Convexity and well-posed problems

By R. Lucchetti. 2006

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Strong uniform continuity (2009, background): “13] as introduced in [25], a general notion of set convergence that subsumes convergence in Hausdorff distance [7,23,27], Attouch–Wets convergence [4,6,7,27], and when restricted to the closed subsets of the space, convergence with respect to the Fell topology [7, Theorem 5.”
Error Bounds and Hölder Metric Subregularity (2014, background): “orm strict q-slope (24). Let x > 0, y = x2, and ρ > 0. Then the nonlocal (q,ρ)-slope (23) of F at (x,y) takes the following form: |∇F|⋄q,ρ (x,y) = sup u6= x [x −u+] + max{|u −x|,ρ|u2 + − x2|} . (49) If u > x, then x− u+ = x − u < 0 and the expression under sup in the right-hand side of (”
Convex Functions: Constructions, Characterizations and Counterexamples (2010, background): “2 (A limiting sandwich example from [297]).”
The Exact Feasibility of Randomized Solutions of Uncertain Convex Programs (2008, background): “To each multiextraction ( δ (1) , . . . , δ ( m ) ) ∈ Δ m , associate the indexes of the corresponding d support constraints (this is always possible except for a probability 0 set because the problem is fully-supported ).”
Strongly Convex Functions, Moreau Envelopes, and the Generic Nature of Convex Functions with Strong Minimizers (2015, background): “While Beer and Lucchetti obtained a similar result on generic well-posedness of convex optimization, their approach relies on epigraphs of convex functions [7, 8, 9].”