Equal Angle Distribution of Polling Directions in Direct-Search Methods

Algorithm 1

The directional direct-search method of [1].

(0) Initialization

   𝑥₀∈Ω,𝛼₀>0,0<𝛽₁≤𝛽₂<1,𝛾≥1

   let 𝒟 be a (possibly infinite) set of positive spanning sets

(1) Iteration  𝑘

   SEARCH

     evaluate 𝑓 on a finite set of points 𝑆_𝑘 (on a mesh if  𝜌=0)

   POLL (optional if SEARCH is successful)

     Choose 𝐷_𝑘∈𝒟, order the set of POLL points

     𝑃_𝑘={𝑥_𝑘+𝛼_𝑘𝑑∶𝑑∈𝐷_𝑘}, and evaluate 𝑓 on 𝑃_𝑘

(2) Updates

   if the iteration is successful (i.e. there exists 𝑥_𝑠∈𝑆_𝑘 or 𝑥_𝑝∈𝑃_𝑘

    such that 𝑓(𝑥_𝑠)<𝑓(𝑥_𝑘)−𝜌(𝛼_𝑘) or 𝑓(𝑥_𝑝)<𝑓(𝑥_𝑘)−𝜌(𝛼_𝑘𝑑_𝑘).)

     𝑥_𝑘+1←𝑥_𝑠 or 𝑥_𝑝

     maintain or increase the step size: 𝛼_𝑘∈[𝛼_𝑘,𝛾𝛼_𝑘]

   else (the iteration fails)

     𝑥_𝑘+1←𝑥_𝑘

     decrease the step size: 𝛼_𝑘∈[𝛽₁𝛼_𝑘,𝛽₂𝛼_𝑘]

   𝑘←𝑘+1

   goto (1) if stopping criteria are not met