This routine implements the dual method of Goldfarb and Idnani (1982, 1983) for solving quadratic programming problems of the form \(\min(-d^T b + 1/2 b^T D b)\) with the constraints \(A^T b >= b_0\).

`solve.QP(Dmat, dvec, Amat, bvec, meq=0, factorized=FALSE)`

Dmat

matrix appearing in the quadratic function to be minimized.

dvec

vector appearing in the quadratic function to be minimized.

Amat

matrix defining the constraints under which we want to minimize the quadratic function.

bvec

vector holding the values of \(b_0\) (defaults to zero).

meq

the first `meq`

constraints are treated as equality
constraints, all further as inequality constraints (defaults to 0).

factorized

logical flag: if `TRUE`

, then we are passing
\(R^{-1}\) (where \(D = R^T R\)) instead of the matrix
\(D\) in the argument `Dmat`

.

a list with the following components:

vector containing the solution of the quadratic programming problem.

scalar, the value of the quadratic function at the solution

vector containing the unconstrained minimizer of the quadratic function.

vector of length 2, the first component contains the number of iterations the algorithm needed, the second indicates how often constraints became inactive after becoming active first.

vector with the Lagragian at the solution.

vector with the indices of the active constraints at the solution.

D. Goldfarb and A. Idnani (1982). Dual and Primal-Dual Methods for Solving Strictly Convex Quadratic Programs. In J. P. Hennart (ed.), Numerical Analysis, Springer-Verlag, Berlin, pages 226--239.

D. Goldfarb and A. Idnani (1983).
A numerically stable dual method for solving strictly convex quadratic
programs.
*Mathematical Programming*, **27**, 1--33.

# NOT RUN { ## ## Assume we want to minimize: -(0 5 0) %*% b + 1/2 b^T b ## under the constraints: A^T b >= b0 ## with b0 = (-8,2,0)^T ## and (-4 2 0) ## A = (-3 1 -2) ## ( 0 0 1) ## we can use solve.QP as follows: ## Dmat <- matrix(0,3,3) diag(Dmat) <- 1 dvec <- c(0,5,0) Amat <- matrix(c(-4,-3,0,2,1,0,0,-2,1),3,3) bvec <- c(-8,2,0) solve.QP(Dmat,dvec,Amat,bvec=bvec) # }