# Optimization over convex polyhedra via Hadamard parametrizations

arXiv: https://arxiv.org/abs/2410.23874 · HTML: https://mlsanity.com/paper/2410.23874/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Oct 2024.

## Built on by (most cited first, 1 of 1)

- [On exploration of an interior mirror descent flow for stochastic nonconvex constrained problem](https://mlsanity.com/paper/2507.15264/) (2025)

## Abstract (arXiv)

In this paper, we study linearly constrained optimization problems (LCP). After applying Hadamard parametrization, the feasible set of the parametrized problem (LCPH) becomes an algebraic variety, with conducive geometric properties which we explore in depth. We derive explicit formulas for the tangent cones and second-order tangent sets associated with the parametrized polyhedra. Based on these formulas, we develop a procedure to recover the Lagrangian multipliers associated with the constraints to verify the optimality conditions of the given primal variable without requiring additional constraint qualifications. Moreover, we develop a systematic way to stratify the variety into a disjoint union of finitely many Riemannian manifolds. This leads us to develop a hybrid algorithm combining Riemannian optimization and projected gradient to solve (LCP) with convergence guarantees. Numerical experiments are conducted to verify the effectiveness of our method compared with various state-of-the-art algorithms.