Efficient Vertex-Oriented Polytopic Projection for Web-scale Applications

TL;DR

This paper introduces a vertex-oriented incremental algorithm for projecting points onto polytopes, significantly improving efficiency in web-scale applications like online marketplaces where such projections are computational bottlenecks.

Contribution

The paper presents a novel vertex-oriented incremental projection algorithm for polytopes, optimized for structures common in large-scale applications, reducing computational costs.

Findings

Projections mostly lie on polytope vertices under certain structures.

The new algorithms outperform existing methods in efficiency.

Applications in internet marketplaces benefit from faster projection computations.

Abstract

We consider applications involving a large set of instances of projecting points to polytopes. We develop an intuition guided by theoretical and empirical analysis to show that when these instances follow certain structures, a large majority of the projections lie on vertices of the polytopes. To do these projections efficiently we derive a vertex-oriented incremental algorithm to project a point onto any arbitrary polytope, as well as give specific algorithms to cater to simplex projection and polytopes where the unit box is cut by planes. Such settings are especially useful in web-scale applications such as optimal matching or allocation problems. Several such problems in internet marketplaces (e-commerce, ride-sharing, food delivery, professional services, advertising, etc.), can be formulated as Linear Programs (LP) with such polytope constraints that require a projection step in the overall optimization process. We show that in the very recent work, the polytopic projection is the most expensive step and our efficient projection algorithms help in gaining massive improvements in performance.