Generic optimality conditions for semialgebraic convex programs

Jérôme Bolte, Aris Daniilidis, Adrian S. Lewis

Research output: Contribution to journalArticleResearchpeer-review

26 Citations (Scopus)


We consider linear optimization over a nonempty convex semialgebraic feasible region F . Semidefinite programming is an example. If F is compact, then for almost every linear objective there is a unique optimal solution, lying on a unique "active" manifold, around which F is "partly smooth," and the second-order sufficient conditions hold. Perturbing the objective results in smooth variation of the optimal solution. The active manifold consists, locally, of these perturbed optimal solutions; it is independent of the representation of F and is eventually identified by a variety of iterative algorithms such as proximal and projected gradient schemes. These results extend to unbounded sets F. © 2011 INFORMS.
Original languageEnglish
Pages (from-to)55-70
JournalMathematics of Operations Research
Issue number1
Publication statusPublished - 1 Feb 2011


  • Active sets
  • Convex optimization
  • Generic
  • Identifiable surface
  • Partial smoothness
  • Second-order sufficient conditions
  • Semialgebraic
  • Sensitivity analysis


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