An Interior Point Method for Solving Semidefinite Programs Using Cutting Planes and Weighted Analytic Centers

Algorithm 1

WAC-NEWTON.

INPUT: point 𝑦₀∈int(ℛ), weight vector 𝜔∈ℝ^𝑞, tolerance WTOL>0, and maximum

number of iterations MAX

Set 𝑙=0

while  𝑙<MAX  do

 1.Compute the direction vector 𝑠_𝑙=−(∇²𝜙_𝜔(𝑦_𝑙))⁻¹(∇𝜙_𝜔(𝑦_𝑙))

 2.Compute the Newton decrement 𝑑=𝑠^𝑇𝑙(∇²𝜙_𝜔(𝑦_𝑙))𝑠_𝑙

 3.Compute stepsize 𝛼_𝑙

 4.𝑦_𝑙+1=𝑦_𝑙+𝛼_𝑙𝑠_𝑙

 if  𝑑<WTOL  then

  break while

 5.𝑙←𝑙+1

OUTPUT: 𝑥_ac(𝜔)=𝑦_𝑙