Journal of Optimization / 2015 / Article / Alg 1

Research Article

Constraint Consensus Methods for Finding Strictly Feasible Points of Linear Matrix Inequalities

Algorithm 1

Original-DBmax (OD) constraint consensus algorithm.
INPUT: An initial point , a feasibility distance tolerance , a movement tolerance , maximum number of iterations and
Phase 1: Do the original basic constraint consensus method, Algorithm 1 in [9], using the parameters , and
and starting from to get a near-feasible point.
Let be the last iterate of Phase 1. (Remark: is the starting point of Phase 2.)
Phase 2: (uses DBmax consensus method directions given in [9])
Set
Set
While   and is infeasible  do
 Set , , , , for each variable
for every constraint   do
  if constraint is violated then
   Calculate feasibility vector
   for every variable in th constraint do
    if    then
     
     if    then
      
    else if    then
     
     if    then
      
for every variable : do
  if    then
   
  else if    then
   
  else
   
Determine the LMI crossing points , with , on the consensus ray , and let denote
the constraint of the crossing point .
If there are no crossing points (i.e. ), set .
Set .
Set
Set
Set
for    do
 Update by flipping , the th bit of
if    then
  replace with
  Set
Set
If , then is feasible
.