NL2sol.jl: Non-linear least squares optimization

NL2sol.jl solves the non-linear least squares problem. That is, it finds an x that minimizes $ \sum_{i=1}^{n} {{r}_{i}}^{2}(x) $ where x is a vector of size p. It returns a struct of type Optim.MultivariateOptimizationResults that contains the relevant info (see the Julia Optim module docs for further info). It does this by wrapping the FORTRAN version of the code.

The residual and the jacobian functions are expected to take args that have been preallocated for those values. These arrays are actually allocated in the Julia function nl2sol before passing to the Fortran subroutine nl2sol.

Pkg.add("NL2sol")

using NL2sol

function rosenbrock_res(x, r)
    r[1] = 10.0 * (x[2] - x[1]^2 )
    r[2] = 1.0 - x[1]
    return r
end

function rosenbrock_jac(x, jac)
    jac[1, 1] = -20.0 * x[1]
    jac[1, 2] =  10.0
    jac[2, 1] =  -1.0
    jac[2, 2] =   0.0
   return jac
end

function main()
    println("NL2SOL on Rosenbrock")
    result = nl2sol(rosenbrock_res, rosenbrock_jac, [-1.2, 1.0], 2; quiet=true)
    println(result)
end

main()

Alternatively, if you do not have or do not want to write a jacobian, you can use nl2sno, which uses a finite difference approximation to the jacobian. Even if you do have the jacobian available, nl2sno can be used to check for its correctness. In this case, you must provide the iv and v arrays (see below). A complete example would look like:

using NL2sol

function rosenbrock_res(x, r)
    r[1] = 10.0 * (x[2] - x[1]^2 )
    r[2] = 1.0 - x[1]
    return r
end

function main()
    println("NL2nso on Rosenbrock")
    iv, v = nl2_set_defaults(2, 2)
    result = nl2sno(rosenbrock_res, [-1.2, 1.0], 2, iv, v)
    println(result)
end

main()

The wrapped Fortran code is the original netlib version of NL2SOL, a non-linear, least-squares optimization program. It is detailed in two Transactions on Mathematical Software (TOMS) papers. They are:

J.E. Dennis, D.M. Gay, R.E. Welsch, "An Adaptive Nonlinear Least-Squares Algorithm", ACM Transactions on Mathematical Software (TOMS), Volume 7 Issue 3, Sept. 1981, pp 348-368, ACM New York, NY, USA see here

J.E. Dennis, D.M. Gay, R.E. Welsch, "Algorithm 573: NL2SOL—An Adaptive Nonlinear Least-Squares Algorithm", ACM Transactions on Mathematical Software (TOMS), Volume 7 Issue 3, Sept. 1981, pp 369-383, ACM New York, NY, USA see here

Here we use the original NL2SOL Fortran 77 source code which appears as TOMS ALgorithm 573 (NL2SOL Version 2.2). The code was downloaded from netlib and is archived in the deps/src/nl2sol directory as a single blob in the file named nl2sol.netlib.orig.f

This blob has also been broken up into the individual source files and commented out the "c/6" code for the "c/7" code, which enables the f77 version. Also added are cmake files for building the code and running the tests. Running the Fortran tests and coverage is manual and not part of the installation. (The coverage is a very respectable 87%) The original fortran test code now lives in a separate subdirectory (.../deps/src/tests) as well. To learn how to build and run the tests with coverage, see NL2sol.jl/deps/src/CMakeLists.txt

Wrapper code has been added using the C interface facilities of Julia. (ie ccall and cfunction etc), so that nl2sol can be called directly from Julia.

The runtests.jl in the test directory has many examples of using Julia to call nl2sol and using Julia functions to calculate the residual and the jacobian.

There are two calling signitures for nl2sol. One is the simplified version used above an its complete version is given by:

function nl2sol(res::Function, jac::Function, init_x, n; 
                maxIter=df_maxIter, maxFuncCall=df_maxFuncCall, 
                tolX=df_tolX,  tolAbsFunc=df_tolAbsFunc,
                tolRelFunc=df_tolRelFunc, quiet=true)

The required arguments are the function that calculates the residual vector, the funtion that calculates the jacobian, the initial starting guess for the non-linear parameters, and the number of 'measurements' that we are fitting (this is also the length of the residual vector returned by the res::Function). The optional arguments control some (but not all) of the convergence criteria.

The alternative version requires that you first call a function to set the defaults and that function returns an integer and a real array which must then be passed to nl2sol. A calling sequence would look like

iv, v = set_defaults(n, p)
## change default values inside of iv, v
results = nl2sol(res, jac, init_x, n, iv, v)

The advantage of this form is that all of the control and tuning parameters of NL2sol are available by changing some of the values in the iv and/or v arrays. Also available are more status values in these arrays. They are well documented in the 'program paper' above.

As an optimization solution, this would compete most directly with the levenberg_marquardt from the LsqFit module. It differs from that algorithm in that NL2SOL is a quasi-Newton method (not BFGS but rather DFP for those who care). Because of that you would expect NL2SOL to perform better on those models that have large(r) residuals at the optimum. It will also generally perform better if the starting guess is far from the optimim point.

• Only supported in Julia 1.0+

• nl2itr, which uses "reverse communication" to request residual and jacobian updates, has not been exported.

• NL2sol uses a different convergence testing strategy than Optim.levenberg_marquardt. This makes doing apples to apples comparisons challenging.

Note that we let the Julia wrapper for nl2sol allocates the memory for both the residual and the jacobian.

nl2sol can print detailed iteration summaries. This is turned on by setting the keyword parameter quiet to false, ie

    result = nl2sol(rosenbrock_res, rosenbrock_jac, [-1.2, 1.0], 2; quiet=false)