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

Algorithm 2

SDP-CUT.

INPUT: 𝑥^∗0∈int(ℛ), weight 𝑤∈ℝ₊, 𝜖>0, STOL>0, and MAX

Set 𝑘=0

While  𝑘<MAX  do

 1.Compute cutting plain constraint 𝑐^𝑇𝑥−𝑐^𝑇𝑥^∗𝑘+𝜖≥0

 2.Let 𝑥^∗𝑘+1 be the 𝑤𝑒𝑖𝑔ℎ𝑡𝑒𝑑𝑎𝑛𝑎𝑙𝑦𝑡𝑖𝑐𝑐𝑒𝑛𝑡𝑒𝑟 of new system with barrier function 𝜙^∗𝑤(𝑥)

 (3.1) to be computed by WAC-NEWTON* starting from the point 𝑥^∗𝑘

 Compute 𝜕_𝑘+1=𝑐^𝑇𝑥^∗𝑘−𝑐^𝑇𝑥^∗𝑘+1

 if  𝜕_𝑘<STOL  then

  break while

 3.𝑘←𝑘+1

OUTPUT: 𝑥^∗_cut=𝑥^∗𝑘, 𝑝^∗_cut=𝑐^𝑇𝑥^∗𝑘