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)
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 |
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 |
factorized |
logical flag: if |
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. |
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.
## ## 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)
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