Equal Angle Distribution of Polling Directions in Direct-Search Methods

Algorithm 3

The EADGSS algorithm.

(0) Initialization

   𝑥₀∈Ω,₀←0,𝑘←0,𝑡₀←1

(1) Iteration  𝑘

   SEARCH (optional)

     evaluate 𝑓 on a finite set of points 𝑆_𝑘

   POLL (optional if SEARCH is successful)

     if the POLL size is the smallest one so far

       𝑡_𝑘←_𝑘+𝑡₀

     else

       𝑡_𝑘←1+max{𝑡_𝑗∶𝑗=0,1,…,𝑘}

     find 𝑐_𝑡𝑘,

     compute 𝑄_𝑘 and 𝐷_𝑘=[𝑄_𝑘−𝑄_𝑘] or 𝑉_𝑘 and 𝐷_𝑘=[𝑉_𝑘−∑^𝑛_𝑖=1𝑣_𝑖],

     evaluate 𝑓 on 𝑃_𝑘={𝑥_𝑘+2^−𝑘𝑑∶𝑑∈𝐷_𝑘}

(2) Updates

   if the iteration is successful (i.e. there exists 𝑥_𝑠∈𝑆_𝑘 or 𝑥_𝑝=𝑥_𝑘+2^𝑘𝑑∈𝑃_𝑘

    such that 𝑓(𝑥_𝑠)<𝑓(𝑥_𝑘)−𝜌(2^−𝑘) or 𝑓(𝑥_𝑝)<𝑓(𝑥_𝑘)−𝜌(2^𝑘‖𝑑‖).)

     𝑥_𝑘+1←𝑥_𝑠 or 𝑥_𝑝

     _𝑘+1←_𝑘−1

   else (the iteration fails)

     𝑥_𝑘+1←𝑥_𝑘

     _𝑘+1←_𝑘+1

   𝑘←𝑘+1

   goto (1) if stopping criteria are not met