[Paper Review] Lifting for Simplicity: Concise Descriptions of Convex Sets

Hamza Fawzi, João Gouveia+3 more|arXiv (Cornell University)|Feb 22, 2020

Photochromic and Fluorescence Chemistry4 citations

TL;DR

This paper presents a unified framework for constructing and analyzing lifts of convex sets using structured factorizations of their slack operators, with a focus on spectrahedral lifts via sums of squares. It establishes that the existence of low-complexity lifts (polyhedral or spectrahedral) is equivalent to specific factorization properties, and provides tools to construct such lifts or prove their nonexistence using facial structure and algebraic obstructions.

ABSTRACT

This paper presents a selected tour through the theory and applications of lifts of convex sets. A lift of a convex set is a higher-dimensional convex set that projects onto the original set. Many convex sets have lifts that are dramatically simpler to describe than the original set. Finding such simple lifts has significant algorithmic implications, particularly for optimization problems. We consider both the classical case of polyhedral lifts, described by linear inequalities, as well as spectrahedral lifts, defined by linear matrix inequalities, with a focus on recent developments related to spectrahedral lifts. Given a convex set, ideally we would either like to find a (low-complexity) polyhedral or spectrahedral lift, or find an obstruction proving that no such lift is possible. To this end, we explain the connection between the existence of lifts of a convex set and certain structured factorizations of its associated slack operator. Based on this characterization, we describe a uniform approach, via sums of squares, to the construction of spectrahedral lifts of convex sets and illustrate the method on several families of examples. Finally, we discuss two flavors of obstruction to the existence of lifts: one related to facial structure, and the other related to algebraic properties of the set in question. Rather than being exhaustive, our aim is to illustrate the richness of the area. We touch on a range of different topics related to the existence of lifts, and present many examples of lifts from different areas of mathematics and its applications.

Motivation & Objective

• To develop a systematic approach to constructing concise lifts of convex sets, particularly spectrahedral lifts, to improve algorithmic efficiency in optimization.
• To establish a connection between the existence of lifts and structured factorizations of the slack operator associated with a convex set.
• To provide constructive methods—especially via sums of squares—for generating spectrahedral lifts of convex sets.
• To identify and analyze obstructions to the existence of lifts, based on facial structure and algebraic properties of the set.
• To unify and extend existing results on lifts, including polyhedral and spectrahedral lifts, across diverse mathematical and applied contexts.

Proposed method

• Use the slack operator of a convex set as a central object, defined via its nonnegative entries representing distances from extreme points to supporting hyperplanes.
• Characterize the existence of polyhedral lifts via nonnegative factorizations of the slack matrix, generalizing Yannakakis’ theorem.
• Characterize spectrahedral lifts via positive semidefinite (psd) factorizations of the slack operator, linking to sums of squares representations.
• Construct spectrahedral lifts using sums of squares decompositions, particularly for convex sets defined by polynomials or algebraic constraints.
• Apply hierarchical constructions (e.g., theta bodies) to approximate convex hulls of algebraic sets through spectrahedral lifts.
• Use facial structure analysis and degree bounds in polynomial ideals to derive obstructions to the existence of small lifts.

Experimental results

Research questions

• RQ1 When does a convex set admit a spectrahedral lift that is significantly simpler than its original description?
• RQ2 What structural properties of a convex set determine whether it admits a low-complexity polyhedral or spectrahedral lift?
• RQ3 How can sums of squares methods be systematically applied to construct spectrahedral lifts of convex sets?
• RQ4 What algebraic or geometric obstructions prevent the existence of small lifts for certain convex sets?
• RQ5 What is the relationship between the facial structure of a convex set and the complexity of its possible lifts?

Key findings

• A convex set admits a spectrahedral lift if and only if its slack operator admits a positive semidefinite factorization, providing a precise algebraic criterion.
• Spectrahedral lifts can be systematically constructed using sums of squares decompositions of polynomials, particularly for convex sets defined by polynomial inequalities.
• Obstructions to lifts exist based on facial structure: for example, a set with too many faces or complex face lattices may not admit a small lift.
• Algebraic obstructions arise from degree bounds in polynomial ideals; for instance, Scheiderer’s result shows that certain nonnegative polynomials cannot be SOS, implying no spectrahedral lift exists.
• The complexity of deciding whether a convex set admits a lift is NP-hard, as shown by reductions from nonnegative and positive semidefinite rank computation.
• The theory unifies diverse examples—from chain polytopes and Horn cones to epigraphs of convex polynomials—demonstrating the broad applicability of lift-based representations.