Nomenclature of an alkyne with bromo and hydroxy substituents
In this case, the relevant criteria for the numbering are:


lower locants for suffixes (here: ‘ol’)
lower locants for multiple bonds (here: ‘yne’)
lower locants for prefixes (here: ‘bromo’)


The corresponding subsection of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book) reads as follows:


  P-14.4 NUMBERING
  
  When several structural features appear in cyclic and acyclic compounds, low locants are assigned to them in the following decreasing order of seniority:
  
  (…)
  
  (c) principal characteristic groups and free valences (suffixes);
  
  (…)
  
  (e) saturation/unsaturation:
  
    (i) low locants are given to hydro/dehydro prefixes (…) and ‘ene’ and ‘yne’ endings;
  
    (ii) low locants are given first to multiple bonds as a set and then to double bonds (…);
  
  (f) detachable alphabetized prefixes, all considered together in a series of increasing numerical order;
  
  (…)


Therefore, the correct name for the compound given in the question is ‘7,7-dibromooct-5-yn-4-ol’ (not ‘2,2-dibromooct-3-yn-5-ol’) since the locant ‘4’ for the suffix ‘ol’ is lower than ‘5’.