Adjusted Sublevel Sets, Normal Operator, and Quasi-convex Programming
Didier Aussel, Nicolas Hadjisavvas · SIAM Journal on Optimization · 2005
A new notion of "adjusted sublevel set" of a function is introduced and studied. These sets lie between the sublevel and strict sublevel sets of the function. In contrast to the normal operators to sublevel or strict sublevel sets that were studied in the literature so far, the normal operator to the adjusted sublevel sets is both quasi-monotone and, in the case of quasi-convex functions, cone upper-semicontinuous. This makes this new notion appropriate for all kinds of quasi-convex functions and, in particular, for quasi-convex functions whose graph presents a "flat part." Application is given to quasi-convex optimization through the study of an associated variational inequality problem.