Searching for, and quantifying, non-convexity of functions

TL;DR

This paper introduces a method to identify convex regions and quantify non-convexity in functions by decomposing symmetric matrices like the Hessian, with applications demonstrated in finance and insurance.

Contribution

It proposes a novel decomposition-based approach to analyze convexity and non-convexity in functions, providing practical tools for complex problem domains.

Findings

Effective identification of convex regions in functions

Quantification of non-convexity in various examples

Application to risk measurement in finance and insurance

Abstract

Convexity plays a prominent role in a number of problems, but practical considerations frequently give rise to non-convex functions. We suggest a method for determining convex regions, and also for assessing the lack of convexity in the other regions. The method relies on a specially constructed decomposition of symmetric matrices, such as the Hessian. We illustrate theoretical results using several examples, one of which analyses a problem arising in risk measurement and management in insurance and finance.