and you want to find parameters $x\left(1\right)$ and $x\left(2\right)$ to fit a model of the form

`$\text{ydata}=x\left(1\right)\mathrm{exp}\left(x\left(2\right)\text{xdata}\right).$`

Input the observation times and responses.

```xdata = ... [0.9 1.5 13.8 19.8 24.1 28.2 35.2 60.3 74.6 81.3]; ydata = ... [455.2 428.6 124.1 67.3 43.2 28.1 13.1 -0.4 -1.3 -1.5];```

Create a simple exponential decay model. The model computes a vector of differences between predicted values and observed values.

`fun = @(x)x(1)*exp(x(2)*xdata)-ydata;`

Fit the model using the starting point `x0 = [100,-1]`. First, use the default `'trust-region-reflective'` algorithm.

```x0 = [100,-1]; options = optimoptions(@lsqnonlin,'Algorithm','trust-region-reflective'); x = lsqnonlin(fun,x0,[],[],options)```
```Local minimum possible. lsqnonlin stopped because the final change in the sum of squares relative to its initial value is less than the value of the function tolerance. ```
```x = 1×2 498.8309 -0.1013 ```

See if there is any difference using the `'levenberg-marquardt'` algorithm.

```options.Algorithm = 'levenberg-marquardt'; x = lsqnonlin(fun,x0,[],[],options)```
```Local minimum possible. lsqnonlin stopped because the relative size of the current step is less than the value of the step size tolerance. ```
```x = 1×2 498.8309 -0.1013 ```

The two algorithms found the same solution. Plot the solution and the data.

```plot(xdata,ydata,'ko') hold on tlist = linspace(xdata(1),xdata(end)); plot(tlist,x(1)*exp(x(2)*tlist),'b-') xlabel xdata ylabel ydata title('Exponential Fit to Data') legend('Data','Exponential Fit') hold off```

Find the $x$ that minimizes

$\sum _{k=1}^{10}{\left(2+2k-{e}^{k{x}_{1}}-{e}^{k{x}_{2}}\right)}^{2}$,

and find the value of the minimal sum of squares.

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

${F}_{k}\left(x\right)=2+2k-{e}^{k{x}_{1}}-{e}^{k{x}_{2}}$,

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

The `myfun` function, which computes the 10-component vector F, appears at the end of this example.

Find the minimizing point and the minimum value, starting at the point `x0 = [0.3,0.4]`.

```x0 = [0.3,0.4]; [x,resnorm] = lsqnonlin(@myfun,x0)```
```Local minimum possible. lsqnonlin stopped because the size of the current step is less than the value of the step size tolerance. ```
```x = 1×2 0.2578 0.2578 ```
```resnorm = 124.3622 ```

The `resnorm` output is the squared residual norm, or the sum of squares of the function values.

The following function computes the vector-valued objective function.

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

Examine the solution process both as it occurs (by setting the `Display` option to `'iter'`) and afterward (by examining the `output` structure).

Suppose that you have observation time data `xdata` and observed response data `ydata`, and you want to find parameters $x\left(1\right)$ and $x\left(2\right)$ to fit a model of the form

`$\text{ydata}=x\left(1\right)\mathrm{exp}\left(x\left(2\right)\text{xdata}\right).$`

Input the observation times and responses.

```xdata = ... [0.9 1.5 13.8 19.8 24.1 28.2 35.2 60.3 74.6 81.3]; ydata = ... [455.2 428.6 124.1 67.3 43.2 28.1 13.1 -0.4 -1.3 -1.5];```

Create a simple exponential decay model. The model computes a vector of differences between predicted values and observed values.

`fun = @(x)x(1)*exp(x(2)*xdata)-ydata;`

Fit the model using the starting point `x0 = [100,-1]`. Examine the solution process by setting the `Display` option to `'iter'`. Obtain an `output` structure to obtain more information about the solution process.

```x0 = [100,-1]; options = optimoptions('lsqnonlin','Display','iter'); [x,resnorm,residual,exitflag,output] = lsqnonlin(fun,x0,[],[],options);```
``` Norm of First-order Iteration Func-count Resnorm step optimality 0 3 359677 2.88e+04 Objective function returned Inf; trying a new point... 1 6 359677 11.6976 2.88e+04 2 9 321395 0.5 4.97e+04 3 12 321395 1 4.97e+04 4 15 292253 0.25 7.06e+04 5 18 292253 0.5 7.06e+04 6 21 270350 0.125 1.15e+05 7 24 270350 0.25 1.15e+05 8 27 252777 0.0625 1.63e+05 9 30 252777 0.125 1.63e+05 10 33 243877 0.03125 7.48e+04 11 36 243660 0.0625 8.7e+04 12 39 243276 0.0625 2e+04 13 42 243174 0.0625 1.14e+04 14 45 242999 0.125 5.1e+03 15 48 242661 0.25 2.04e+03 16 51 241987 0.5 1.91e+03 17 54 240643 1 1.04e+03 18 57 237971 2 3.36e+03 19 60 232686 4 6.04e+03 20 63 222354 8 1.2e+04 21 66 202592 16 2.25e+04 22 69 166443 32 4.05e+04 23 72 106320 64 6.68e+04 24 75 28704.7 128 8.31e+04 25 78 89.7947 140.674 2.22e+04 26 81 9.57381 2.02599 684 27 84 9.50489 0.0619927 2.27 28 87 9.50489 0.000462261 0.0114 Local minimum possible. lsqnonlin stopped because the final change in the sum of squares relative to its initial value is less than the value of the function tolerance. ```

`output`
```output = struct with fields: firstorderopt: 0.0114 iterations: 28 funcCount: 87 cgiterations: 0 algorithm: 'trust-region-reflective' stepsize: 4.6226e-04 message: 'Local minimum possible....' bestfeasible: [] constrviolation: [] ```

For comparison, set the `Algorithm` option to `'levenberg-marquardt'`.

```options.Algorithm = 'levenberg-marquardt'; [x,resnorm,residual,exitflag,output] = lsqnonlin(fun,x0,[],[],options);```
``` First-order Norm of Iteration Func-count Resnorm optimality Lambda step 0 3 359677 2.88e+04 0.01 Objective function returned Inf; trying a new point... 1 13 340761 3.91e+04 100000 0.280777 2 16 304661 5.97e+04 10000 0.373146 3 21 297292 6.55e+04 1e+06 0.0589933 4 24 288240 7.57e+04 100000 0.0645444 5 28 275407 1.01e+05 1e+06 0.0741266 6 31 249954 1.62e+05 100000 0.094571 7 36 245896 1.35e+05 1e+07 0.0133606 8 39 243846 7.26e+04 1e+06 0.0094431 9 42 243568 5.66e+04 100000 0.0082162 10 45 243424 1.61e+04 10000 0.00777935 11 48 243322 8.8e+03 1000 0.0673933 12 51 242408 5.1e+03 100 0.675209 13 54 233628 1.05e+04 10 6.59804 14 57 169089 8.51e+04 1 54.6992 15 60 30814.7 1.54e+05 0.1 196.939 16 63 147.496 8e+03 0.01 129.795 17 66 9.51503 117 0.001 9.96069 18 69 9.50489 0.0714 0.0001 0.080486 19 72 9.50489 5.23e-05 1e-05 5.07043e-05 Local minimum possible. lsqnonlin stopped because the relative size of the current step is less than the value of the step size tolerance. ```

The `'levenberg-marquardt'` converged with fewer iterations, but almost as many function evaluations:

`output`
```output = struct with fields: iterations: 19 funcCount: 72 stepsize: 5.0704e-05 cgiterations: [] firstorderopt: 5.2319e-05 algorithm: 'levenberg-marquardt' message: 'Local minimum possible....' bestfeasible: [] constrviolation: [] ```

## Input Arguments

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Function whose sum of squares is minimized, specified as a function handle or the name of a function. For the `'interior-point'` algorithm, `fun` must be a function handle. `fun` is a function that accepts an array `x` and returns an array `F`, the objective function evaluated at `x`.

Note

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

The function `fun` can be specified as a function handle to a file:

`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);`

`lsqnonlin` passes `x` to your objective function in the shape of the `x0` argument. For example, if `x0` is a 5-by-3 array, then `lsqnonlin` passes `x` to `fun` as a 5-by-3 array.

If the Jacobian can also be computed and the `'SpecifyObjectiveGradient'` option is `true`, set by

`options = optimoptions('lsqnonlin','SpecifyObjectiveGradient',true)`

then the function `fun` must return a second output argument with 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 an array of `m` components and `x` has `n` elements, where `n` is the number of elements 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`.)

Example: `@(x)cos(x).*exp(-x)`

Data Types: `char` | `function_handle` | `string`

Initial point, specified as a real vector or real array. Solvers use the number of elements in `x0` and the size of `x0` to determine the number and size of variables that `fun` accepts.

Example: `x0 = [1,2,3,4]`

Data Types: `double`

Lower bounds, specified as a real vector or real array. If the number of elements in `x0` is equal to the number of elements in `lb`, then `lb` specifies that

`x(i) >= lb(i)` for all `i`.

If `numel(lb) < numel(x0)`, then `lb` specifies that

`x(i) >= lb(i)` for ```1 <= i <= numel(lb)```.

If `lb` has fewer elements than `x0`, solvers issue a warning.

Example: To specify that all x components are positive, use ```lb = zeros(size(x0))```.

Data Types: `double`

Upper bounds, specified as a real vector or real array. If the number of elements in `x0` is equal to the number of elements in `ub`, then `ub` specifies that

`x(i) <= ub(i)` for all `i`.

If `numel(ub) < numel(x0)`, then `ub` specifies that

`x(i) <= ub(i)` for ```1 <= i <= numel(ub)```.

If `ub` has fewer elements than `x0`, solvers issue a warning.

Example: To specify that all x components are less than 1, use ```ub = ones(size(x0))```.

Data Types: `double`

Linear inequality constraints, specified as a real matrix. `A` is an `M`-by-`N` matrix, where `M` is the number of inequalities, and `N` is the number of variables (number of elements in `x0`). For large problems, pass `A` as a sparse matrix.

`A` encodes the `M` linear inequalities

`A*x <= b`,

where `x` is the column vector of `N` variables `x(:)`, and `b` is a column vector with `M` elements.

For example, consider these inequalities:

x1 + 2x2 ≤ 10
3x1 + 4x2 ≤ 20
5x1 + 6x2 ≤ 30,

Specify the inequalities by entering the following constraints.

```A = [1,2;3,4;5,6]; b = [10;20;30];```

Example: To specify that the x components sum to 1 or less, use ```A = ones(1,N)``` and `b = 1`.

Data Types: `double`

Linear inequality constraints, specified as a real vector. `b` is an `M`-element vector related to the `A` matrix. If you pass `b` as a row vector, solvers internally convert `b` to the column vector `b(:)`. For large problems, pass `b` as a sparse vector.

`b` encodes the `M` linear inequalities

`A*x <= b`,

where `x` is the column vector of `N` variables `x(:)`, and `A` is a matrix of size `M`-by-`N`.

For example, consider these inequalities:

x1 + 2x2 ≤ 10
3x1 + 4x2 ≤ 20
5x1 + 6x2 ≤ 30.

Specify the inequalities by entering the following constraints.

```A = [1,2;3,4;5,6]; b = [10;20;30];```

Example: To specify that the x components sum to 1 or less, use ```A = ones(1,N)``` and `b = 1`.

Data Types: `double`

Linear equality constraints, specified as a real matrix. `Aeq` is an `Me`-by-`N` matrix, where `Me` is the number of equalities, and `N` is the number of variables (number of elements in `x0`). For large problems, pass `Aeq` as a sparse matrix.

`Aeq` encodes the `Me` linear equalities

`Aeq*x = beq`,

where `x` is the column vector of `N` variables `x(:)`, and `beq` is a column vector with `Me` elements.

For example, consider these inequalities:

x1 + 2x2 + 3x3 = 10
2x1 + 4x2 + x3 = 20,

Specify the inequalities by entering the following constraints.

```Aeq = [1,2,3;2,4,1]; beq = [10;20];```

Example: To specify that the x components sum to 1, use `Aeq = ones(1,N)` and `beq = 1`.

Data Types: `double`

Linear equality constraints, specified as a real vector. `beq` is an `Me`-element vector related to the `Aeq` matrix. If you pass `beq` as a row vector, solvers internally convert `beq` to the column vector `beq(:)`. For large problems, pass `beq` as a sparse vector.

`beq` encodes the `Me` linear equalities

`Aeq*x = beq`,

where `x` is the column vector of `N` variables `x(:)`, and `Aeq` is a matrix of size `Me`-by-`N`.

For example, consider these equalities:

x1 + 2x2 + 3x3 = 10
2x1 + 4x2 + x3 = 20.

Specify the equalities by entering the following constraints.

```Aeq = [1,2,3;2,4,1]; beq = [10;20];```

Example: To specify that the x components sum to 1, use `Aeq = ones(1,N)` and `beq = 1`.

Data Types: `double`

Nonlinear constraints, specified as a function handle. `nonlcon` is a function that accepts a vector or array `x` and returns two arrays, `c(x)` and `ceq(x)`.

• `c(x)` is the array of nonlinear inequality constraints at `x`. `lsqnonlin` attempts to satisfy

 `c(x) <= 0` for all entries of `c`. (1)
• `ceq(x)` is the array of nonlinear equality constraints at `x`. `lsqnonlin` attempts to satisfy

 `ceq(x) = 0` for all entries of `ceq`. (2)

For example,

`x = lsqnonlin(@myfun,x0,lb,ub,A,b,Aeq,beq,@mycon,options)`

where `mycon` is a MATLAB function such as

```function [c,ceq] = mycon(x) c = ... % Compute nonlinear inequalities at x. ceq = ... % Compute nonlinear equalities at x.```
If the Jacobians (derivatives) of the constraints can also be computed and the `SpecifyConstraintGradient` option is `true`, as set by
`options = optimoptions('','SpecifyConstraintGradient',true)`
then `nonlcon` must also return, in the third and fourth output arguments, `GC`, the Jacobian of `c(x)`, and `GCeq`, the Jacobian of `ceq(x)`. The Jacobian G(x) of a vector function F(x) is

`${G}_{i,j}\left(x\right)=\frac{\partial {F}_{i}\left(x\right)}{\partial {x}_{j}}.$`

`GC` and `GCeq` can be sparse or dense. If `GC` or `GCeq` is large, with relatively few nonzero entries, save running time and memory in the `'interior-point'` algorithm by representing them as sparse matrices. For more information, see Nonlinear Constraints.

Data Types: `function_handle`

Optimization options, specified as the output of `optimoptions` or a structure as `optimset` returns.

Some options apply to all algorithms, and others are relevant for particular algorithms. See Optimization Options Reference for detailed information.

Some options are absent from the `optimoptions` display. These options appear in italics in the following table. For details, see View Optimization Options.

 All Algorithms `Algorithm` Choose between `'trust-region-reflective'` (default), `'levenberg-marquardt'`, and `'interior-point'`.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 number of elements of `F` returned by `fun` must be at least as many as the length of `x`.The `'interior-point'` algorithm is the only algorithm that can solve problems with linear or nonlinear constraints. If you include these constraints in your problem and do not specify an algorithm, the solver automatically switches to the `'interior-point'` algorithm. The `'interior-point'` algorithm calls a modified version of the `fmincon` `'interior-point'` algorithm.For more information on choosing the algorithm, see Choosing the Algorithm. `CheckGradients` Compare user-supplied derivatives (gradients of objective or constraints) to finite-differencing derivatives. Choices are `false` (default) or `true`. For `optimset`, the name is `DerivativeCheck` and the values are `'on'` or `'off'`. See Current and Legacy Option Names. Diagnostics Display diagnostic information about the function to be minimized or solved. Choices are `'off'` (default) or `'on'`. DiffMaxChange Maximum change in variables for finite-difference gradients (a positive scalar). The default is `Inf`. DiffMinChange Minimum change in variables for finite-difference gradients (a positive scalar). The default is `0`. `Display` Level of display (see Iterative Display): `'off'` or `'none'` displays no output.`'iter'` displays output at each iteration, and gives the default exit message.`'iter-detailed'` displays output at each iteration, and gives the technical exit message.`'final'` (default) displays just the final output, and gives the default exit message.`'final-detailed'` displays just the final output, and gives the technical exit message. `FiniteDifferenceStepSize` Scalar or vector step size factor for finite differences. When you set `FiniteDifferenceStepSize` to a vector `v`, the forward finite differences `delta` are`delta = v.*sign′(x).*max(abs(x),TypicalX);`where `sign′(x) = sign(x)` except `sign′(0) = 1`. Central finite differences are`delta = v.*max(abs(x),TypicalX);`Scalar `FiniteDifferenceStepSize` expands to a vector. The default is `sqrt(eps)` for forward finite differences, and `eps^(1/3)` for central finite differences. For `optimset`, the name is `FinDiffRelStep`. See Current and Legacy Option Names. `FiniteDifferenceType` Finite differences, used to estimate gradients, are either `'forward'` (default), or `'central'` (centered). `'central'` takes twice as many function evaluations, but should be more accurate.The algorithm is careful to obey bounds when estimating both types of finite differences. So, for example, it could take a backward, rather than a forward, difference to avoid evaluating at a point outside bounds. For `optimset`, the name is `FinDiffType`. See Current and Legacy Option Names. `FunctionTolerance` Termination tolerance on the function value, a positive scalar. The default is `1e-6`. See Tolerances and Stopping Criteria. For `optimset`, the name is `TolFun`. See Current and Legacy Option Names. 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. `MaxFunctionEvaluations` Maximum number of function evaluations allowed, a positive integer. The default is `100*numberOfVariables` for the `'trust-region-reflective'` algorithm, `200*numberOfVariables` for the `'levenberg-marquardt'` algorithm, and `3000` for the `'interior-point'` algorithm. See Tolerances and Stopping Criteria and Iterations and Function Counts.For `optimset`, the name is `MaxFunEvals`. See Current and Legacy Option Names. `MaxIterations` Maximum number of iterations allowed, a positive integer. The default is `400` for the `'trust-region-reflective'` and `'levenberg-marquardt'` algorithms, and `1000` for the `'interior-point'` algorithm. See Tolerances and Stopping Criteria and Iterations and Function Counts.For `optimset`, the name is `MaxIter`. See Current and Legacy Option Names. `OptimalityTolerance` Termination tolerance on the first-order optimality (a positive scalar). The default is `1e-6`. See First-Order Optimality Measure.Internally, the `'levenberg-marquardt'` algorithm uses an optimality tolerance (stopping criterion) of `1e-4` times `FunctionTolerance` and does not use `OptimalityTolerance`. For `optimset`, the name is `TolFun`. See Current and Legacy Option Names. `OutputFcn` Specify one or more user-defined functions that an optimization function calls at each iteration. Pass a function handle or a cell array of function handles. The default is none (`[]`). See Output Function and Plot Function Syntax. `PlotFcn` Plots various measures of progress while the algorithm executes; select from predefined plots or write your own. Pass a name, a function handle, or a cell array of names or function handles. For custom plot functions, pass 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. Custom plot functions use the same syntax as output functions. See Output Functions for Optimization Toolbox and Output Function and Plot Function Syntax.For `optimset`, the name is `PlotFcns`. See Current and Legacy Option Names. `SpecifyObjectiveGradient` If `false` (default), the solver approximates the Jacobian using finite differences. If `true`, the solver uses a user-defined Jacobian (defined in `fun`), or Jacobian information (when using `JacobMult`), for the objective function. For `optimset`, the name is `Jacobian`, and the values are `'on'` or `'off'`. See Current and Legacy Option Names. `StepTolerance` Termination tolerance on `x`, a positive scalar. The default is `1e-6` for the `'trust-region-reflective'` and `'levenberg-marquardt'` algorithms, and `1e-10` for the `'interior-point'` algorithm. See Tolerances and Stopping Criteria.For `optimset`, the name is `TolX`. See Current and Legacy Option Names. `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)`. The solver uses `TypicalX` for scaling finite differences for gradient estimation. `UseParallel` When `true`, the solver estimates gradients in parallel. Disable by setting to the default, `false`. See Parallel Computing. Trust-Region-Reflective Algorithm `JacobianMultiplyFcn` Jacobian multiply function, specified as a function handle. 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. The solver uses `Jinfo` to compute the preconditioner. See Passing Extra Parameters for information on how to supply values for any additional parameters `jmfun` needs.Note`'SpecifyObjectiveGradient'` must be set to `true` for the solver to pass `Jinfo` from `fun` to `jmfun`.See Minimization with Dense Structured Hessian, Linear Equalities and Jacobian Multiply Function with Linear Least Squares for similar examples.For `optimset`, the name is `JacobMult`. See Current and Legacy Option Names. JacobPattern Sparsity pattern of the Jacobian for finite differencing. Set `JacobPattern(i,j) = 1` when `fun(i)` depends on `x(j)`. Otherwise, set ```JacobPattern(i,j) = 0```. In other words, `JacobPattern(i,j) = 1` when you can have ∂`fun(i)`/∂`x(j)` ≠ 0.Use `JacobPattern` when it is inconvenient to compute the Jacobian matrix `J` in `fun`, though you can determine (say, by inspection) when `fun(i)` depends on `x(j)`. The solver can approximate `J` via sparse finite differences when you give `JacobPattern`.If the structure is unknown, do not set `JacobPattern`. The default behavior is as if `JacobPattern` is a dense matrix of ones. Then the solver computes a full finite-difference approximation in each iteration. This can be expensive for large problems, so it is usually better 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 Large Scale Nonlinear Least Squares. 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. `SubproblemAlgorithm` Determines how the iteration step is calculated. The default, `'factorization'`, takes a slower but more accurate step than `'cg'`. See Trust-Region-Reflective Least Squares. TolPCG Termination tolerance on the PCG iteration, a positive scalar. The default is `0.1`. Levenberg-Marquardt Algorithm InitDamping Initial value of the Levenberg-Marquardt parameter, a positive scalar. Default is `1e-2`. For details, see Levenberg-Marquardt Method. ScaleProblem `'jacobian'` can sometimes improve the convergence of a poorly scaled problem; the default is `'none'`. Interior-Point Algorithm `BarrierParamUpdate ` Specifies how `fmincon` updates the barrier parameter (see fmincon Interior Point Algorithm). The options are:`'monotone'` (default)`'predictor-corrector'`This option can affect the speed and convergence of the solver, but the effect is not easy to predict. `ConstraintTolerance` Tolerance on the constraint violation, a positive scalar. The default is `1e-6`. See Tolerances and Stopping Criteria.For `optimset`, the name is `TolCon`. See Current and Legacy Option Names. InitBarrierParam Initial barrier value, a positive scalar. Sometimes it might help to try a value above the default `0.1`, especially if the objective or constraint functions are large. `SpecifyConstraintGradient` Gradient for nonlinear constraint functions defined by the user. When set to the default, `false`, `lsqnonlin` estimates gradients of the nonlinear constraints by finite differences. When set to `true`, `lsqnonlin` expects the constraint function to have four outputs, as described in `nonlcon`.For `optimset`, the name is `GradConstr` and the values are `'on'` or `'off'`. See Current and Legacy Option Names. `SubproblemAlgorithm` Determines how the iteration step is calculated. The default, `'factorization'`, is usually faster than `'cg'` (conjugate gradient), though `'cg'` might be faster for large problems with dense Hessians. See fmincon Interior Point Algorithm.For `optimset`, the values are `'cg'` and `'ldl-factorization'`. See Current and Legacy Option Names.

Example: ```options = optimoptions('lsqnonlin','FiniteDifferenceType','central')```

Problem structure, specified as a structure with the following fields:

Field NameEntry

`objective`

Objective function

`x0`

Initial point for `x`

`Aineq`

Matrix for linear inequality constraints

`bineq`

Vector for linear inequality constraints

`Aeq`

Matrix for linear equality constraints

`beq`

Vector for linear equality constraints
`lb`Vector of lower bounds
`ub`Vector of upper bounds

`nonlcon`

Nonlinear constraint function

`solver`

`'lsqnonlin'`

`options`

Options created with `optimoptions`

You must supply at least the `objective`, `x0`, `solver`, and `options` fields in the `problem` structure.

Data Types: `struct`

## Output Arguments

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Solution, returned as a real vector or real array. The size of `x` is the same as the size of `x0`. Typically, `x` is a local solution to the problem when `exitflag` is positive. For information on the quality of the solution, see When the Solver Succeeds.

Squared norm of the residual, returned as a nonnegative real. `resnorm` is the squared 2-norm of the residual at `x`: `sum(fun(x).^2)`.

Value of objective function at solution, returned as an array. In general, `residual = fun(x)`.

Reason the solver stopped, returned as an integer.

 `1` Function converged to a solution `x`. `2` Change in `x` is less than the specified tolerance, or Jacobian at `x` is undefined. `3` Change in the residual is less than the specified tolerance. `4` Relative magnitude of search direction is smaller than the step tolerance. `0` Number of iterations exceeds `options.MaxIterations` or number of function evaluations exceeded `options.MaxFunctionEvaluations`. `-1` A plot function or output function stopped the solver. `-2` No feasible point found. The bounds `lb` and `ub` are inconsistent, or the solver stopped at an infeasible point.

Information about the optimization process, returned as a structure with fields:

 `firstorderopt` Measure of first-order optimality `iterations` Number of iterations taken `funcCount` The number of function evaluations `cgiterations` Total number of PCG iterations (`'trust-region-reflective'` and `'interior-point'` algorithms) `stepsize` Final displacement in `x` `constrviolation` Maximum of constraint functions (`'interior-point'` algorithm) `bestfeasible` Best (lowest objective function) feasible point encountered (`'interior-point'` algorithm). A structure with these fields:`x``fval``firstorderopt``constrviolation`If no feasible point is found, the `bestfeasible` field is empty. For this purpose, a point is feasible when the maximum of the constraint functions does not exceed `options.ConstraintTolerance`.The `bestfeasible` point can differ from the returned solution point `x` for a variety of reasons. For an example, see Obtain Best Feasible Point. `algorithm` Optimization algorithm used `message` Exit message

Lagrange multipliers at the solution, returned as a structure with fields:

 `lower` Lower bounds corresponding to `lb` `upper` Upper bounds corresponding to `ub` `ineqlin` Linear inequalities corresponding to `A` and `b` `eqlin` Linear equalities corresponding to `Aeq` and `beq` `ineqnonlin` Nonlinear inequalities corresponding to the `c` in `nonlcon` `eqnonlin` Nonlinear equalities corresponding to the `ceq` in `nonlcon`

Jacobian at the solution, returned as a real matrix. `jacobian(i,j)` is the partial derivative of `fun(i)` with respect to `x(j)` at the solution `x`.

For problems with active constraints at the solution, `jacobian` is not useful for estimating confidence intervals.

## Limitations

• The trust-region-reflective algorithm 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, `lsqnonlin` uses the Levenberg-Marquardt algorithm.

• `lsqnonlin` can solve complex-valued problems directly. Note that constraints do not make sense for complex values, because complex numbers are not well-ordered; asking whether one complex value is greater or less than another complex value is nonsensical. For a complex problem with bound constraints, split the variables into real and imaginary parts. Do not use the `'interior-point'` algorithm with complex data. See Fit a Model to Complex-Valued Data.

• The preconditioner computation used in the preconditioned conjugate gradient part of the trust-region-reflective 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, `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.

You can use the trust-region reflective algorithm in `lsqnonlin`, `lsqcurvefit`, and `fsolve` with small- to medium-scale problems without computing the Jacobian in `fun` or providing the Jacobian sparsity pattern. (This also applies to using `fmincon` or `fminunc` without computing the Hessian or supplying the Hessian sparsity pattern.) How small is small- to medium-scale? No absolute answer is available, as it depends on the amount of virtual memory in your computer system configuration.

Suppose your problem has `m` equations and `n` unknowns. If the command `J = sparse(ones(m,n))` causes an `Out of memory` error on your machine, then this is certainly too large a problem. If it does not result in an error, the problem might still be too large. You can find out only by running it and seeing if MATLAB runs within the amount of virtual memory available on your system.

## Algorithms

The Levenberg-Marquardt and trust-region-reflective methods are based on the nonlinear least-squares algorithms also used in `fsolve`.

• The default trust-region-reflective 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-Reflective Least Squares.

• The Levenberg-Marquardt method is described in references [4], [5], and [6]. See Levenberg-Marquardt Method.

The `'interior-point'` algorithm uses the `fmincon` `'interior-point'` algorithm with some modifications. For details, see Modified fmincon Algorithm for Constrained Least Squares.

## Alternative Functionality

### App

The Optimize Live Editor task provides a visual interface for `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, 1996, pp. 418–445.

[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, 1994, pp. 189–224.

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

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

[5] Marquardt, D. “An Algorithm for Least-squares Estimation of Nonlinear Parameters.” SIAM Journal Applied Mathematics, Vol. 11, 1963, pp. 431–441.

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

[7] Moré, J. J., B. S. Garbow, and K. E. Hillstrom. User Guide for MINPACK 1. Argonne National Laboratory, Rept. ANL–80–74, 1980.

[8] Powell, M. J. D. “A Fortran Subroutine for Solving Systems of Nonlinear Algebraic Equations.” Numerical Methods for Nonlinear Algebraic Equations, P. Rabinowitz, ed., Ch.7, 1970.

## Version History

Introduced before R2006a

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