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).
Socp(f, A, b, C, d, N,
x = NULL, z = NULL, w = NULL, control = list())Vector defining linear objective, length(f)==length(x)
Matrix with the \(A_i\) vertically stacked: \(A = [ A_1; A_2; \ldots; A_L]\).
Vector with the \(b_i\) vertically stacked: \(b = [ b_1; b_2; \ldots; b_L]\).
Matrix with the \(c_i'\) vertically stacked: \(C = [ c_1'; c_2'; \ldots; c_L']\).
Vector with the \(d_i\) vertically stacked: \(d = [ d_1; d_2; \ldots; d_L]\).
Vector of size L, defining the size of each constraint.
Primal feasible initial point. Must satisfy: \(|| A_i*x + b_i || < c_i' * x + d_i\) for \(i = 1, \ldots, L\).
Dual feasible initial point.
Dual feasible initial point.
A list of control parameters.
A list-object with the following elements:
Solution to the primal problem.
Solution to the dual problem.
Number of iterations performed.
see out_mode in SocpControl.
A logical code. TRUE indicates successful
convergence.
A numerical code. It indicates if the convergence was successful.
A character string giving any additional information returned by the optimiser.
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.
Lobo, M. and Vandenberghe, L. and Boyd, S., SOCP: Software for Second-order Cone Programming, User's Guide, Beta Version, April 1997, Stanford University.