solve.QP.compact function

Solve a Quadratic Programming Problem

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.compact(Dmat, dvec, Amat, Aind, bvec, meq=0, factorized=FALSE)

Arguments

• Dmat: matrix appearing in the quadratic function to be minimized.
• dvec: vector appearing in the quadratic function to be minimized.
• Amat: matrix containing the non-zero elements of the matrix $A$ that defines the constraints. If $m_i$ denotes the number of non-zero elements in the $i$-th column of $A$ then the first $m_i$ entries of the $i$-th column of Amat hold these non-zero elements. (If $maxmi$ denotes the maximum of all $m_i$, then each column of Amat may have arbitrary elements from row $m_i+1$ to row $maxmi$ in the $i$-th column.)
• Aind: matrix of integers. The first element of each column gives the number of non-zero elements in the corresponding column of the matrix $A$. The following entries in each column contain the indexes of the rows in which these non-zero elements are.
• 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.

Returns

a list with the following components: - solution: vector containing the solution of the quadratic programming problem.

• value: scalar, the value of the quadratic function at the solution

• unconstrained.solution: vector containing the unconstrained minimizer of the quadratic function.

• iterations: 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.

• Lagrangian: vector with the Lagragian at the solution.

• iact: vector with the indices of the active constraints at the solution.

References

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.

See Also

solve.QP

Examples

##
## 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.compact as follows:
##
Dmat       <- matrix(0,3,3)
diag(Dmat) <- 1
dvec       <- c(0,5,0)
Aind       <- rbind(c(2,2,2),c(1,1,2),c(2,2,3))
Amat       <- rbind(c(-4,2,-2),c(-3,1,1))
bvec       <- c(-8,2,0)
solve.QP.compact(Dmat,dvec,Amat,Aind,bvec=bvec)