R: Second-order Cone Programming

Socp {parma}

R Documentation

Second-order Cone Programming

Description

The function solves second-order cone problem by primal-dual interior point method. It is a wrapper function to the C-routines written by Lobo, Vandenberghe and Boyd (see reference below).

Usage

Socp(f, A, b, C, d, N,
     x = NULL, z = NULL, w = NULL, control = list())

Arguments

`f`	Vector defining linear objective, `length(f)==length(x)`
`A`	Matrix with the `A_i` vertically stacked: `A = [ A_1; A_2; \ldots; A_L]`.
`b`	Vector with the `b_i` vertically stacked: `b = [ b_1; b_2; \ldots; b_L]`.
`C`	Matrix with the `c_i'` vertically stacked: `C = [ c_1'; c_2'; \ldots; c_L']`.
`d`	Vector with the `d_i` vertically stacked: `d = [ d_1; d_2; \ldots; d_L]`.
`N`	Vector of size `L`, defining the size of each constraint.
`x`	Primal feasible initial point. Must satisfy: `\|\| A_ix + b_i \|\| < c_i' x + d_i` for `i = 1, \ldots, L`.
`z`	Dual feasible initial point.
`w`	Dual feasible initial point.
`control`	A list of control parameters.

Details

The primal formulation of an SOCP is given as:

minimise f' * x

subject to

||A_i*x + b_i|| <= c_i' * x + d_i

for i = 1,\ldots, L. Here, x is the (n \times 1) vector to be optimised. The dual form of an SOCP is expressed as:

maximise \sum_{i = 1}^L -(b' * z_i + d_i * w_i)

subject to

\sum_{i = 1}^L (A_i' * z_i + c_i * w_i) = f

and

||z_i || = w_i

for i = 1,\ldots, L, given strictly feasible primal and dual initial points.

The algorithm stops, if one of the following criteria is met:

abs.tol – maximum absolute error in objective function; guarantees that for any x: abs(f'*x - f'*x\_opt) <= abs\_tol.
rel.tol – maximum relative error in objective function; guarantees that for any x: abs(f'*x - f'*x\_opt)/(f'*x\_opt) <= rel\_tol (if f'*x\_opt > 0). Negative value has special meaning, see target next.
target – if rel\_tol<0, stops when f'*x < target or -b'*z >= target.
max.iter – limit on number of algorithm outer iterations. Most problems can be solved in less than 50 iterations. Called with max_iter = 0 only checks feasibility of x and z, (and returns gap and deviation from centrality).
The target value is reached. rel\_tol is negative and the primal objective p is less than the target.

Value

A list-object with the following elements:

`x`	Solution to the primal problem.
`z`	Solution to the dual problem.
`iter`	Number of iterations performed.
`hist`	see `out_mode` in `SocpControl`.
`convergence`	A logical code. `TRUE` indicates successful convergence.
`info`	A numerical code. It indicates if the convergence was successful.
`message`	A character string giving any additional information returned by the optimiser.

Note

This function has been ported from the Rsocp package contained in the Rmetrics-Project on R-Forge. In contrast to the former implementation, allowance is made for specifying more than one cone constraint.

Author(s)

Bernhard Pfaff

References

Lobo, M. and Vandenberghe, L. and Boyd, S., SOCP: Software for Second-order Cone Programming, User's Guide, Beta Version, April 1997, Stanford University.