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Quadratic Programming with Equality and Inequality Constraints

Syntax	Qpro(level, G, g, A, a, neq, xlow, xup, xout, yAout, yLout, yUout)
See Also	sqp , Qbox

Description
Determines the value of the x that solves the problem
     minimize  (1/2) x' G x + g x  with respect to x
     such that A.row(i) x + a(i) = 0 for i = 1, ..., neq
               A.row(i) x + a(i) < 0 for i = neq + 1, ..., rowdim(A)
               xlow < x < xup
The Lagrangian for this problem is
     L(x, yA, yL, yU) = (1/2) x' G x  + g x
          + yA' (A x + a) + yL' (xlow - x) + yU' (xup - x)
The return value of Qpro is true if it succeeds in solving the problem and false otherwise.

The real or double-precision symmetric positive definite matrix G contains the quadratic term in the objective function. The row vector g specifies the linear term in the objective function. It has the same type and number of columns as G. The matrix A specifies the linear term in the constraints. It has the same type and column dimension as G. The column vector a specifies the constant term in the constraints. It has the same type and row dimension as A.

The input values of the parameters xout, yAout, yLout, and yUout do not matter. If the return value of Qpro is true, the output value of these parameters satisfy the Kuhn-Tucker condition in Equation (9.1.16) of Practical Methods of Optimization, namely:
1. The partial of L with respect to x is zero at the point (xout, yAout, yLout, yUout)
2. The constraints are satisfied at the point xout.
3. All of the elements of yLout and yUout are greater than zero. In addition, the elements of yAout with index greater than neq are greater than zero.
4. The following equations hold  
     0 = yAout' (- A x - a)
     0 = yLout' (x - xlow)
     0 = yUout' (xup - x)
Note that because all the terms in these products are non-negative, this implies each term is zero. The output value of xout, yLout and yUout are a column vectors with the same type and dimensions as xlow. The output value of yAout is a column vector with the same type and row dimension as A.

The parameter level is an integer scalar specifying the level of tracing inside of the Qpro function. If level > 1, the text "Beginning Qpro" and the value of G, g, A, a, xlow, and xup, are printed before attempting to solve this problem. If Qpro cannot solve the problem, the text "Qpro returns false" is printed before it returns. If Qpro does solve the problem, the output values of xout, yAout, yLout, and yUout are printed. In addition the partial of L with respect to x, and A xout + a are printed together with the text "Qpro returns true".

If level > 2, a tracing level of level - 1 is used in the call to Qbox that is used by Qpro.

Example
The program below solves the problem
     minimize   x(1)^2 + x(2)^2 + 2 x(1) + x(2)
     subject to - x(1) - x(2) < 0


clear
level = 1
G     = {[2., 0], [0, 2.]}
g     = [2., 1.]
A     = [-1., -1.]
a     = 0.
neq   = 0
xlow  = {-10., -10.}
xup   = {+10., +10.}
xout  = novalue
yAout = novalue
yLout = novalue
yUout = novalue
ok = Qpro(level, G, g, A, a, neq, xlow, xup, xout, yAout, yLout, yUout)
print "ok =", ok