refine(SDP,Sequence) -- refine an SDP solution

Synopsis

Function: refine
Usage:

(X1,y1) = refine(P,(X0,y0))
Inputs:
- P, an instance of the type SDP, a semidefinite programming problem
- X0y0, primal and dual solutions
Optional inputs:
- Bits => ..., default value null,
- ErrorTolerance => ..., default value null,
- Iterations => ..., default value null,
- ResidualTolerance => ..., default value null,
- SingularConditionNumber => ..., default value null,
- Software => ..., default value null,
Outputs:
- X1y1, refined primal and dual solutions

Description

This function uses Newton's method to improve the precision of an optimal primal/dual pair. The refined solution has relative error bounded by min(ErrorTolerance,2(-Bits)). The number of iterations made is at most Iterations.

i1 : P = sdp(matrix{{1,0},{0,2}}, matrix{{0,1},{1,0}}, matrix{{-1}});

i2 : (X0,y0) = (matrix{{.71, -.5}, {-.5, .35}}, matrix{{-1.41}})

o2 = (| .71 -.5 |, | -1.41 |)
      | -.5 .35 |

o2 : Sequence

i3 : (X1,y1) = refine(P,(X0,y0))

o3 = (| .707107 -.5     |, | -1.41421 |)
      | -.5     .353553 |

o3 : Sequence

Ways to use this method: