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

.