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