The Collaborative International Dictionary of English v.0.48:

minimize \min"i*mize\ (m[i^]n"[i^]*m[imac]z), v. t. [imp. & p.
   p. minimized (m[i^]n"[i^]*m[imac]zd); p. pr. & vb. n.
   minimizing (m[i^]n"[i^]*m[imac]*z[i^]ng).]
   1. To reduce to the smallest part or proportion possible; to
      reduce to a minimum. --Bentham.
      [1913 Webster]

   2. To represent, to oneself or others, as of no importance or
      minimal importance or effect; to belittle or disarage.
      [PJC]

WordNet (r) 3.0 (2006):

minimize
    v 1: make small or insignificant; "Let's minimize the risk"
         [syn: minimize, minimise] [ant: maximise, maximize]
    2: represent as less significant or important [syn:
       understate, minimize, minimise, downplay] [ant:
       amplify, exaggerate, hyperbolise, hyperbolize,
       magnify, overdraw, overstate]
    3: cause to seem less serious; play down; "Don't belittle his
       influence" [syn: minimize, belittle, denigrate,
       derogate]

Moby Thesaurus II by Grady Ward, 1.0:

64 Moby Thesaurus words for "minimize":
   abbreviate, abridge, bedwarf, belittle, bring down,
   bring into discredit, bring low, cry down, curtail, cut down,
   de-emphasize, debase, decrease, decry, degrade, deprecate,
   depreciate, derogate, derogate from, detract from, devaluate,
   devalue, diminish, disapprove of, discount, discredit, disgrace,
   disparage, dispraise, downplay, dwarf, hold in contempt, knock,
   lessen, make light of, make little of, make nothing of, minify,
   misestimate, misprize, play down, prune, put down, reduce,
   reflect discredit upon, run down, sell short, set at naught,
   set little by, shorten, shrink, shrug off, slight, speak ill of,
   submit to indignity, talk down, think little of, think nothing of,
   underestimate, underplay, underprize, underrate, underreckon,
   undervalue