Constraint Consensus Methods for Finding Interior Feasible Points in Second-Order Cones

Algorithm 2

DBmax constraint consensus algorithm.

INPUTS: set of 𝑞 constraints (1.1), an initial point 𝑥, a feasibility distance tolerance 𝛼,

a movement tolerance 𝛽, maximum number of iterations 𝜇.

while steps <𝜇  do

 𝑁𝐼𝑁𝐹=0, for all 𝑗∶𝑛⁺_𝑗=0,  𝑛⁻_𝑗=0,  𝑠⁺_𝑗=0,  𝑠⁻_𝑗=0

 for every constraint 𝑖  do

  if constraint 𝑖 is violated then

   Calculate feasibility vector 𝑓𝑣_𝑖 and feasibility distance ‖𝑓𝑣_𝑖‖

   if  ‖𝑓𝑣_𝑖‖>𝛼  then

    𝑁𝐼𝑁𝐹=𝑁𝐼𝑁𝐹+1

    for every variable 𝑥_𝑗  do

     if  𝑓𝑣_𝑖𝑗>0  then

      𝑛⁺_𝑗←𝑛⁺_𝑗+1

      if  𝑓𝑣_𝑖𝑗>𝑠⁺_𝑗  then

       𝑠⁺_𝑗←𝑓𝑣_𝑖𝑗

     else if  𝑓𝑣_𝑖𝑗<0  then

      𝑛⁻_𝑗←𝑛⁻_𝑗+1

      if  𝑓𝑣_𝑖𝑗<𝑠⁻_𝑗  then

       𝑠⁻_𝑗←𝑓𝑣_𝑖𝑗

 if  𝑁𝐼𝑁𝐹=0  then

  exit successfully

 for every variable 𝑥_𝑗: do

  if  𝑛⁺=𝑛⁻  then

   𝑡_𝑗=(𝑠⁺_𝑗+𝑠⁻_𝑗)/2

  else if  𝑛⁺_𝑗>𝑛⁻_𝑗  then

   𝑡_𝑗=𝑠⁺_𝑗

  else

   𝑡_𝑗=𝑠⁻_𝑗

 if  ‖𝑡‖≤𝛽  then

  exit unsuccessfully

 𝑥=𝑥+𝑡, steps = steps + 1