Sum-of-Squares Conic Approximations

• Sums-of-squares based conic approximations are convex relaxations that certify polynomial and matrix nonnegativity using SOS decompositions and semidefinite programming.
• They leverage alternative formulations like DSOS and SDSOS to replace intractable constraints with tractable LP, SOCP, or smaller SDP blocks, improving scalability.
• Their applications span robust optimization, control synthesis, and combinatorial problems, offering rigorous convergence hierarchies and precise error bounds in structured settings.

Sums-of-Squares Based Conic Approximations are a family of tractable convex relaxations for cones arising in polynomial and matrix nonnegativity—especially the positive semidefinite (PSD) cone, copositive cone, and nonnegativity cones of polynomials restricted over semialgebraic sets. These approximations leverage the algebraic property that a nonnegative polynomial is always a sum of squares (SOS) in certain dimensions and degrees, and in more complex cases, SOS forms arise as tractable certificates of nonnegativity via semidefinite programming (SDP). Sums-of-squares approximations translate intractable nonnegativity constraints, which are often NP-hard, into optimization problems solvable by SDP, linear programming (LP), or second order cone programming (SOCP) by restricting certificates to SOS forms and specialized matrix classes.

1. Foundational SOS Hierarchies and Matrix Cones

The archetypal SOS-based cone is the sum-of-squares cone for polynomials of degree $2d$ in $n$ variables: $$\Sigma_{n,2d} = \left\{ p(x) : p(x) = \sum_{i=1}^m q_i(x)^2, \; q_i\in \mathbb{R}[x]_{d} \right\}$$ A central theme is the distinction between nonnegative polynomials and sums of squares, and the strict containment $\Sigma_{n,2d} \subsetneq P_{n,2d}$ except in the "Hilbert cases" $(n=1)$ or $(2d=2)$ or $(n=2,2d=4)$ (Blekherman, 2010, Goel et al., 2023).

To approximate matrix cones such as the PSD cone $S^n_+$ and copositive cone $\mathrm{COP}_n$, Parrilo introduced nested SOS-based cones, e.g., for copositivity: $$\mathcal{K}_n^{(r)} = \left\{ M \in S^n : (\sum_{i=1}^{n} x_i^2)^r\,(x^{\circ 2})^T M x^{\circ 2} \in \Sigma \right\}$$ where $(x^{\circ 2}) = (x_1^2, ..., x_n^2)$ and $\Sigma$ denotes the classical SOS cone (Laurent et al., 2022, Vargas et al., 2023).

Factor-width-k cones offer refined approximations for the PSD cone: $\mathcal{F}_k^n = \{A \in S^n_+ : \exists V,\, A = VV^T,\, \text{each column of %%%%10%%%% has at most %%%%11%%%% nonzeros} \}$ Their duals enforce PSD on all $k \times k$ principal minors (Gouveia et al., 2019): $$(\mathcal{F}_k^n)^* = \{ X \in S^n : X_K \succeq 0\ \forall K \subseteq \{1,\ldots,n\},\,|K| = k \}$$

2. SOS-Based Conic Relaxations: Structure, Implementation, and Complexity

SOS-based conic approximations are typically implemented as semidefinite programs (SDPs) by encoding:

• Gram matrix certificates for SOS polynomials, $p(x) = z(x)^T Q z(x)$, with $Q \succeq 0$.
• For fixed relaxations, replacing $Q \succeq 0$ with tractable alternatives: diagonally dominant (DSOS, via LP), scaled diagonally dominant (SDSOS, via SOCP), or block-diagonal $k\times k$ PSDs for factor-width-$k$ cones (Ahmadi et al., 2015, Ahmadi et al., 2015).
• For copositive cones, expressing matrix copositivity using SOS representations either on the whole space or restricted subsets: nonnegativity on the simplex, sphere, or nonnegative orthant, yielding Lasserre-type relaxations via Putinar's certificate (Laurent et al., 2022, Vargas et al., 2023).

A typical SDP associated to a conic SOS relaxation involves moment/localizing matrices whose size is polynomial in $n$ and relaxation degree $r$. For DSOS/SDSOS relaxations, constraints decompose into small PSD blocks or rotated second-order cones, yielding tractable (and scalable) LP or SOCP instances but with a relaxation gap (Ahmadi et al., 2015).

Column generation refines LP/SOCP inner approximations of SOS cones: starting from a basic atom set (DD or SDD matrices), new violating atoms are iteratively detected and added, monotonicly improving solution quality and approaching full SDP strength (Ahmadi et al., 2015).

3. Exactness, Hierarchies, and Limitations

SOS-based cones form hierarchies: $$\Sigma_{n,2d} \subseteq \Sigma_{n,2d}^{(1)} \subseteq ... \subseteq P_{n,2d}$$ For the copositive cone, Parrilo's hierarchy $\{\mathcal{K}_n^{(r)}\}$ satisfies $\bigcup_r \mathcal{K}_n^{(r)} = \mathrm{COP}_n$ for $n \leq 5$ (exactness). The exceptional role of the $5\times 5$ Horn matrix $H$ is critical to the finite convergence: $COP_5 = \bigcup_r \mathcal{K}_5^{(r)}$ if and only if every scaled $H$ belongs to some $\mathcal{K}_5^{(r)}$ (Laurent et al., 2022). For $n \geq 6$ the hierarchy never fills the full cone, with counterexamples governed by the structure of extreme rays.

For factor-width-k cones, certificates fail to strengthen via multipliers for symmetric quadratics (all $k$), general quadratics ($k=2$), and quaternary quadratics ($k=3$): multiplier-based SOS relaxations do not fill the gaps in nonnegativity (Gouveia et al., 2019).

Filtrations of cones between the SOS and PSD form allow finer scale approximations, exploiting Gram-matrix representations over Veronese and related projective varieties. collapse occurs for a finite number of steps in the Hilbert cases, but strict inclusions persist for larger $n$ and $d$ (Goel et al., 2023).

4. Sparse and Structured Approximations via SOS

In sparse and semialgebraic settings, SOS-based cones enable highly efficient conic programs:

• For varieties defined by quadratic monomial ideals (e.g., graph-theoretic sparsity), the SOS cone projects to partially specified PSD matrices, replaced by smaller block-wise constraints on maximal cliques (Blekherman et al., 2020).
• Approximation guarantees for sparse block decompositions are quantified by conic distance parameters, giving precise error bounds in terms of graph girth and clique complexity. In chordal graphs, the sparse cone is exact.

SOS-based generalizations also yield structured relaxations for second-order and $\ell_1$-norm cones. The specialized construction reduces per-iteration complexity in interior-point algorithms, and lower-dimensional cones (e.g., SOS-$\ell_2$, SOS-$\ell_1$) allow significant computational speedups compared to generic scalar-SOS representations (Kapelevich et al., 2021).

5. Applications in Optimization and Control

SOS-based conic approximations underlie robust optimization frameworks in discrete and continuous domains:

• Maximum Stable Set problems can be formulated via copositive cones, which are approximated by polyhedral (Pólya-based), SOS (Reznick/Lasserre-type), or factor-width-$k$ hierarchies, each admitting LP, SDP, or SOCP solvers and yielding sequence of upper bounds with finite convergence in favorable cases (Vargas et al., 2023, Ahmadi et al., 2015).
• SOS optimization enables synthesizing polynomial controllers for nonlinear and hybrid dynamic systems. Value functions are upper/lower bounded by polynomials, with nonnegativity certified by SOS decompositions over relevant semialgebraic sets, realized through Gram matrix LMIs or dual cones (Yang et al., 2023).

6. Theoretical Insights, Tropicalization, and Approximability Limits

The power and limitations of SOS-based cones hinge on underlying algebraic-geometric structure:

• In non-Hilbert cases, explicit linear inequalities from Cayley-Bacharach relations separate strictly nonnegative but non-SOS polynomials, furnishing outer approximations and boundary structure (Blekherman, 2010).
• Tropicalizations of the SOS and moment cones yield explicit finite polyhedral approximations. The gap between the moment cone and its SOS dual (pseudo-moment cone) encodes the blind spots of SOS certification: certain binomial inequalities remain unprovable by degree-$d$ SOS (Blekherman et al., 2022). This combinatorial perspective quantifies the limits of sums-of-squares relaxations in approximating nonnegativity.

7. Algorithmic Variants and Numerical Performance

Beyond SDP reformulations, recent developments include nonsymmetric interior-point optimization directly over the SOS cone and its dual via self-concordant barrier representations, interpolation schemes, and primal-dual embedding. This approach achieves dramatically lower time and memory complexity for high-degree instances compared to classical SDP-based methods, and allows explicit recovery of Gram certificates when required (Papp et al., 2017).

Best practices include exploiting specialized SOS cones for structured constraints, leveraging sparse or block-structured formulations whenever possible, and using dynamic atom selection (column generation) or basis pursuit to interpolate between fast LP/SOCP relaxations and expensive SDP exactness (Ahmadi et al., 2015, Ahmadi et al., 2015, Kapelevich et al., 2021).

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Sums-of-squares based conic approximations represent a mathematically rigorous and practically powerful toolkit for convexifying polynomial and matrix nonnegativity, enabling tractable conic programming relaxations with quantifiable tightness/complexity trade-offs, and exhibiting deep connections between convex algebraic geometry, combinatorial optimization, and computational mathematics.