Invex Optimization Revisited

TL;DR

This paper investigates conditions under which non-convex optimization problems guarantee that all KKT points are global optima, focusing on KT-invexity and its implications for interior point methods, especially in power flow problems.

Contribution

It provides necessary conditions for KT-invexity in n-dimensions and shows these are sufficient in two dimensions, applying results to power flow problems.

Findings

Necessary conditions for KT-invexity in n-dimensions.

Sufficiency of conditions in two-dimensional cases.

Power flow problem solutions meet conditions under mild assumptions.

Abstract

Given a non-convex optimization problem, we study conditions under which every Karush-Kuhn-Tucker (KKT) point is a global optimizer. This property is known as KT-invexity and allows to identify the subset of problems where an interior point method always converges to a global optimizer. In this work, we provide necessary conditions for KT-invexity in n-dimensions and show that these conditions become sufficient in the two-dimensional case. As an application of our results, we study the Optimal Power Flow problem, showing that under mild assumptions on the variable's bounds, our new necessary and sufficient conditions are met for problems with two degrees of freedom.