SOCR: Second-Order Cone Representability

• SOCR sets are convex sets defined by second-order cone constraints and linear equations, providing tractable models in optimization.
• They bridge the gap between polyhedral, spectrahedral, and nonconvex quadratic programming through precise representability limits.
• SOCR formulations enable efficient convex relaxations in global optimization, polynomial certification, and optimal experimental design.

A set is called second-order cone representable (SOCR) if it can be written as the intersection, projection, or image of linear equations and second-order cone (SOC) constraints—typically in the form $\{x : \|B^T x\| \leq b^T x\}$, where $B$ and $b$ are appropriately chosen. SOCR sets occupy a central role in convex analysis and optimization, as they combine algebraic and geometric tractability with expressive modeling capability. The theory and applications of SOCR sets, as well as their relationship to linear, semidefinite, and disjunctive programming, constitute a rapidly developing area with connections across mathematical optimization, convex algebraic geometry, and algorithmic design.

1. Fundamental Definitions and Algebraic Structure

The basic object of paper is the closed convex set defined by the standard (rotated) SOC in $\mathbb{R}^{n}$: $$K = \{ x \in \mathbb{R}^n : \|B^T x\| \leq b^T x \}$$ with $B \in \mathbb{R}^{n \times d}$ (full column rank) and $b \notin \operatorname{range}(B)$. This form includes the canonical Lorentz (ice-cream) cone $$Q = \{ (x, t) \in \mathbb{R}^{n-1} \times \mathbb{R} : \|x\| \leq t \}$$ as a special case.

A set $S \subseteq \mathbb{R}^n$ is SOCR if it is the linear image of an intersection of finitely many SOCs and affine spaces; equivalently, if $S$ can be formulated via a finite number of conic quadratic constraints and linear equations. The concept extends naturally to cones, polytopes, spectrahedra, and their projections, with a central distinction being that every SOCP-feasible set is SOCR, but not every spectrahedral (semidefinite representable) set is SOCR (Fawzi, 2016).

The SOCR property can also be characterized using supporting functionals and the existence of a small semidefinite extension degree $\operatorname{sxdeg}(S) \leq 2$ (Scheiderer, 2020); i.e., $S$ can always be represented by 2$\times$2 linear matrix inequalities under suitable linear embeddings.

2. Expressiveness: SOCR Sets versus LP and SDP

SOCR sets include all polyhedra, the (rotated or non-rotated) second-order cones, their images and preimages under linear maps, and any finite intersection or Minkowski sum of such sets. SOCR sets are strictly more expressive than polyhedra, but strictly less so than spectrahedra (semidefinite representable sets). This strictness is quantitatively shown by the inability to represent the $3 \times 3$ positive semidefinite cone $S_+^3$ as SOCR, as a consequence of the unbounded growth of the SOC rank of its slack matrix (Fawzi, 2016). However, certain slices of $S_+^3$, such as $\{X \in S_+^3 : X_{11} = X_{22}\}$, are SOCR, and all slices of $S_+^3$ of dimension at most four are SOCR; this provides a precise boundary for the expressiveness gap (Averkov, 2019).

Every closed convex semialgebraic subset of $\mathbb{R}^2$ is SOCR, so low-dimensional geometry is covered completely by second-order cones (Scheiderer, 2020). Moreover, every Nash-smooth hyperbolicity cone (a generalization of the classically smooth hyperbolicity cones) is SOCR, and the same holds for compact convex semialgebraic sets with Nash-smooth strictly positively curved boundary (Scheiderer, 21 Sep 2025).

3. SOCR in Convexification of Nonconvex Quadratic Sets

SOCs (and thus SOCR sets) are crucial in the convexification and relaxation of nonconvex quadratic programming. The intersection of a SOC and a nonconvex quadratic set—such as $K \cap \{x : x^T A_1 x \leq 0\}$, or a trust region with an additional quadratic constraint—can be tightly convexified using SOC aggregation and block diagonalization techniques (Burer et al., 2014). This involves introducing aggregated matrices $A_t = (1 - t)A_0 + tA_1$, finding a critical parameter $s$ at which $A_t$ first becomes singular, and writing the convex hull as $K \cap S$ with $S$ another SOC. The approach applies to trust-region subproblems, intersections of ellipsoids and linear or quadratic disjunctions, and generalizes classical disjunctive cutting-plane techniques, all through SOCR relaxations.

For quadratically constrained quadratic programming (QCQP), it is shown that the exact convex hull of a quadratic equation intersected with any bounded polyhedron is always SOCR, via a constructive dimension-reduction argument (Santana et al., 2018). This provides a general method for formulating tight, finite-dimensional conic relaxations, even for highly nonconvex sets, and is of particular relevance in global branch-and-bound algorithms.

4. SOCR in Polynomial and Signomial Optimization

A major subclass of polynomial nonnegativity cones—Sums of Nonnegative Circuits (SONC) and their relatives (SAGE, $\mathcal{S}$-cones)—admits explicit SOCR representations (Wang et al., 2019, Magron et al., 2020, Naumann et al., 2020). In particular, every SONC cone (comprising sparse polynomials nonnegative via the arithmetic-geometric mean inequality) can be written as an affine projection of a product of rotated SOCs. This fundamentally alters the computational landscape for nonnegativity certification and unconstrained polynomial optimization: as opposed to sums-of-squares, which require SDP, the associated SOCPs are both scalable and numerically efficient, and can be extended to provide hybrid numeric-symbolic exact nonnegativity certificates.

Furthermore, the process of constructing minimal SOCR representations for certain classes of polynomial inequalities, such as weighted geometric mean inequalities, is intimately related to the combinatorics of mediated sets; the optimal number of SOC constraints correlates with the minimal length of such mediating sequences (Wang, 2022). This link enables the design of efficient algorithms for compact conic representations of polynomial inequalities, relevant to both optimization and quantum information.

5. SOCR in Experimental Design and Conic Optimization

In the context of statistical optimal design, canonical criteria such as $D$-, $A$-, $G$-, and $I$-optimality are all SOCR (Sagnol et al., 2013). For example, maximizing the $D$-optimality criterion (the $m$th root of the determinant of an information matrix) is equivalent to maximizing a geometric mean of certain auxiliary variables subject to linear and SOC constraints. When the experimental design is required to be exact (integer allocations), the resulting mixed-integer SOCP (MISOCP) can be solved to optimality using standard branch-and-cut tools, demonstrating practical advantages over both semidefinite programming and classical heuristics for highly constrained designs. This modeling framework extends seamlessly to subsystem-optimality and constrained experimental problems.

In conic programming itself, SOCR sets are central: they permit high-throughput, large-scale second-order cone programs (SOCPs), which are efficiently solvable by interior-point algorithms due to the explicit self-concordant barrier on the cone $-\log f(x)$, whenever $f$ defines a Nash-smooth SOCR cone (Scheiderer, 21 Sep 2025). Sensitivity and variational analysis for SOCPs rely on the geometric and smooth structure of SOCR sets, particularly in deducing properties like uniqueness of multipliers, error bounds, and calmness, all of which can be derived from second-order epi-differentiability and explicit tangent/normal cone characterizations (Hang et al., 2017).

6. Hierarchies, Limitations, and Future Directions

Despite their broad utility, SOCR sets are provably less general than spectrahedra: the full $3\times3$ positive semidefinite cone $S_+^3$ does not admit a SOCR representation (Fawzi, 2016), with the failure characterized by an unbounded SOC rank of the associated slack matrix. However, all slices of $S_+^3$ of dimension at most four or slices orthogonal to nonzero singular matrices are SOCR (Averkov, 2019). In $\mathbb{R}^2$, every convex semialgebraic set is SOCR (Scheiderer, 2020). For Nash-smooth hyperbolicity cones, the $2\times2$ block structure is both necessary and sufficient for SOCR (Scheiderer, 21 Sep 2025). This suggests a stratification of convex sets by their semidefinite extension degree and curvature/smoothness properties; cones with "stricter" algebraic-geometric regularity admit SOCR representations, while certain high-dimensional spectrahedra do not.

Continued research targets the precise characterization for which spectrahedral slices and convex bodies in higher dimensions are SOCR, the combinatorial structure of minimal representations, the design of efficient cutting-plane or decomposition algorithms that provably preserve SOCR, and their application in large-scale nonconvex, stochastic, or composite optimization settings.

7. Practical Applications and Impact

SOCR sets and formulations have substantial impact in practice:

• Global optimization: SOC relaxations are used to convexify nonconvex QCQP and enable efficient mixed-integer programming solvers for problems as varied as process systems engineering, power flow, and signal processing (Santana et al., 2018, Rhodes et al., 2 Jun 2025, Ruan et al., 17 Aug 2025).
• Polynomial optimization: SONC–SOCR techniques dramatically improve tractability for sparse, high-degree polynomials (Wang et al., 2019, Magron et al., 2020).
• Experimental design: Mixed-integer SOCP (MISOCP) formulations for exact $D$-optimal designs yield solutions and optimality certificates beyond the reach of traditional heuristics (Sagnol et al., 2013).
• Control, machine learning, and statistics: The characterization of SOCR sets guides modelers in selecting tractable constraints, efficient solvers, and representable fairness or robustness metrics (Sundar et al., 6 Dec 2024).
• Convex geometry and algebraic analysis: The reduction of general LMI-based representations to $2 \times 2$ block-diagonal (SOC) cases (particularly under Nash-smoothness and positive curvature) enables new lines of research in algebraic convexity and computational real algebraic geometry (Scheiderer, 21 Sep 2025).

The foundational and practical importance of SOCR sets makes them indispensable in the landscape of convex optimization, with ongoing work extending their applicability and sharpening the understanding of their boundaries relative to the broader spaces of spectrahedra and semialgebraic convex bodies.