The Flattened Aggregate Constraint Homotopy Method for Nonlinear Programming Problems with Many Nonlinear Constraints

Algorithm 1

The FACH-S-N procedure.

Input  Give 𝜃∈(0,1], 𝑐₁>0, 𝑐₂>0, 𝛼>1, 𝑡_end and 𝑡_𝑐; starting point 𝑤⁰∈Ω×𝑅₊₊; initial steplength ℎ₁, steplength

   contraction factor 𝐵_min, steplength expansion factor 𝐵_max; tracking tolerances 𝐻_tol and 𝐻_final for correction. Set

   𝑡₀=1, 𝑘=1, IT=0, 𝑁_good=2, 𝛽¹=0.

Step  1. The predictor step.

  Step  1.1. Compute the predictor direction.

     If 𝑘=1, compute 𝑑 by solving

                   𝐻^′𝑥⁰,𝜆⁰𝑑,1_0𝑇𝑑=−𝑑⁰,

     where 𝑑⁰=(0,…,0,−1)∈𝑅^𝑛+2, set the predictor direction 𝑑¹=𝑑/‖𝑑‖;

     else, compute the predictor direction 𝑑^𝑘, the angle 𝛽^𝑘 between 𝑑^𝑘 and 𝑑^𝑘−1:

              𝑑^𝑘=(𝑤^𝑘−1,𝑡_𝑘−1)−(𝑤^𝑘−2,𝑡_𝑘−2)‖(𝑤^𝑘−1,𝑡_𝑘−1)−(𝑤^𝑘−2,𝑡_𝑘−2)‖,𝛽^𝑘=arccos𝑑^𝑘𝑇𝑑^𝑘−1.

  Step  1.2. Adjust the steplength.

       Adjust the steplength ℎ_𝑘 as follows:

       if the predictor point is infeasible, set ℎ_𝑘=𝐵_min(1)ℎ_𝑘, 𝑁_good=0;

       else if the corrector step fails or 𝛽^𝑘>𝜋/4, set ℎ_𝑘=𝐵_min(1)ℎ_𝑘−1, 𝑁_good=0;

       else if 𝑖=5, set ℎ_𝑘=𝐵_min(2)ℎ_𝑘−1, 𝑁_good=0;

       else if 𝑖=4, set ℎ_𝑘=ℎ_𝑘−1, 𝑁_good=𝑁_good+1;

       else if 𝑖=3, set 𝑁_good=𝑁_good+1,

         if 𝑁_good>2, set ℎ_𝑘=min{1,𝐵_max(2)ℎ_𝑘−1}, else, set ℎ_𝑘=ℎ_𝑘−1;

       else, set 𝑁_good=𝑁_good+1,

         if 𝑁_good>2, set ℎ_𝑘=min{1,𝐵_max(1)ℎ_𝑘−1}, else, set ℎ_𝑘=ℎ_𝑘−1.

      If ℎ_𝑘<10⁻¹⁰, stop the algorithm with an error flag.

  Step  1.3. Compute the predictor point and check its feasibility.

        Compute the predictor point

                  𝑤^(𝑘,0),𝑡_(𝑘,0)=𝑤^𝑘−1,𝑡_𝑘−1+ℎ_𝑘𝑑^𝑘.

     If 𝑤^(𝑘,0)∉Ω_𝜃(𝑡_(𝑘,0))⁰×𝑅₊₊ or 𝑡_(𝑘,0)≥1,

       the predictor point is infeasible, goto Step  1.2;

     else if 𝑡_(𝑘,0)≤𝑡_end,

       adjust the steplength ℎ_𝑘, compute the point (𝑤^(𝑘,0),0), if 𝑤^(𝑘,0)∈Ω×𝑅₊, goto Step  3, else, the predictor

       point is infeasible, goto Step  1.2;

     else, goto Step  2.

Step  2. The corrector step.

   Set 𝑖=0.

   Repeat

     If 𝑖=5, set IT=IT+5, 𝑤^𝑘=𝑤^𝑘−1, 𝑡_𝑘=𝑡_𝑘−1, 𝑑^𝑘+1=𝑑^𝑘, 𝑘=𝑘+1, the corrector step fails, goto Step  1.2.

     Compute the Newton step 𝑑^{(𝑘,𝑖+1)} by solving

               𝐻^′𝑤^(𝑘,𝑖),𝑡_(𝑘,𝑖)𝑑^𝑘𝑇𝑑^{(𝑘,𝑖+1)}=−𝐻𝑤^(𝑘,𝑖),𝑡_(𝑘,𝑖)0,

     the corrector point

               𝑤^{(𝑘,𝑖+1)},𝑡_{(𝑘,𝑖+1)}=𝑤^(𝑘,𝑖),𝑡_(𝑘,𝑖)+𝑑^{(𝑘,𝑖+1)};

     set 𝑖=𝑖+1.

     If 𝑤^(𝑘,𝑖)∉Ω_𝜃(𝑡_(𝑘,𝑖))⁰×𝑅₊₊ or 𝑡_(𝑘,𝑖)≥1,

       set IT=IT+𝑖, 𝑤^𝑘=𝑤^𝑘−1, 𝑡_𝑘=𝑡_𝑘−1, 𝑑^𝑘+1=𝑑^𝑘, 𝑘=𝑘+1, the corrector step fails, goto Step  1.2;

     else if 𝑡_(𝑘,𝑖)<0,

       compute the point (𝑤^(𝑘,0),0) by a damping Newton step, set IT=IT+𝑖, if 𝑤^(𝑘,0)∈Ω×𝑅₊, goto Step  3,

       else, set 𝑤^𝑘=𝑤^𝑘−1, 𝑡_𝑘=𝑡_𝑘−1, 𝑑^𝑘+1=𝑑^𝑘, 𝑘=𝑘+1, the corrector step fails, goto Step  1.2.

  Until ‖𝐻(𝑤^(𝑘,𝑖),𝑡_(𝑘,𝑖))‖_inf≤𝐻_tol and ‖𝑑^(𝑘,𝑖)‖≤𝐻_tol.

  If 𝑡_𝑘<𝑡_𝑐, return with 𝑥^∗=𝑥^𝑘, stop the algorithm; else, set 𝐻_tol=min{𝐻_tol,𝑡_𝑘}, 𝑘=𝑘+1, goto Step  1.

Step  3. The end game.

   Set 𝑙=0.

   Repeat

     If 𝑙=5 or ‖𝑑^(𝑘,𝑙)‖>‖𝑑^{(𝑘,𝑙−1)}‖ for 𝑙>1, or 𝑤^(𝑘,𝑙)∉Ω×𝑅₊, set 𝑡_end=0.3∗𝑡_end, 𝑤^𝑘=𝑤^𝑘−1, 𝑡_𝑘=𝑡_𝑘−1,

     𝑑^𝑘+1=𝑑^𝑘, IT=IT+𝑙, 𝑘=𝑘+1, the corrector step fails, goto Step  1.2.

     Compute the Newton step 𝑑^{(𝑘,𝑙+1)} by solving 𝐹^′_𝑡𝑐(𝑤^(𝑘,𝑙))𝑑^{(𝑘,𝑙+1)}=−𝐹_𝑡𝑐(𝑤^(𝑘,𝑙)), the corrector point

                   𝑤^{(𝑘,𝑙+1)}=𝑤^(𝑘,𝑙)+𝑑^{(𝑘,𝑙+1)};

     set 𝑙=𝑙+1.             

  Until ‖𝐹_𝑡𝑐(𝑤^(𝑘,𝑙))‖_inf≤𝐻_final and ‖𝑑^(𝑘,𝑙)‖≤𝐻_final.             

  Set IT=IT+𝑙, return with 𝑥^∗=𝑥^(𝑘,𝑙), stop the algorithm.