Optimization Problems with Nearly Convex Objective Functions and Nearly Convex Constraint Sets

To every nearly convex optimization problem, that is a minimization problem with a nearly convex objective function and a nearly convex constraint set, we associate a uniquely defined convex optimization problem with a lower semicontinuous objective function and a closed constraint set. Interesting relationships between the original nearly convex problem and the associated convex problem are established. Optimality conditions in the form of Fermat’s rules are obtained for both problems. We then get a Lagrange multiplier rule for a nearly convex optimization problem under a geometrical constraint and functional constraints from the Kuhn-Tucker conditions for the associated convex optimization problem. The obtained results are illustrated by concrete examples.

💡 Research Summary

The paper investigates optimization problems whose objective functions and constraint sets are “nearly convex” – a notion that generalizes classical convexity. Two distinct definitions of nearly convex sets are examined: Minty’s definition (a set Ω lies between a convex set C and its closure (\bar C)) and Ho’s pointwise definition (for any two points in Ω there exists a sequence of points on the open segment approaching one endpoint while staying in Ω). Through a series of examples the authors demonstrate that these definitions are independent; a set may satisfy one but not the other. Throughout the rest of the work, the Minty-type definition is adopted.

A function is called nearly convex if its epigraph is a nearly convex set. The authors recall known properties: the effective domain of a nearly convex function is itself nearly convex, the sum of two nearly convex functions is nearly convex provided their relative interiors intersect, and the subdifferential sum rule holds under the same interior‑intersection condition. Max‑type functions preserve near convexity, and their subdifferential can be expressed as the convex hull of the subdifferentials of the active functions.

The central contribution is the construction of a uniquely associated convex‑type problem for any given nearly convex problem. Given a nearly convex objective (f) and a nearly convex feasible set (D), the authors define \