SOC Representable Relaxation

• SOC representable relaxation is a convex reformulation that uses second-order cone constraints to approximate or exactly capture nonconvex quadratic and polynomial structures.
• It replaces complex nonconvex conditions with tractable SOCP formulations, enhancing efficiency in applications like power flow analysis and nonconvex QCQP.
• This approach balances linear and semidefinite relaxations while exploiting sparsity and structure to improve scalability and computational performance.

A second-order cone (SOC) representable relaxation is a convex reformulation or outer approximation whose feasible region admits a description by linear constraints together with second-order cone constraints, or, equivalently in several settings, by a lift to a product of second-order cones followed by linear projection (Magron et al., 2020). Within nonconvex optimization, SOC representability occupies an intermediate position between linear and semidefinite formulations: it is rich enough to encode rotated quadratic inequalities, $2\times 2$ positive semidefinite conditions, and many convex envelopes of quadratic or polynomial structures, while remaining substantially more tractable than general semidefinite programming in many applications (Jiang et al., 2016). In some important cases the relaxation is not merely an outer approximation but the exact convex hull; notably, for a quadratic equation intersected with a bounded polyhedron, the convex hull is second-order cone representable (Santana et al., 2018).

1. Conceptual basis and geometric form

The core object is the second-order cone, typically used to encode inequalities of the form

$\|u\|_2 \le t,$

together with rotated variants that represent quadratic-over-linear and hyperbolic constraints. A recurring primitive is the $2\times 2$ positive semidefinite block: $$\begin{pmatrix} a & b\ b & c \end{pmatrix} \succeq 0 \iff \|(2b,\; a-c)\|_2 \le a+c,$$ which makes small semidefinite conditions directly SOC representable (Naumann et al., 2020). This mechanism underlies SDSOS formulations, geometric-mean constructions, and numerous lifted relaxations (Ahmadi et al., 2015).

In optimization, an SOC representable relaxation usually arises by replacing a nonconvex equality or inequality with a convex conic surrogate. Typical transformations include relaxing quadratic equalities to one-sided quadratic inequalities, introducing lifted variables such as $X=xx^\top$, or replacing nonlinear right-hand sides by concave envelopes (Jiang et al., 2016, Du et al., 1 Nov 2025). Because these constructions preserve conic structure, the resulting models can be solved as second-order cone programs (SOCPs), and several papers emphasize that SOCP is often computationally more practical than SDP while remaining significantly tighter than purely polyhedral relaxations (Santana et al., 2018, Jiang et al., 2016).

A further conceptual distinction is between exact SOC representability and approximate SOC relaxation. Exact SOC representability means that the convex hull itself has an SOC description; approximate SOC relaxation means that the model is a tractable outer bound whose tightness depends on additional assumptions, structural cuts, or post hoc feasibility checks (Santana et al., 2018, Larroux et al., 18 Feb 2026).

2. Exact convexification of quadratic sets

A central theorem states that for

$$S := \{ x \in \mathbb{R}^n : x^\top Q x + \alpha^\top x = g,\; x \in P \},$$

where $Q$ is symmetric and $P=\{x:Ax\le b\}$ is a bounded polyhedron, the convex hull $\operatorname{conv}(S)$ is second-order cone representable (Santana et al., 2018). The proof is constructive rather than existential: after spectral decomposition, affine change of variables, and completing the square, the quadratic equation is reduced to a canonical form with positive quadratic, negative quadratic, linear, and redundant components. The argument then proceeds by covering the extreme points through a finite union of disjunctive pieces and convexifying those pieces recursively over the facets of the polytope (Santana et al., 2018).

The geometry of the quadratic surface drives the construction. One case corresponds to surfaces that split into two convex sheets; there, convexification follows from SOC descriptions of quadratic inequalities intersected with polyhedra. The other case is ruled geometry, where every point lies on a line contained in the surface; then no interior point of the polytope can be extreme, so all extreme points lie on facet intersections, which leads to an induction on dimension (Santana et al., 2018). The construction yields explicit formulations, including standard SOC epigraph forms such as

$$\left\{(w,t): \|2w,\; t-1\| \le t+1,\; (w,t)\in P\right\},$$

for subcases of $\|u\|_2 \le t,$0 (Santana et al., 2018).

A related convexification framework studies intersections $\|u\|_2 \le t,$1 and $\|u\|_2 \le t,$2, where $\|u\|_2 \le t,$3 is an SOC representable cone, $\|u\|_2 \le t,$4 is a nonconvex homogeneous quadratic set, and $\|u\|_2 \le t,$5 is an affine hyperplane (Burer et al., 2014). The key device is conic aggregation: $\|u\|_2 \le t,$6 with $\|u\|_2 \le t,$7 chosen as the smallest positive value where $\|u\|_2 \le t,$8 becomes singular. Under verifiable eigenvalue and apex conditions, the convex hull is captured exactly by intersecting the original cone $\|u\|_2 \le t,$9 with a new SOC representable cone $2\times 2$0 (Burer et al., 2014). This unifies earlier split and disjunctive convexifications and shows that SOC representability can persist well beyond ellipsoids or sign-definite quadratics.

3. Reformulation mechanisms and canonical constructions

Beyond single quadratic equations, SOC representable relaxations appear through several recurrent reformulation paradigms. For the generalized trust region subproblem,

$2\times 2$1

a simultaneous block diagonalization via congruence reduces the pair $2\times 2$2 to a canonical form with $2\times 2$3 and $2\times 2$4 blocks (Jiang et al., 2016). When the optimal value is bounded from below, all such GTRSs are SOCP representable. The resulting SOCP introduces $2\times 2$5 for scalar blocks and cross-term variables $2\times 2$6 for $2\times 2$7 blocks, eliminating the original nonconvex quadratic constraint in favor of linear and SOC-tractable constraints (Jiang et al., 2016). The same machinery extends to equality- and interval-constrained variants and yields simplified versions of the S-lemma, the S-lemma with equality, and the S-lemma with interval bounds (Jiang et al., 2016).

For general nonconvex QCQP, another construction decomposes an indefinite matrix as

$2\times 2$8

rewrites each nonconvex quadratic constraint as two SOC-type constraints with an auxiliary variable $2\times 2$9, and then linearizes products of SOC constraints with linear constraints, as well as products of two SOC constraints (Jiang et al., 2016). The resulting GSRT-A and GSRT-B relaxations augment the basic SDP, RLT, and SOC-RLT constraints with additional valid inequalities derived from SOC $$\begin{pmatrix} a & b\ b & c \end{pmatrix} \succeq 0 \iff \|(2b,\; a-c)\|_2 \le a+c,$$0 linear, SOC $$\begin{pmatrix} a & b\ b & c \end{pmatrix} \succeq 0 \iff \|(2b,\; a-c)\|_2 \le a+c,$$1 SOC, Hadamard-product, and Kronecker-product constructions (Jiang et al., 2016). The paper establishes dominance relationships showing that these relaxations strictly strengthen standard SDP+RLT+SOC-RLT formulations for nonconvex QCQP (Jiang et al., 2016).

Constraint-wise convexification yields a different pattern. In bipartite bilinear programs, the convex hull of the feasible set defined by a single bilinear equality together with $$\begin{pmatrix} a & b\ b & c \end{pmatrix} \succeq 0 \iff \|(2b,\; a-c)\|_2 \le a+c,$$2 over a box is SOC representable in extended $$\begin{pmatrix} a & b\ b & c \end{pmatrix} \succeq 0 \iff \|(2b,\; a-c)\|_2 \le a+c,$$3-space (Dey et al., 2018). Intersecting the convex hulls of the individual constraints produces an overall SOCP relaxation that is tighter than the standard SDP relaxation intersected with the boolean quadratic polytope (Dey et al., 2018). In a related sparse quadratic setting over the unit hypercube, an extension of the Reformulation Linearization Technique to continuous quadratic sets yields SOC inequalities based on perspective terms and multilinear lifted variables; if the nodes carrying positive diagonal terms form a stable set, then the convex hull is SOC representable, and under additional tree-decomposition conditions the formulation has polynomial size and can be constructed in polynomial time (Dey et al., 25 Aug 2025).

Mixed-integer conic sets with nonlinear right-hand sides admit yet another exact convexification principle. For

$$\begin{pmatrix} a & b\ b & c \end{pmatrix} \succeq 0 \iff \|(2b,\; a-c)\|_2 \le a+c,$$4

the convex hull is exactly characterized by replacing $$\begin{pmatrix} a & b\ b & c \end{pmatrix} \succeq 0 \iff \|(2b,\; a-c)\|_2 \le a+c,$$5 with its concave envelope $$\begin{pmatrix} a & b\ b & c \end{pmatrix} \succeq 0 \iff \|(2b,\; a-c)\|_2 \le a+c,$$6 over $$\begin{pmatrix} a & b\ b & c \end{pmatrix} \succeq 0 \iff \|(2b,\; a-c)\|_2 \le a+c,$$7, provided a fibre condition and a column-span condition hold (Du et al., 1 Nov 2025). When $$\begin{pmatrix} a & b\ b & c \end{pmatrix} \succeq 0 \iff \|(2b,\; a-c)\|_2 \le a+c,$$8 for quadratic $$\begin{pmatrix} a & b\ b & c \end{pmatrix} \succeq 0 \iff \|(2b,\; a-c)\|_2 \le a+c,$$9, the relaxation

$X=xx^\top$0

is SOC representable and serves as a strong tractable surrogate (Du et al., 1 Nov 2025).

4. Power systems and network-flow formulations

Optimal power flow has become a major testing ground for SOC representable relaxations. In meshed AC-OPF, one SOC formulation linearizes branch KVL relations such as

$X=xx^\top$1

by replacing the sine term with the linearized relation $X=xx^\top$2, and relaxes the branch loss equality to an SOC inequality (Larroux et al., 18 Feb 2026). Tightness is then assessed by whether the relaxed inequalities are active and whether branch angle differences satisfy the cycle consistency condition

$X=xx^\top$3

The numerical evidence is sharply topology-dependent: in the IEEE 33-bus radial case, residuals for cycle consistency are numerically zero, whereas in the IEEE 39-bus meshed case the maximum residual is $X=xx^\top$4 degrees and the resulting inconsistency translates to power-flow errors of approximately $X=xx^\top$5 p.u. (Larroux et al., 18 Feb 2026). Thus, zero relaxation gap in the conic sense does not guarantee AC feasibility in meshed networks (Larroux et al., 18 Feb 2026).

To address the loss of cycle physics in standard principal-minor SOC relaxations, a separate line of work derives 3-cycle SOC constraints from $X=xx^\top$6 Hermitian PSD matrices (Geth et al., 2021). Standard PM SOC constraints enforce only

$X=xx^\top$7

which is sufficient on trees but weak on loopy networks. The 3-cycle construction samples quadratic forms of a $X=xx^\top$8 PSD matrix and generates a family of SOC inequalities parameterized by $X=xx^\top$9 and $$S := \{ x \in \mathbb{R}^n : x^\top Q x + \alpha^\top x = g,\; x \in P \},$$0; these constraints encode nontrivial three-bus relationships corresponding to Kirchhoff’s Voltage Law without introducing an explicit angle relaxation (Geth et al., 2021). In reported experiments, Kim+PM SOC attains exactness on the 14-bus case with $$S := \{ x \in \mathbb{R}^n : x^\top Q x + \alpha^\top x = g,\; x \in P \},$$1 optimality gap, while PM SOC alone leaves a $$S := \{ x \in \mathbb{R}^n : x^\top Q x + \alpha^\top x = g,\; x \in P \},$$2 gap (Geth et al., 2021).

For active distribution networks, SOC relaxation of the branch-flow equation

$$S := \{ x \in \mathbb{R}^n : x^\top Q x + \alpha^\top x = g,\; x \in P \},$$3

is written as

$$S := \{ x \in \mathbb{R}^n : x^\top Q x + \alpha^\top x = g,\; x \in P \},$$4

and then further approximated by a polyhedron to facilitate projection and region construction (Li et al., 2021). Combined with an adaptive constraint generation algorithm, this produces a dispatchable-region approximation whose effective percentage is $$S := \{ x \in \mathbb{R}^n : x^\top Q x + \alpha^\top x = g,\; x \in P \},$$5 on the 33-bus, 2-renewable case, versus $$S := \{ x \in \mathbb{R}^n : x^\top Q x + \alpha^\top x = g,\; x \in P \},$$6 for a linearized model (Li et al., 2021). The same paper reports that computation time is about $$S := \{ x \in \mathbb{R}^n : x^\top Q x + \alpha^\top x = g,\; x \in P \},$$7 seconds for the proposed method versus more than $$S := \{ x \in \mathbb{R}^n : x^\top Q x + \alpha^\top x = g,\; x \in P \},$$8 seconds for exhaustive sampling on that instance (Li et al., 2021).

5. Polynomial nonnegativity, algebraic certificates, and conic hierarchies

SOC representable relaxations also play a foundational role in polynomial optimization. A constructive theorem shows that the cone of sums of nonnegative circuits (SONC) admits an SOC lift (Magron et al., 2020). The proof uses $$S := \{ x \in \mathbb{R}^n : x^\top Q x + \alpha^\top x = g,\; x \in P \},$$9-rational mediated sets and a decomposition of nonnegative circuit polynomials into sums of binomial squares with rational exponents; each binomial-square term is modeled through a rotated SOC condition of the form

$Q$0

This yields SOCP formulations whose number of second-order cones grows linearly in the number of polynomial terms (Magron et al., 2020). The same framework supports an unconstrained polynomial optimization algorithm and a rounding-projection scheme for exact symbolic nonnegativity certificates (Magron et al., 2020).

The $Q$1-cone extends this perspective to a common framework for SONC and SAGE-type certificates. For rational supports, both the primal $Q$2-cone and its dual admit explicit generalized second-order descriptions obtained by decomposing arithmetic-geometric inequalities into chains of $Q$3 PSD constraints (Naumann et al., 2020). The cone is then assembled from reduced circuits, with the primal realized as a projected lift of sums of circuit cones and the dual as a projected lift of intersections of their duals (Naumann et al., 2020).

A separate bounded-degree hierarchy for global polynomial optimization replaces SOS polynomials by SDSOS polynomials, whose scaled diagonally dominant Gram matrices are SOCP representable (Chuong et al., 2017). The $Q$4-th level relaxation combines Krivine-Stengle positivity certificates with a fixed-degree SDSOS term, producing a convergent SOCP hierarchy whose conic block sizes depend only on the fixed degree $Q$5, not on the hierarchy level $Q$6 (Chuong et al., 2017). For SOCP-convex polynomials and for polynomials with essentially non-positive coefficients, the first SOCP relaxation is exact (Chuong et al., 2017).

The same SOC/SDSOS logic carries to nonsmooth semi-algebraic convexity. First-order SDSOS-convex semi-algebraic functions are defined as pointwise suprema of first-order SDSOS-convex polynomials over SOCP-representable uncertainty sets, and under suitable assumptions the optimal value and optimal solutions of the associated optimization problems are obtained exactly by solving an SOCP (Yang et al., 9 Sep 2025). The class includes the Euclidean norm, the $Q$7-norm, and least-squares functions with $Q$8-type structure (Yang et al., 9 Sep 2025).

6. Sparsity, scalability, and cross-domain applications

A major practical advantage of SOC representable relaxation is its compatibility with sparsity. For QCQPs lifted to a matrix variable $Q$9, replacing full PSD by all $P=\{x:Ax\le b\}$0 principal minors gives the standard SOCP relaxation; exploiting aggregate sparsity then reduces the number of second-order cones from all index pairs to only those appearing in the aggregate sparsity graph (Sheen et al., 2019). The sparse SOCP formulation requires no extra equality constraints for clique overlaps, admits a trivial zero-fill completion, and preserves the max-determinant completion property (Sheen et al., 2019). The paper proves that the sparse and dense SOCP relaxations have the same optimal value, and reports that much larger lattice-graph and pooling instances can be handled than with the SDP relaxations (Sheen et al., 2019).

The same conic scalability appears in quantum many-body optimization. For quantum Max Cut, optimizing over mutually consistent three-qubit reduced density matrices reduces, after symmetry arguments, to variables $P=\{x:Ax\le b\}$1 constrained on each triangle by a linear Lieb-Mattis inequality and a second-order cone representable Parekh-Thompson triangle inequality (Huber et al., 2024). With Pauli level-1 of the quantum Lasserre hierarchy, this SOC relaxation achieves an approximation ratio of $P=\{x:Ax\le b\}$2 to the ground-state energy and is reported to be solvable on systems with hundreds of qubits (Huber et al., 2024).

Biochemical process design provides an example where the relaxation can be exact for structural reasons. In gradostat optimization, the nonconvex microbial-growth equality $P=\{x:Ax\le b\}$3 is relaxed to $P=\{x:Ax\le b\}$4, and for Contois kinetics this inequality has an SOC representation; for Monod kinetics, SOC representations arise either under constant biomass or through SOC-representable convex envelopes (Taylor et al., 2021). The relaxation is exact when the gradostat is outflow connected, the system matrix is irreducible, and the growth rate satisfies a derivative condition stated in the paper (Taylor et al., 2021).

SOC representability has also entered learning theory. Classical ReLU-ICNNs are equivalent to LP value functions and therefore represent only continuous piecewise-linear convex functions, whereas SOC-ICNN augments the architecture with quadratic and Euclidean-norm branches so that the network output is the optimal value of an SOCP (Liu et al., 24 Apr 2026). The paper proves the strict inclusion

$P=\{x:Ax\le b\}$5

and states that this expansion of representational capacity occurs without increasing the asymptotic order of forward-pass complexity (Liu et al., 24 Apr 2026). This suggests that SOC representable relaxation is no longer confined to classical optimization models but can also serve as a representational prior in convex surrogate learning.

7. Exactness, limitations, and recurring misconceptions

A common misconception is that an SOC relaxation is automatically an exact model whenever the numerical gap is small. The meshed OPF results show otherwise: even when the conic loss inequalities are tight, angle recovery can fail because cycle consistency is violated, so the formulation is then an approximation rather than a true relaxation of AC-OPF (Larroux et al., 18 Feb 2026). Another misconception is that SOC strengthening always improves a base conic model. For separable power-cone relaxations over the simplex, the paper on $P=\{x:Ax\le b\}$6 proves that without RLT constraints, SOCP and SDP liftings do not strengthen the basic power-cone relaxation at all: $P=\{x:Ax\le b\}$7 whereas adding RLT does strengthen it (Dey et al., 18 Oct 2025).

The size of an exact SOC description can also be prohibitive. The convex hull construction for quadratic equations over polytopes is exact but may have exponential complexity in the number of polyhedral facets (Santana et al., 2018). Likewise, the sparse quadratic-hypercube formulation can be exponential in general, and polynomial-size SOC formulations require explicit structural conditions involving stable plus-loop sets and tree decompositions of logarithmic width and spread (Dey et al., 25 Aug 2025). Exact convexification results for mixed-integer conic sets likewise depend crucially on fibre and column-span assumptions; when those fail, the concave-envelope formula need not be tight (Du et al., 1 Nov 2025).

Finally, SOC representable relaxation should not be conflated with universal dominance over semidefinite relaxations. In QCQP, the standard SDP relaxation is at least as strong as the standard SOCP relaxation in general, although equality holds for special matrix classes such as non-positive off-diagonal or diagonal-zero data (Sheen et al., 2019). The significance of SOC representability is therefore more specific: it provides a large class of exact convex hulls, strong outer approximations, and scalable conic formulations whose practical value depends on structure, sparsity, and the geometry of the original nonconvex set (Jiang et al., 2016, Santana et al., 2018).