Week 8 and 9: The ordering of lambda in quadprog

I was trying to dig through __qp__.cc to figure out why the ordering of lambda in [1] does not match that of quadprog in Matlab.
The example shows how the values were different:

C = [0.9501    0.7620    0.6153    0.4057
    0.2311    0.4564    0.7919    0.9354
    0.6068    0.0185    0.9218    0.9169
    0.4859    0.8214    0.7382    0.4102
    0.8912    0.4447    0.1762    0.8936];
d = [0.0578
    0.3528
    0.8131
    0.0098
    0.1388];
A =[0.2027    0.2721    0.7467    0.4659
    0.1987    0.1988    0.4450    0.4186
    0.6037    0.0152    0.9318    0.8462];
b =[0.5251
    0.2026
    0.6721];
Aeq = [3 5 7 9];
beq = 4;
lb = -0.1*ones(4,1);
ub = 1*ones(4,1);
H = C ' * C;
f = -C ' * d;

[x, obj_qp, INFO, lambda] = qp ([],H,f,Aeq,beq,lb,ub,[],A,b);
lambda = 
   0.01654
   0.06743
   0.00000
   0.24993
   0.00000
   0.00000
   0.00000
   0.00000
   0.00000
   0.00000
   0.49819
   0.00000

Reordering lambda based on the length of constraints resulted in
lambda.eqlin = 
   0.01654
lambda.lower =
   0.06743
   0.00000
   0.24993
   0.00000
lambda.upper =
   0.00000
   0.00000
   0.00000
   0.00000
lambda.ineqlin =
   0.00000
   0.49819
   0.00000

Matlab however,  gave lambda.eqlin= -0.0165 for this example and lambda.lower =
    0.0674
    0.2499
         0
         0

There were two issues with the ordering:

1.  the sign for Lagrange multipliers corresponding to linear equality constraint is always different from Matlab's
2. The multipliers corresponding to the bound constraints as underlined above are swapped.

I tried several different examples to understand what is going on. For all the examples, the sign for lambda.eqlin was consistently different. Although, I still can't pinpoint why but for now I am just multiplying lambda.eqlin by -1.

For the swapping issue, I tried the same example with just the bound constraints:

[x, obj_qp, INFO, lambda] = qp ([],H,f,[],[],lb,ub)
lambda= 
   0.07377
   0.00000
   0.29791
   0.00000
   0.00000
   0.00000
   0.00000
   0.00000
but only considering lower bound constraints gave:
[x, obj_qp, INFO, lambda] = qp ([],H,f,[],[],lb,[])
lambda=
   0.07377
   0.29791
   0.00000
   0.00000

which is how it is supposed to be. Tracing back, I found out that the ordering of lambda vector in qp.m [2] is not [equality constraints; lower bounds; upper bounds; other inequality constraints] like I previously assumed. From lines 287 and 288 in qp.m [2], the bound constraints are added to the inequality constraint matrix alternatively. So the issue wasn't swapping but understanding how the constraints are passed to __qp__.

In my code in [1], I had to make significant changes in the original qp.m code such as:
- Inequality constraint matrix has the order: [Linear inequality constraints; lower bounds; upper bounds]
- Check for too close bounds forming an equality constraint- This brings indexing issues as now the Lagrange multiplier value corresponding to bounds is in place of multipliers corresponding to linear equality constraints.
Also, Matlab only accepts too close bounds when using a medium scale algorithm and since the lower bound is approximately equal to the upper bound and it is considered as a single equality constraint, the single Lagrange multiplier is placed in the corresponding lambda.upper field while the corresponding lambda.lower value is zero.

Continuing the above example:

lb(4) = 0.3;
ub = 0.3*ones(4,1);

[x, obj_qp, INFO, lambda] = qp ([],H,f,[],[],lb,ub)
lambda =

  -0.04828
   0.04154
   0.00000
   0.28615
   0.00000
   0.00000
   0.01546

Here, lb(4) = ub(4) and hence the constraint is treated as an equality constraint so the value for corresponding Lagrange multipliers is present on top (underlined)
I added checks for such cases and now my code in [1] gives same results as Matlab:

[x,obj,flag,op,lambda]=quadprog(H,f,[],[],[],[],lb,ub)

lambda =
  scalar structure containing the fields:
    lower =
       0.04154
       0.28615
       0.00000
       0.00000
    upper =
       0.000000
       0.000000
       0.015456
       0.048279

- qp.m strips off the -Inf constraints before passing them to __qp__. I am doing the same in quadprog. I have added further checks to make sure the multipliers are placed in the right positions in their respective fields. 

Plans for the next weeks:
- Get feedback from my mentor on the changes in quadprog.
- Begin intial work on optimoptions.

[1] https://github.com/AsmaAfzal/octave_workspace/blob/master/quadprog.m
[2] http://fossies.org/dox/octave-4.0.0/qp_8m_source.html

Week 7: quadprog wrapping __qp__

I have wrapped quadprog on __qp__ instead of qp.m in [1].

Main differences between quadprog in [1] and qp.m.

- Input argument placement
  quadprog(H, f, A, b, Aeq, beq, lb, ub, x0, options)  =  qp (x0, H, f, Aeq, beq, lb, ub, [], A, b, options) 

- Check for empty inputs A and b
qp ([],H,f,Aeq,beq,lb,ub,[],A,[])

This works. qp simply ignores inequality constraints due to if checks
in lines  258, 266 and 275 of qp.m. Matlab gives an error if A is empty and b is not and vice versa.

Example:
quadprog (H, f, A, [], Aeq, beq, lb, ub)
Error: The number of rows in A must be the same as the length of b. I have added this check in line 181 in [1].

- Lambda output as a structure instead of a vector as in qp.m.

Ordering of lambda:

• The order of lambda vector output (qp_lambda) from __qp__(in my code) is [equality constraints; inequality constraints; lower bounds; upper bounds]. 
• The multipliers are present if the constraints are given as inputs so the size of qp_lambda depends on the size of constraints. 
• Variables idx_ineq, idx_lb and idx_ub make sure I pick the right values. 

Example:

H = diag([1; 0]);

f = [3; 4];

A = [-1 -3; 2 5; 3 4];

b = [-15; 100; 80];

l = zeros(2,1);

[x, obj, info, qp_lambda] = qp ([], H, f, [], [],l,[],[], A, b)

[x,fval,exitflag,output,lambda] = quadprog (H, f, A, b,[],[],l,[])

qp_lambda =

   1.66667

   0.00000

   1.33333

   0.00000

   0.00000

lambda =

scalar structure containing the fields:

    lower =

       1.66667

       0.00000

    upper = [](0x0)

    eqlin = [](0x0)

    ineqlin =

       1.33333

       0.00000

       0.00000

Things to do:

• Check the sign issue for lambda.eqlin (qp gives values -1* Matlab's)
• Check if __qp__ changes the order of constraints. The values of lambda from qp.m in the last example in [2] are there but not coinciding with the respective constraints.
• Move on to optimoptions.

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

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

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