Quadratic Programming quadprog, qp

Just a post to review the things I've learned over the past week:

Intuitive explanation of Lagrange multipliers [1]. Given a D-dimensional function to optimize and a set of a few equality constraints.

• The vector normal to the function should be a scaled version of the vector normal to the constraint function (In other words, both normals are parallel to each other). This scaling factor is the Lagrange multiplier corresponding a particular constraint. 
• $\nabla f=\lambda_1\nabla g+\lambda_2 \nabla h$, where $g$ and $h$ are two constraint functions.

KKT Conditions: Generalized method of Lagrange multipliers to be applicable on Inequality constraints.  $g_i(x)-b_i\geq 0$

• Feasibility- $g_i(x^*)-b_i\geq 0$
• Stationarity- $\nabla f(x^*)-\sum\limits_i\lambda_i^*\nabla g_i(x^*)=0$
• Complementary Slackness $\lambda_i^*(g_i(x^*)-b_i)=0$
• Positive Lagrange multipliers $\lambda_i \geq 0, \forall i$

In case we obtain negative Lagrange multiplier, the constraint corresponding to the most negative multiplier is removed and the optimization is performed again until all multipliers are positive.

A bit about active set algorithm [2]:

• Possibility that none of the constraints are active or may be some are active. We only need to solve for equality constraints that are active at the optimum (binding).
• When we have an active set $S^*, x^* \in F$, where $F=\{x|A x^*\leq b\}$,$\lambda^* \geq0$, where $\lambda^*$ is the set of Lagrange multipliers for equality constraints $Ax=b$ 

Algorithm:

• Start at $x=x_0$ and initial active set
• Calculate $x^*_{EQP}, \lambda_{EQP}^*$ which minimizes the EQP defined by the current active set. Two possible outcomes:
1. $x^*_{EQP}$ is feasible  ($x_{EQP}^* \in F$). Set $x=x_{EQP}$ and check Lagrange mulitpliers $\lambda_{EQP}^*$. If positive, solution found! Otherwise, remove constraint with $\min(\lambda_{EQP}^*)$ and repeat.
2. $x^*_{EQP}$is infeasible. We move as far as possible along the line segment from $x_0$ to $x^*_{EQP}$ while staying feasible. Add to $S$ the constraint we encounter that prevents further progress. This is the blocking constraint.

Quadratic programming:

$\min\limits_{x}\frac{1}{2} x^THx + x^Tq$

s.t.

$A_{eq} x = b_{eq}$

$lb \leq x \leq ub$

$A_{lb} \leq A_{in}x \leq A_{ub}$

What qp.m does:

 [x, obj, info, lambda] = qp (x0, H, q, Aeq, beq, lb, ub, A_lb, A_in, A_ub)     

• Checks feasibility of initial guess $x_0$
• Checks size of inputs and that they make sense.
• Checks if bounds lb,ub too close or A_lb or A_ub too close. If they are very close then the inequality is treated as an equality constraint instead.
• Checks if any bound is set to Inf or -Inf. qp simply strikes it off.
• Calls backend solver __qp__ using null space active set algorithm. 

The ordering of lambda

• quadprog returns Lagrange multipliers in a structure (with fields upper, lower, eqlin, ineqlin) and the multipliers corresponding to the constraints not provided are left empty.  
• In qp, lambda is a column vector with Lagrange multipliers associated to the constraints in the following order: [equality constraints; lower bounds; upper bounds; other inequality constraints]. 
• The length of lambda vector output from qp depends on the number of different constraints provided as  input.
• Two issues in wrapping qp.m
1.  the order, i.e. the position of the bounds constraints within the inequality constraints) is not specified by qp.m. The code could change and the ordering too. 
2. qp.m strips off the INF constraints before calling __qp__ but does not process the lambda (returned by __qp__) accordingly.
• Solution:
• If this order is "specified" then we could extract parts of lambda. Patch for Inf checks in qp output lambda will make things easier but is not critical.

References
[1] http://www.slimy.com/~steuard/teaching/tutorials/Lagrange.html
[2] https://www.youtube.com/user/ucmath352/videos