A unified approach for projections onto the intersection of $\ell_1$ and $\ell_2$ balls or spheres

Hongying Liu; Hao Wang; Mengmeng Song

Home/Archive/arXiv

arXiv · 1911.03946

A unified approach for projections onto the intersection of $\ell_1$ and $\ell_2$ balls or spheres

Hongying Liu·Hao Wang·Mengmeng Song

View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper focuses on designing a unified approach for computing the projection onto the intersection of an $\ell_1$ ball/sphere and an $\ell_2$ ball/sphere. We show that the major computational efforts of solving these problems all rely on finding the root of the same piecewisely quadratic function, and then propose a unified numerical method to compute the root. In particular, we design breakpoint search methods with/without sorting incorporated with bisection, secant and Newton methods to find the interval containing the root, on which the root has a closed form. It can be shown that our proposed algorithms without sorting possess $O(n log n)$ worst-case complexity and $O(n)$ in practice. The efficiency of our proposed algorithms are demonstrated in numerical experiments.

Explore related subjects

math.OC cs.NA math.NA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Hongying Liu, Hao Wang, Mengmeng Song. 2019-11-10. A unified approach for projections onto the intersection of $\ell_1$ and $\ell_2$ balls or spheres. https://arxiv.org/abs/1911.03946

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Deterministic and Random Bipartite Matching on General Networks: Convex Flow Reformulation, Asymptotic Properties, and Fast Algorithms

Minimum-distance bipartite matching on general networks has numerous applications various fields. This paper first focuses on deterministic problems and presents an exact edgewise-separable convex-flow reformulation. By introducing a smooth monotone rearrangement approximation of the edge-wise imbalance profiles, the convex-flow reformulation's can be solved efficiently. If we further conduct a first-order resistance-based approximation of the convex program, a one-step Laplacian-based estimator can be analytically derived in closed forms. The paper also studies random problems where supply and demand points are randomly distributed. We show that the expected optimal matching distance scales with the square root of the number of points if the supply/demand point distributions are identical, or linearly otherwise. In the former case, the optimal flow is proven to be centered, symmetric, and sub-Gaussian. In the latter case, the limiting resistance network characterizes how supply-demand imbalance is redistributed and motivates a fast algorithm that approximate the optimal flow based on the limiting resistance. Numerical experiments show that the proposed estimators closely approximate the exact matching cost while substantially reducing computation time. The proven theoretical properties of the random matching solution are numerically verified by large-scale Monte Carlo simulations.

math.OC↗

arXiv2026

Conformal-DRO: Distributionally Robust Optimization with Conformalized Ambiguity Set

Data-driven distributionally robust optimization (DRO) typically treats the conditional outcome law as fixed and uses ambiguity sets to capture estimation error. This paper studies latent distributional heterogeneity, where each instance has an unobserved law but contributes only one observation, so uncertainty persists even if the mixture law is known. We propose Conformal-DRO, which uses nested conformal regions to construct an ambiguity set for the future latent law. Under exchangeability, the set covers this law with probability at least $1-\alpha$ in finite samples, without estimating underlying latent laws or their mixing mechanism. The conformal path induces a data-driven transport geometry, while $\alpha$ determines the radius. The worst-case problem reduces to a finite linear program over conformal shells and admits sparse adversarial solutions. The resulting robust value provides a finite-sample certificate for the selected decision's expected cost.

math.OC↗

arXiv2026

The best approximation tuple: an extension of the Cheney-Goldstein algorithm and results to the multiple sets case

In this paper we extend the algorithm and several results published in the celebrated 1959 paper of Cheney and Goldstein about the best approximation pair (BAP) problem in two separate directions. One is the consideration of more than two sets. The other is the ability to handle each set as an intersections of a finite family of sets. We call the resulting problem the "Best Approximation Tuple (BAT) problem". The fundamental observation that leads to this generalizations is to recognize and handle one set (the "pivot set") as different from the remaining sets (the "satellite sets") instead of seeking cycles as the minimizers of a target functional. This enable us to overcome a certain theoretical obstacle related to cycles and minimizers of general functionals. We prove the convergence of the algorithm to the unique solution of the problem in the Euclidean case with strictly convex and compact satellite sets. Because of the lack of Fej\'er monotonicity, our convergence analysis is not standard, and is based on almost unknown properties of orthogonal projections regarding equality and inequality in the definition of nonexpansiveness.

math.OC↗