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.
- Active sets
- Convex optimization
- Identifiable surface
- Partial smoothness
- Second-order sufficient conditions
- Sensitivity analysis
Bolte, J., Daniilidis, A., & Lewis, A. S. (2011). Generic optimality conditions for semialgebraic convex programs. Mathematics of Operations Research, 36(1), 55-70. https://doi.org/10.1287/moor.1110.0481