ipop function

Quadratic Programming Solver

ipop solves the quadratic programming problem :

$\min(c'*x + 1/2 * x' * H * x)$

subject to:

$b <= A * x <= b + r$

$l <= x <= u$

ipop(c, H, A, b, l, u, r, sigf = 7, maxiter = 40, margin = 0.05,
     bound = 10, verb = 0)

Arguments

• c: Vector or one column matrix appearing in the quadratic function
• H: square matrix appearing in the quadratic function, or the decomposed form $Z$ of the $H$ matrix where $Z$ is a $n x m$ matrix with $n > m$ and $ZZ' = H$.
• A: Matrix defining the constrains under which we minimize the quadratic function
• b: Vector or one column matrix defining the constrains
• l: Lower bound vector or one column matrix
• u: Upper bound vector or one column matrix
• r: Vector or one column matrix defining constrains
• sigf: Precision (default: 7 significant figures)
• maxiter: Maximum number of iterations
• margin: how close we get to the constrains
• bound: Clipping bound for the variables
• verb: Display convergence information during runtime

Details

ipop uses an interior point method to solve the quadratic programming problem.

The $H$ matrix can also be provided in the decomposed form $Z$

where $ZZ' = H$ in that case the Sherman Morrison Woodbury formula is used internally.

Returns

An S4 object with the following slots - primal: Vector containing the primal solution of the quadratic problem

• dual: The dual solution of the problem

• how: Character string describing the type of convergence

all slots can be accessed through accessor functions (see example)

References

R. J. Vanderbei

LOQO: An interior point code for quadratic programming

Optimization Methods and Software 11, 451-484, 1999

https://vanderbei.princeton.edu/ps/loqo5.pdf

Author(s)

Alexandros Karatzoglou (based on Matlab code by Alex Smola)

alexandros.karatzoglou@ci.tuwien.ac.at

See Also

solve.QP, inchol, csi

Examples

## solve the Support Vector Machine optimization problem
data(spam)

## sample a scaled part (500 points) of the spam data set
m <- 500
set <- sample(1:dim(spam)[1],m)
x <- scale(as.matrix(spam[,-58]))[set,]
y <- as.integer(spam[set,58])
y[y==2] <- -1

##set C parameter and kernel
C <- 5
rbf <- rbfdot(sigma = 0.1)

## create H matrix etc.
H <- kernelPol(rbf,x,,y)
c <- matrix(rep(-1,m))
A <- t(y)
b <- 0
l <- matrix(rep(0,m))
u <- matrix(rep(C,m))
r <- 0

sv <- ipop(c,H,A,b,l,u,r)
sv
dual(sv)