# An exact penalty approach for optimization with nonnegative orthogonality constraints

## TL;DR

This paper introduces an exact penalty method for solving optimization problems with nonnegative orthogonality constraints, providing a practical algorithm with convergence guarantees and demonstrating effectiveness on various applications.

## Contribution

It develops a novel exact penalty framework for nonnegative orthogonality constrained optimization and proposes an efficient algorithm with theoretical convergence analysis.

## Key findings

- The penalty model is exact with sufficiently large penalty parameters.
- The proposed algorithm converges to weakly stationary points.
- Numerical experiments confirm the method's effectiveness on multiple problems.

## Abstract

Optimization with nonnegative orthogonality constraints has wide applications in machine learning and data sciences. It is NP-hard due to some combinatorial properties of the constraints. We first propose an equivalent optimization formulation with nonnegative and multiple spherical constraints and an additional single nonlinear constraint. Various constraint qualifications, the first- and second-order optimality conditions of the equivalent formulation are discussed. By establishing a local error bound of the feasible set, we design a class of (smooth) exact penalty models via keeping the nonnegative and multiple spherical constraints. The penalty models are exact if the penalty parameter is sufficiently large other than going to infinity. A practical penalty algorithm with postprocessing is then developed. It uses a second-order method to approximately solve a series of subproblems with nonnegative and multiple spherical constraints. We study the asymptotic convergence of the penalty algorithm and establish that any limit point is a weakly stationary point of the original problem and becomes a stationary point under some additional mild conditions. Extensive numerical results on the projection problem, orthogonal nonnegative matrix factorization problems and the K-indicators model show the effectiveness of our proposed approach.