Hamiltonian Convexity & Extended Lifting

• Hamiltonian Convexity is a framework that exploits latent convexity through K-lifts and cone factorization, enabling simpler representations of complex optimization sets.
• It examines both polyhedral and spectrahedral lifts, highlighting how nonnegative and psd-ranks determine the minimal size of convex representations.
• This approach applies across mixed-integer, variational, and control problems, offering strategies to overcome nonconvexities with efficient convex reformulations.

Extended Convex Lifting (ECL) is a general framework for representing complex convex or nonconvex sets and optimization problems as projections of simpler (often higher-dimensional) convex sets, thereby revealing latent convexity. ECL is rooted in the study of cone factorizations, operator theory, and optimization, and applies across convex geometry, combinatorial optimization, variational methods, mixed-integer optimization, and modern robust control.

1. Core Definitions and Factorization Theorems

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

The central result is the factorization theorem:

• $C$ admits a proper $K \subset \mathbb{R}^m$0-lift if and only if its slack operator $K \subset \mathbb{R}^m$1 admits a $K \subset \mathbb{R}^m$2-factorization: there exist $K \subset \mathbb{R}^m$3 and $K \subset \mathbb{R}^m$4 such that $K \subset \mathbb{R}^m$5.
• Conversely, any $K \subset \mathbb{R}^m$6-factorization induces a (possibly improper) $K \subset \mathbb{R}^m$7-lift (Gouveia et al., 2011, Thomas, 2018, Fawzi et al., 2020).

Cone rank is defined for a family of cones $K \subset \mathbb{R}^m$8, closed under faces: for a convex body $K \subset \mathbb{R}^m$9, its cone rank is the minimal $K$0 so that $K$1 admits a $K$2-factorization.

When $K$3, $K$4 is the nonnegative rank, and for $K$5 (the cone of $K$6 positive semidefinite matrices), the psd-rank.

2. Polyhedral and Spectrahedral Lifts: Complexity, Examples, and Rank

Polyhedral lifts ($K$7) and spectrahedral (psd) lifts ($K$8) correspond to representing $K$9 as the projection of a polyhedron or spectrahedron, respectively:

• Nonnegative rank governs the minimal size of a polyhedral lift for polytopes (Yannakakis theorem).
• Psd-rank governs the minimal size of a spectrahedral lift (Gouveia et al., 2011, Thomas, 2018, Fawzi et al., 2020).

Key examples:

• Cross-polytope: $C$0 has a size-$C$1 polyhedral lift, exponentially smaller than the native $C$2-facet description (Gouveia et al., 2011, Fawzi et al., 2020).
• Elliptope: Set of correlation matrices can be lifted to a spectrahedron of polynomial size (Fawzi et al., 2020).
• Stable set polytopes for perfect graphs admit psd-lifts of size $C$3 via Lovász’s theta-body (Gouveia et al., 2011).

The minimal lift size is lower-bounded by algebraic and combinatorial properties: e.g., a full-dimensional polytope in $C$4 has psd-rank at least $C$5 (Thomas, 2018, Fawzi et al., 2020).

3. ECL in Optimization and Variational Problems

ECL facilitates the reformulation of nonconvex, non-smooth, or high-dimensional problems as convex programs:

• Convex Optimization via Epigraph Lifting: The epigraph of a convex $C$6, $C$7, is convex. Lifting the unconstrained minimization of $C$8 to a feasibility problem in $C$9 enables solution by projections onto convex sets, providing a globally convergent algorithm for a wide class of $(L, \pi)$0 (Cetin et al., 2013).
• Infinite-dimensional and PDE Lifting: In the Calderón problem, ECL replaces a nonlinear map $(L, \pi)$1 by a linear map $(L, \pi)$2 on rank-one operators $(L, \pi)$3 in Bochner or Hilbert–Schmidt operator space, and nuclear norm relaxation yields convex recovery guarantees under a dual certificate (non-degenerate source condition) (Alberti et al., 1 Jul 2025).
• Variational Problems on Manifolds: For manifold-valued variational problems, ECL lifts $(L, \pi)$4 to $(L, \pi)$5 (space of probability measures), bringing the problem into the field of convex-concave saddle-point optimization, and generalizing sublabel-accurate approaches (Vogt et al., 2019, Möllenhoff et al., 2019).

4. ECL in Mixed-Integer, Disjunctive, and Quadratic Programs

ECL underpins the systematic derivation of strong convex hull descriptions in nonconvex or discrete settings:

• Mixed-Integer Lifting: Minimal valid inequalities for mixed-integer linear programs are lifted by constructing cuts in augmented spaces, using gauge functions of maximal $(L, \pi)$6-free sets and their convex extensions, yielding order-independent minimal coefficients (Basu et al., 2017).
• Disjunctive Programming: For the convex hull of $(L, \pi)$7 polytopes in $(L, \pi)$8, full optimal big-$(L, \pi)$9 lifting suffices for $L \subset \mathbb{R}^m$0 or with common-constraint structure; for $L \subset \mathbb{R}^m$1, extra rounds (e.g., MIR cuts) may be necessary (Qu et al., 2024).
• Bipartite Bilinear Sets: Sequential and subadditive (two-slope) liftings derive second-order cone representable (SOC) constraints for QCQPs, unifying and generalizing lifting approaches for strong convex relaxations (Gu et al., 2021).

5. Applications in Control, RL, and Hidden Convexity

Modern control theory has leveraged ECL to reconcile nonconvex policy-search with classical LMI-based design:

• LQR, LQG, $L \subset \mathbb{R}^m$2 Control: Despite nonconvexity in the controller space, ECL provides diffeomorphic parameterizations onto convex sets defined by Riccati or LMI certificates. Under standard nondegeneracy, every stationary point is globally optimal, and convex SDPs certify strong duality and gradient dominance (Zheng et al., 2023, Zheng et al., 2024, Watanabe et al., 14 Mar 2025, Pai et al., 5 Mar 2026).
• Mixed $L \subset \mathbb{R}^m$3 Control: ECL yields scalable policy-update schemes, convexifies the Riccati/Lyapunov inequality constraints, and underlies the benign landscape of stationary points (Pai et al., 5 Mar 2026).
• Distributed Control under QI: ECL gives convexification in cases with quadratic invariance (QI) by mapping policies to Youla parameters, enabling convex analysis of structured control synthesis (Zheng et al., 2024).

6. Obstructions and Limitations: When Small Lifts Fail

In some situations, obstructions preclude the existence of small polyhedral or spectrahedral lifts:

• Facial Structure: Count of faces, chain-length, and neighborliness in the face poset can forbid small lifts; e.g., the number of Boolean lattice faces puts lower bounds on nonnegative rank (Fawzi et al., 2020).
• Algebraic Degree: The degree of the algebraic boundary of $L \subset \mathbb{R}^m$4 controls the minimal LMI size: e.g., for $L \subset \mathbb{R}^m$5-ellipse or high-degree semialgebraic sets, no spectrahedral lift of polynomial size exists (Fawzi et al., 2020, Thomas, 2018).
• Symmetry: Symmetric lifts can require much higher dimension than (possibly non-symmetric) unconstrained lifts (Gouveia et al., 2011).

7. Extensions, Open Problems, and Future Directions

ECL continues to be a focal point for advances at the intersection of convex optimization, real algebraic geometry, combinatorial optimization, and control:

• Open questions include exact characterization of families with small psd-lifts, the role of symmetry, computational complexity of deciding cone rank and lift size, and further generalizations to nonlinear, hybrid, or purely data-driven settings (Thomas, 2018, Gouveia et al., 2011, Fawzi et al., 2020, Zheng et al., 2024).
• The ECL perspective unifies mixed-integer, variational, and control-theoretic techniques, provides algorithmic tools for face enumeration (fixed-d polyhedra), and motivates convex relaxation strategies via operator and cone factorization.

In summary, Extended Convex Lifting is a foundational geometric and algebraic strategy for uncovering and exploiting hidden convexity in a range of problems, with broad implications for optimization complexity, algorithmic design, and theoretical understanding (Gouveia et al., 2011, Thomas, 2018, Fawzi et al., 2020, Zheng et al., 2023, Zheng et al., 2024, Alberti et al., 1 Jul 2025).