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A perturbation view of level-set methods for convex optimization. (English) Zbl 07311801
Summary: Level-set methods for convex optimization are predicated on the idea that certain problems can be parameterized so that their solutions can be recovered as the limiting process of a root-finding procedure. This idea emerges time and again across a range of algorithms for convex problems. Here we demonstrate that strong duality is a necessary condition for the level-set approach to succeed. In the absence of strong duality, the level-set method identifies $$\epsilon$$-infeasible points that do not converge to a feasible point as $$\epsilon$$ tends to zero. The level-set approach is also used as a proof technique for establishing sufficient conditions for strong duality that are different from Slater’s constraint qualification.
##### MSC:
 90C25 Convex programming 90C46 Optimality conditions and duality in mathematical programming
##### Keywords:
convex analysis; duality; level-set methods
##### Software:
mctoolbox; PDCO; PhaseLift; SPGL1
Full Text:
##### References:
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