Active-Normal IEOS: A Low-Dimensional Feasible-Ascent Method for Inside-Ellipsoid Outside-Sphere Problems with Linear Constraints

Abstract

This technical note develops Active-Normal IEOS (AN-IEOS), a low-dimensional feasible-ascent method for Inside-Ellipsoid Outside-Sphere (IEOS) problems with homogeneous linear inequalities and equalities. The method removes equality constraints exactly through an orthonormal null-space parametrization and constructs feasible ascent directions from a minimum-norm combination of active ellipsoidal and linear-constraint normals. This construction yields a scalar stationarity gap, an explicit family of strict feasible-ascent directions, an analytical maximal feasible step, and a radial reactivation mechanism when no ellipsoidal constraint is active. A finite-precision working-set implementation is also described. The numerical study investigates the effects of ambient problem dimension, distance of the initial point from the IEOS sphere threshold, active-set dimension, and direction-parameter selection. The results indicate that AN-IEOS is particularly suited to warm-started and repeated IEOS problems. In the tested near-threshold cases with small active sets, the prototype implementation achieved approximately 3–5× lower computation times than the full sequential-linearized IEOS approach at dimensions 200–400. The note also documents the method's limitations, including possible convergence to stationary points below the IEOS feasibility threshold. AN-IEOS is therefore intended as a computational component for structured, warm-started IEOS feasibility search rather than as a globally convergent replacement for general nonconvex optimization methods. This document is a non-peer-reviewed technical note. The accompanying implementation and reproducibility material are available at:https://github.com/Roozbeh-Abolpour/active-normal-ieos