Extended Convex Lifting

• Extended Convex Lifting (ECL) is a framework that represents complex convex sets and functions as projections of simpler higher-dimensional convex sets.
• ECL establishes an equivalence between K-lifts and slack operator factorizations, linking conic factorizations with both nonnegative and semidefinite ranks.
• The methodology is pivotal in optimization, control, and statistics by enabling tractable reformulations of nonconvex problems into convex programs.

Extended Convex Lifting (ECL) is the mathematical framework characterizing when a convex set, function, or optimization problem admits a representation as the projection of a higher-dimensional "simpler" convex set, often enabling reformulation of nonconvex or intractable problems as tractable convex programs. At its core is the equivalence between such liftings and conic (or operator) factorizations of objects derived from the problem data, such as slack operators or observables. ECL plays a central role in convex geometry, convex algebraic geometry, optimization, combinatorial optimization, control, statistics, inverse problems, and variational analysis.

1. Foundational Principles and Theorem

Let $C \subset \mathbb{R}^n$ be a full-dimensional convex body and $K \subset \mathbb{R}^m$ a full-dimensional closed convex cone. A K-lift of $C$ is data $(L, \pi)$ where $L \subset \mathbb{R}^m$ is affine and $\pi: \mathbb{R}^m \to \mathbb{R}^n$ is linear such that $C = \pi(K \cap L)$; the lift is proper if $L \cap \operatorname{int} K \neq \emptyset$ (Gouveia et al., 2011).

The central result (generalized Yannakakis–Gouveia–Parrilo–Thomas Theorem) states:

• $C$ admits a proper $K$-lift if and only if its slack operator $K \subset \mathbb{R}^m$0 admits a K-factorization:
  • There exist maps $K \subset \mathbb{R}^m$1, $K \subset \mathbb{R}^m$2 such that

  $K \subset \mathbb{R}^m$3

  for all $K \subset \mathbb{R}^m$4 extreme in $K \subset \mathbb{R}^m$5, $K \subset \mathbb{R}^m$6 extreme in the polar $K \subset \mathbb{R}^m$7.

Conversely, existence of such a factorization gives a (not necessarily proper) K-lift (Gouveia et al., 2011, Thomas, 2018, Fawzi et al., 2020).

For polytopes, this reduces to nonnegative factorization of the slack matrix and recovers Yannakakis’s equivalence of polyhedral lifts and nonnegative rank.

2. Cone Factorizations, psd-rank, and Lifting Complexity

Given a sequence of cones $K \subset \mathbb{R}^m$8 closed under faces, one defines cone rank (or, for spectrahedral lifts, the psd-rank). For a nonnegative matrix $K \subset \mathbb{R}^m$9, the psd-rank is the minimal $C$0 such that there exist $C$1 in $C$2 with $C$3 (Gouveia et al., 2011, Thomas, 2018).

Key bounds:

• $C$4.
• For polytopes, $C$5; for general convex sets, further lower bounds involve algebraic degree and combinatorial structure (Fawzi et al., 2020).

This formalism enables a complexity theory of lifts: sets requiring large nonnegative or psd-rank cannot admit small polyhedral or semidefinite lifts (Thomas, 2018, Fawzi et al., 2020).

3. Methodology: Explicit Constructions and Examples

ECL provides both existence results and constructive tools for lifts:

• Polyhedral lifts: For $C$6, one constructs a $C$7-lift as

$C$8

whose projection onto $C$9 recovers $(L, \pi)$0 (Gouveia et al., 2011, Fawzi et al., 2020).

• Spectrahedral lifts (psd-lifts): For the unit disk $(L, \pi)$1, the slack operator is factorized using

$(L, \pi)$2

(Gouveia et al., 2011, Thomas, 2018). For the Lovász theta body,

$(L, \pi)$3

gives a psd-lift of size $(L, \pi)$4 for perfect graphs (Gouveia et al., 2011, Fawzi et al., 2020).

• Lifting for variational and inverse problems: ECL is extended to PDE-constrained inverse problems (e.g., the Calderón problem) by lifting bilinear terms $(L, \pi)$5 to rank-one operators and relaxing to nuclear-norm regularization, producing infinite-dimensional analogues of PhaseLift with exactness characterized by source conditions (Alberti et al., 1 Jul 2025). In variational imaging, ECL relaxes nonconvex labeling to convex programs on probability measures via sublabel-accurate and manifold-valued liftings (Möllenhoff et al., 2019, Vogt et al., 2019).

4. Applications in Optimization, Control, and Statistics

• Combinatorial optimization: ECL underlies the theory of extended formulations, notably for the permutahedron (via the Birkhoff polytope), matching polytopes, and cut polytopes. Combinatorial and algebraic lower bounds for extension complexity follow from ECL obstructions (Fawzi et al., 2020).
• Policy optimization and control: ECL provides a geometric lens for hidden convexity in nonconvex policy search for LQR, LQG, $(L, \pi)$6 and mixed $(L, \pi)$7 control. Nonconvex, bilinear, or quadratic formulations are lifted to convex LMI or semidefinite programs via change-of-variables and auxiliary variables (e.g., Lyapunov gramians). The principal result is that every non-degenerate stationary point is globally optimal, certifying the "benign landscape" property in these problems (Zheng et al., 2023, Zheng et al., 2024, Watanabe et al., 14 Mar 2025, Pai et al., 5 Mar 2026).
• Statistics and random sets: The lift zonoid and lift expectation of a random convex body encode all one-dimensional marginals of its support function; ECL provides a convex geometric description of depth regions, stochastic orders, and statistical risk bounds (Diaye et al., 2018).
• Mixed-integer programming and disjunctive sets: ECL unifies big-M and subadditive lifting of inequalities, with full characterization of the facet structure of lifted convex hulls in low dimension and sequence-independent lifting schemes for bipartite bilinear sets (Gu et al., 2021, Qu et al., 2024, Basu et al., 2017).

5. Theoretical Limitations and Obstructions

ECL identifies precise obstructions to the existence of compact lifts:

• Facial structure: The face poset of any lift $(L, \pi)$8 must embed that of the original set $(L, \pi)$9, which restricts the possible lifting cones $L \subset \mathbb{R}^m$0 (e.g., neighborliness and chain-length arguments prohibit small-size lifts) (Fawzi et al., 2020).
• Algebraic degree bounds: The degree of the algebraic boundary of $L \subset \mathbb{R}^m$1 limits the minimal size of a spectrahedral lift; certain semialgebraic convex sets admit no LMI representation of any finite size (Scheiderer’s counterexamples) (Thomas, 2018, Fawzi et al., 2020).
• Parameterization and numerical obstacles: In infinite-dimensional and function spaces (e.g., PDEs), the existence and construction of dual certificates (source conditions) for exactness of nuclear-norm relaxations, and identifiability from finite data, can be highly nontrivial (Alberti et al., 1 Jul 2025).

6. Algorithmic and Practical Aspects

ECL-based reformulations enable a range of algorithmic frameworks:

Domain	ECL Construction	Resulting Program Type
Polyhedral combinatorics	Nonnegative-rank slack factorization	Linear/LP, extended formulation
Semidefinite optimization	PSD-rank/slack factorization	SDP, LMI
Quadratic inverse/PDE	Operator-lifting + nuclear norm	Convex (nuclear norm) minimization
Policy optimization	LMI reformulation via variable change	SDP/LMI, strong duality
Mixed-integer programming	Lifting valid inequalities	Piecewise linear/SOC cuts

ECL facilitates solution via standard LP/SDP solvers when factorization is achieved, or motivates sublabel-accurate, block-coordinate, and primal-dual methods for variational relaxations (Cetin et al., 2013, Möllenhoff et al., 2019, Gu et al., 2021, Pai et al., 5 Mar 2026).

7. Symmetry, Uniqueness, and Further Directions

Symmetry constrains small symmetric lifts; asymmetric (non-equivariant) factorizations may yield exponentially smaller lifts (e.g., for regular polygons) (Gouveia et al., 2011). The uniqueness of lifting in minimal cut-generating functions for integer programming is characterized via covering of parameter space by finitely many polyhedral regions (Basu et al., 2017).

Open problems include determining minimal psd-rank for families of polytopes, explicit construction of small lifts for high degree/semi-algebraic sets, generalizing ECL to non-quadratic/non-affine/nonlinear settings, and understanding the quantitative limits of lifting for practical large-scale optimization (Fawzi et al., 2020, Zheng et al., 2024, Alberti et al., 1 Jul 2025).

---

References:

• "Lifts of convex sets and cone factorizations" (Gouveia et al., 2011)
• "Spectrahedral Lifts of Convex Sets" (Thomas, 2018)
• "Lifting for Simplicity: Concise Descriptions of Convex Sets" (Fawzi et al., 2020)
• "Benign Nonconvex Landscapes in Optimal and Robust Control, Part II: Extended Convex Lifting" (Zheng et al., 2024)
• "Policy Optimization of Mixed H2/H-infinity Control: Benign Nonconvexity and Global Optimality" (Pai et al., 5 Mar 2026)
• "A convex lifting approach for the Calderón problem" (Alberti et al., 1 Jul 2025)
• "Lift expectations of random sets" (Diaye et al., 2018)
• "Lifting Vectorial Variational Problems" (Möllenhoff et al., 2019)
• "Lifting methods for manifold-valued variational problems" (Vogt et al., 2019)
• "Lifting convex inequalities for bipartite bilinear programs" (Gu et al., 2021)
• "Unique lifting of integer variables in minimal inequalities" (Basu et al., 2017)
• "Projections Onto Convex Sets (POCS) Based Optimization by Lifting" (Cetin et al., 2013)