Week 5 and 6: Refining fmincon

So my fmincon implementation is coming in shape [1]. 

[x,fval,exitflag,output,lambda,grad,hessian] = 

fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)

I came across a few issues which turned out to be bugs. Olaf pushed fixes in the central repository. Listing the issues for the record:

- Setting gradc (the gradient of general equality/inequality functions). A bug in nonlin_min.m (and __nonlin_residmin__.m)

Example:

      objective_function = @ (p) p(1)^2 + p(2)^2;
 pin = [-2; 5];
 constraint_function = @ (p) p(1)^2 + 1 - p(2);
 gradc = @(p) [2*p(1);-1];
 [p, objf, cvg, outp] = nonlin_min (objective_function, pin, optimset
("equc", {constraint_function, gradc}))

error: function handle type invalid as index value

- Giving linear inequality/ equality constraints to lm_feasible. A bug in nonlin_min.m

Example:

      f = @(x) -x(1) * x(2) * x(3);
S = [1  -1;   2  -2;   2  -2]
b = [0;72];
x0 = [10;10;10];
[x,fval] = nonlin_min( f, x0, optimset ("inequc",{S,b}) )

      error: __lm_feasible__: operator -: nonconformant arguments (op1 is
3x1, op2 is 3x0)

 

-Any zero value in initial guess vector for nonin_residmin/nonlin_min gave an error. Required a minor change of sign(0)==0 in __dfdp__.m.

Example:

      k = 1:10;
func = @(x) 2 + 2 * k - exp (k * x(1)) - exp (k * x(2));
x0 = [0;0.5];
x = nonlin_residmin(func,x0)

warning: division by zero
warning: called from
    __dfdp__ at line 367 column 21
    __nonlin_residmin__> at line -1 column -1
    __lm_svd__ at line 191 column 9
    __nonlin_residmin__ at line 1125 column 21
    nonlin_residmin at line 98 column 21
    runlsqnonlin at line 9 column 3
error: svd: cannot take SVD of matrix containing Inf or NaN values

Functionality for Returning Hessian and Gradient

- New options "ret_objf_grad" and "ret_hessian" to be introduced in nonlin_min (by Olaf). If anyone of these options is set to true, the 'outp' structure output argument of nonlin_min will contain additional fields .objf_grad and .hessian. My code currently checks this. 

Rearranging values of lambda in the fields of a structure.

- For lm_feasible, outp will contain an additional field lambda, a structure which contains Lagrange multipliers in fields separated by constraint type.

I added an additional feature in [1] to cater for the non linear constraint function set using deal();

Example:
      
      c = @(x) [x(1) ^ 2 / 9 + x(2) ^ 2 / 4 - 1;
        x(1) ^ 2 - x(2) - 1];
ceq = @(x) tanh (x(1)) - x(2);
nonlinfcn = @(x) deal (c(x), ceq(x));
obj = @(x) cosh (x(1)) + sinh (x(2));
z = fmincon (obj, [0;0], [], [], [], [], [], [], nonlinfcn)

z =
   -0.6530
   -0.5737

To do:
1- Write test cases/refined examples for for lsqnonlin, lsqcurvefit, nlinfit and fmincon.
2- Start wrapping quadprog to __qp__ instead of qp.m (because of the ordering of the lambda output).

[1] https://github.com/AsmaAfzal/octave_workspace/blob/master/fmincon.m

Progress Update: Midterm Evaluation

Objective:

Adding functions to the Optim package for Octave using existing back-ends.

Expected deliverables before midterm:

• 'lsqnonlin' using 'nonlin_residmin'

Done in [1]. Differences in backends, nonlin_residmin uses "lm_svd" algorithm for optimization as currently the only backend. However, lsqnonlin in Matlab can choose between "trust-region-reflective" and "Levenberg-Marquardt" (LM) algorithms.

Another difference is in complex inputs. lm_svd does not support complex valued inputs whereas Matlab's LM algorithm can accept complex input arguments. One way of providing complex inputs to lsqnonlin in Octave is to split the real and imaginary parts into separate variables and running the optimization.  

• 'lsqcurvefit' using 'nonlin_curvefit', 'nonlin_residmin', or 'lsqnonlin'

Done in [2] using nonlin_curvefit. lsqcurvefit is very similar to lsqnonlin with only a few minor interface differences. lsqcurvefit explicitly takes independent variables and the observations as inputs while these values can be wrapped inside the objective function while using lsqnonlin. Additional bounds for the optimized parameters can be specified. 

• 'nlinfit' using 'leasqr',

I wrapped nlinfit on nonlin_curvefit and curvefit_stat as leasqr repeats the optimization to compute the additional statistics (Jacobian and Covariance matrix) while curvefit_stat saves this computation overhead. I have partially implemented nlinfit in [3] (It hasn't been thoroughly reviewed yet). Two missing features are: 1) Error models and Error parameters estimation, and 2) Robust Weight function. Meanwhile, no such functionality exists in the Octave's optimization backends for the missing features. My current implementation supports the input of scalar positive array of weights for robust regression.  

Since nlinfit is from the statistics toolbox of Matlab, it uses statset and statget to create and get options respectively. I created additional functions statset, statget and __all_stat_opts__ with minor changes to the code in optimset, optimget and __all_opts__.

• 'fmincon' using 'nonlin_min',

In progress [4].


Future goals:

1. Complete fmincon implementation.
2. Create solver specific options using optimoptions and desirably still be able to use optimget. 
3. Arranging lambda output for quadprog by wrapping it on __qp__ instead of qp.m
4. Test cases for all the implemented functions.

[1] https://github.com/AsmaAfzal/octave_workspace/blob/master/lsqnonlin.m

[2] https://github.com/AsmaAfzal/octave_workspace/blob/master/lsqcurvefit.m

[3] https://github.com/AsmaAfzal/octave_workspace/blob/master/nlinfit.m
[4] https://github.com/AsmaAfzal/octave_workspace/blob/master/fmincon.m

Week4: fmincon wrapping nonlin_min

Time flies.. A third of the way through already..

fmincon/nonlin_min is the most elaborate function of all that I have previously 

implemented so before actual coding I would like to thoroughly check the the 

mapping of arguments and options. 

[x,fval,exitflag,output,lambda,grad,hessian] = fmincon (fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)

[x,fval,exitflag,output] = nonlin_min (fun,x0,settings)

A,b Aeq, beq- Linear inequality and equality constraints

Inequality constraints:

Matlab standard: A * x - b <= 0 ,

Octave standard: m.' * x  + v >= 0. 

This implies: m=-A.' , v=b.

Set in Octave using 

optimset ("inequc", {m,v})

Similar for equality constraints:

optimset ("equc", {m,v})

lb, ub- Lower and upper bounds

Set in Octave using optimset

 optimset ("lbound", ..., "ubound",...)

nonlcon- Nonliner constraint function handle 

In Matlab, nonlinear constraints are given in a function with the following format

function [c,ceq,gradc,gradceq] = mycon(x)

c = ...     % Compute nonlinear inequalities at x.  

ceq = ...   % Compute nonlinear equalities at x.

% Optional output arguments:

gradc = ...   % Gradient of c(x).

gradceq = ...   % Gradient of ceq(x).

options = optimoptions('fmincon','GradConstr','on')

Alternative to nonlcon in nonlin_min

  optimset ("equc", {constraint_function})

Options- Options common to both fmincon and nonlin_min

User-supplied Gradient

In Matlab, the objective function must return the second output when the 

GradObj option is set:

optimset('GradObj','on')

Octave:

optimset("objf_grad",@fun)

User-supplied Hessian

In Matlab, the objective function must return the third output when the Hessian 

option is set:

optimset('Hessian','user-supplied')

Octave:

optimset("objf_hessian",@fun)

Lambda- returned as a structure in Matlab but as a vector field of the "output" 

ouput argument in Octave.

Things to investigate:

1. Algorithm mapping. 

2. Exitflag mapping.

3. Setting gradc (the gradient of general equality/inequality functions). The second 

entry for the equc/inequc setting implements this feature in Octave as stated by 

optim_doc but I was unable to make it work properly.

4. Returning additional statistics (Hessian, Gradient)- I used 

residmin_stat/curvefit_stat previously for lsqnonlin and lsqcurvefit. No such function for nonlin_min.

5. Rearranging values of lambda in the fields of a structure.

P.s. this week I spent quite some time with Mercurial and how to possibly work in the optim package. There was a slight miscommunication/confusion but it is clear now and I will continue to publish my code on github..

Week3: nlinfit, statset, stat_get and __all_stat_opts__

So this week I achieved the following milestones:

Wrapping nlinfit on nonlin_curvefit

[beta,R,J,CovB,MSE,ErrorModelInfo] = nlinfit(X,Y,modelfun,beta0,options,Name,Value)

Implementation [1]:

I chose not to wrap nlinfit on lsqcurvefit because 

1. We might end up wrapping lsqcurvefit on lsqnonlin eventually so it is undecided.
2. The default options for lsqnonlin/lsqcurvefit are different from nlinfit. 

Missing features:

1. RobustWgtFun - The field RobustWgtFun in options can be provided with a function handle which computes the residuals at every iteration. The backend optimization algorithm in Octave currently does not support this functionality.  
2. Name-Value pairs. Currently the only implemented one is "weights",  which takes an array of weights for the weighted optimization. "ErrorModelInfo" and "ErrorParameters" are not implemented. The possible error models include, "constant", "proportional" and "combined". The error model also translates to a weight function which helps in reducing the effect of outliers.   
3. ErrorModelInfo- output field which gives information about the error variance, and estimates the parameters of Error models. 

Setting options using statset

In Matlab, options for statistics toolbox are set using statset [2].  The functionality is almost identical to optimset but the different functions in Matlab are because of different toolboxes (statset for statistics and optimset for optimization toolbox).

Added functions:

statset.m, statget.m and __all_stat_opts__.m [3]-[5]

Creating these functions was pretty straight forward.

Still to do:

1. Have to check if the weighted residual and weighted Jacobian output in octave is consistent with Matlab and further refine the functions with the feedback from my mentors.
2. Move on to fmincon wrapping nonlin_min.

[1] https://github.com/AsmaAfzal/octave_workspace/blob/master/nlinfit.m
[2] http://uk.mathworks.com/help/stats/statset.html
[3] https://github.com/AsmaAfzal/octave_workspace/blob/master/statset.m
[4] https://github.com/AsmaAfzal/octave_workspace/blob/master/statget.m
[5] https://github.com/AsmaAfzal/octave_workspace/blob/master/__all_stat_opts__.m

Week 2: lsqnonlin and lsqcurvefit

A bit late blogging about week 2. 

Almost completed functions lsqnonlin and lsqcurvefit. 
Successfully mapped user-specified Jacobian. 
In Matlab, if the Jacobian option is set to "on",  the model function must return 
a second output which is the Jacobian function.
In Octave, the Jacobian function handle is given to the dfdp option using optimset.

Lsqnonlin function description:

[x,resnorm,residual,exitflag,output,lambda,jacobian] = ...
lsqnonlin(fun,x0,lb,ub,options)

This function maps on:

[x, residual, exitflag, output] = nonlin_residmin (fun, x0, options)

Features of lsqnonlin in Octave:

• Input arguments: Acceptable forms are lsqnonlin(fun,x0), lsqnonlin(fun,x0,lb,ub) and lsqnonlin(fun,x0,lb,ub,options)
• Outputs
• x, exitflag, residual and output currently same as nonlin_residmin.
• resnorm=sum(residual.^2)
• Lambda is computed using the complementary pivoting in __lm_svd__.
  It's values differ from Matlab's due to the difference in backends.
• Jacobian is computed using the function residmin_stat ().

Lsqcurvefit function description:

[x,resnorm,residual,exitflag,output,lambda,jacobian] = ...
lsqcurvefit(fun,x0,xdata,ydata,lb,ub,options)

This function maps on

[x, fy, exitflag, output] = nonlin_curvefit (fun, x0, xdata, ydata, options)

Features of lsqcurvefit in Octave:

• Input arguments: Acceptable forms are lsqcurvefit (fun,x0,xdata,ydata), lsqcurvefit (fun,x0,xdata,ydata,lb,ub) and lsqcurvefit (fun,x0,lb,ub,xdata,ydata,options)
• Outputs
• x, exitflag, residual and output currently same as nonlin_curvefit.
• residual = fy-ydata, resnorm = sum(residual.^2)
• Lambda and Jacobian same as in lsqnonlin.

There are only minor interface differences between lsqcurvefit and lsqnonlin.

This week's plan:

• Hopefully, with lsqnonlin and lsqcurvefit wrapped up, I'll move on to nlinfit
• Three key challenges need to be addressed when wrapping nlinfit using nonlin_curvefit
  and curvefit_stat:
• Weight functions: Currently, no such functionality exists in nonlin_curvefit,
   where a user can specify weight functions to perform Robust regression
  (weights computed using the specified function in every iteration).
• Error Models and ErrorModelInfo
• Setting options using statset instead of optimset or optimoptions. 

optimoptions

Answering some questions to better understand the behavior of optimoptions.

Although the answers have been discussed in [1], I am pasting some examples for clarity.

*I was using the terms solver/algorithm in the wrong context before. Solver names are functions such as lsqnonlin, fmincon, etc. and one solver can have multiple algorithms such as interior-point, lev-mar, etc. *

1) Is there an error or warning if optimoptions is used to assign an option not contained in the specified algorithm? 

If the option belongs to a different algorithm of the solver, Matlab stores it as options "not" used by the current algorithm so if we change the algorithm to which the option belongs, we do not need to set it again. 

For eg. The default algorithm of lsqnonlin is trust-region-reflective but when I try to set the option 'ScaleProblem' which is specific to 'levenberg-marquardt' algorithm, I get: 

opts = optimoptions ('lsqnonlin', 'ScaleProblem', 'Jacobian')

opts = 

  lsqnonlin options:

   Options used by current Algorithm ('trust-region-reflective'):
   (Other available algorithms: 'levenberg-marquardt')

   Set by user:
     No options set by user.

   Default:

 Algorithm: 'trust-region-reflective'

    DerivativeCheck: 'off'

     .               .  

     .               .  

     .               .  

     .               .  

    TolX: 1.0000e-06

   Show options not used by current Algorithm ('trust-region-reflective')

Set by user:

    ScaleProblem: 'jacobian'

This gives the same result using dot notation:

opts = optimoptions ('lsqnonlin')

opts.ScaleProblem = 'Jacobian'

2) ... or to assign an option that does not exist in the specified solver or any solver?

This gives an error. Trying to set the option for SQP Algorithm, which is not used by lsqnonlin:

opts=optimoptions('lsqnonlin', 'MaxSQPIter', 100)

Error using optimoptions

'MaxSQPIter' is not an option for LSQNONLIN

using dot notation:

opts.MaxSQPIter = 100;
No public field MaxSQPIter exists for class optim.options.Lsqnonlin.
 

3) If options are transfered to a different solver with optimoptions, are there errors or warnings if the new solver does not have some of these options?

No errors or warnings. The options common to both solvers are copied. They could be options of different algorithms. For eg. 

opts = optimoptions ('fmincon', 'Algorithm', 'sqp', 'TolX', 1e-10) 
opts_lsq = optimoptions ('lsqnonlin', opts) 

Options set in opts_lsq are: 
                PrecondBandWidth: 0 
                TolX: 1.0000e-10 

The option PrecondBandwidth belongs to the trust-region algorithm of fmincon solver. 
Another option copied from opts in opts_lsq belongs to the lev-mar algorithm of lsqnonlin. It is the stored option as mentioned in 1) 

  ScaleProblem: 'none' 
 

4) Can options returned by optimoptions be used with optimset, and   vice versa? 

This returns an error in both cases:

opts = optimoptions ('lsqnonlin','MaxIter',100);
opts = optimset (opts, 'TolX', 1e-6);
Error using optimset
Cannot use OPTIMSET to alter options created using OPTIMOPTIONS.
Use OPTIMOPTIONS instead.

opts = optimset ('TolX', 1e-6);
opts = optimoptions ('lsqnonlin', opts);
Error using optimoptions
Invalid option name specified. Provide a string (such as 'Display').

References: 

[1] http://octave.1599824.n4.nabble.com/lsqnonlin-and-nonlin-residmin-tp4670540p4670569.html

lsqnonlin wrapping nonlin_residmin

So the first week of GSoC is officially over.

I was working on lsqnonlin. My code is accessible here:

https://github.com/AsmaAfzal/octave_workspace/blob/master/lsqnonlin.m

[x, resnorm, residual, exitflag, output, lambda, jacobian]...

= lsqnonlin (fun, x0, lb, ub, options)

[x, resid, cvg, outp] = nonlin_residmin (f, x0, settings)

A recap of encountered problems:

1. output lambda- it wasn't previously returned from the backend [1]
2. options- In addition to optimset, Matlab uses optimoptions to set options specific to a certain optimization function. optimoptions
   
   • Creates and modifies only the options that apply to a solver
   • Shows your option choices and default values for a specific solver/algorithm
   • Displays links for more information on solver options and other available solver algorithms [2].
   octave currently does not have this functionality.  For more discussion on this, check [3]

Things to do:

1. check how user-specified jacobian is to be provided.
2. check for matrix/complex inputs.
3. come up with a plan for writing optimoptions in octave.

 References:
[1] http://octave.1599824.n4.nabble.com/lsqnonlin-and-nonlin-residmin-tp4670540p4670557.html
[2] http://uk.mathworks.com/help/optim/ug/optimoptions-and-optimset.html
[3] http://octave.1599824.n4.nabble.com/lsqnonlin-and-nonlin-residmin-tp4670540p4670560.html