Geodesic
An upper bound for a condition number theorem of variational inequalities
Serdica Mathematical Journal, Tome 49 (2023) no. 1–3, pp. 33-48

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Nonlinear variational inequalities in Banach spaces are considered. A notion of (absolute) condition number with respect to the right-hand side is introduced. A distance among variational inequalities is defined. We prove that the distance to suitably restricted ill-conditioned variational inequalities is bounded from above by a multiple of the reciprocal of the condition number. By using an analogous lower bound of the companion paper [14], we obtain a full condition number theorem for variational inequalities. The particular case of convex optimization problems is also considered. Known results dealing with optimization problems are thereby generalized.
Keywords: variational inequalities, condition number theorems, conditioning in convex optimization, 49J40, 49K40, 49J53, 90C31 
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Zolezzi, Tullio. An upper bound for a condition number theorem of variational inequalities. Serdica Mathematical Journal, Tome 49 (2023) no. 1–3, pp. 33-48. http://geodesic.mathdoc.fr/item/SMJ2_2023_49_1_3_a2/