SDP Hierarchies for Inner Approximations

• The paper introduces an SDP-based hierarchy that systematically replaces nonconvex constraints with LMIs, ensuring convergent inner approximations for optimization problems.
• It leverages sum-of-squares relaxations, moment representations, and conic duality to approximate polynomial, copositive, and combinatorial sets with increasing precision.
• Applications in control theory, combinatorial optimization, and dynamical systems demonstrate practical methods with proven convergence guarantees and reduced conservatism.

The SDP-based hierarchy of inner approximations is a foundational technique in modern convex and polynomial optimization, combining sum-of-squares relaxations, conic duality, and moment-theoretic representations to construct tractable and systematically improving inner approximations to challenging nonconvex objects—such as feasible regions defined by polynomial inequalities, copositive cones, or combinatorial polytopes. This methodology replaces intractable nonconvex constraints by more manageable linear matrix inequalities (LMIs) or semidefinite programs (SDPs), yielding convergent sequences of convex approximations with provable guarantees in the limit. The hierarchy appears in various forms, including the Lasserre hierarchy for polynomial optimization, hierarchies for the copositive and completely positive cones, and elaborate constructs for dynamical systems and robust control, enabling rigorous approximations to sets, cones, and solution regions otherwise inaccessible to direct optimization.

1. Fundamental Principles of SDP-Based Hierarchies

At the core of the SDP-based hierarchy is the idea that hard membership constraints—typically defined by nonconvex conditions such as polynomial positivity, copositivity, or combinatorial set representations—can be replaced by a sequence of convex restrictions derived from polynomial sum-of-squares (SOS) representations and moment relaxations. Each level $d$ of the hierarchy corresponds to matrices or polynomials of degree up to $2d$, imposing positive semidefiniteness constraints (LMIs) on matrices of moments or localizing polynomials.

For optimization over polynomials, the Lasserre hierarchy (Doherty et al., 2012, Jeyakumar et al., 2013) systematically lifts a scalar polynomial optimization problem to an LMI-based optimization over truncated moment sequences, exploiting Putinar's Positivstellensatz for compact sets and extended quadratic modules for the noncompact case (Jeyakumar et al., 2013). For cones such as the copositive or completely positive cones, the hierarchy constructs nested spectrahedra or their duals, yielding respectively outer or inner approximations (Lasserre, 2010, Kuryatnikova et al., 2018).

This approach is underpinned by duality (between moment and SOS representations) and by the existence of moment matrices that are nearly representable by probability measures—guaranteed quantitatively in the case of polynomial optimization over the hypersphere via a de Finetti-type theorem (Doherty et al., 2012). Increasing the level of the hierarchy yields tighter approximations; in the limit, these converge to the true nonconvex set.

2. Constructions for Key Classes of Problems

The formulation of SDP-based hierarchies is problem dependent. For the copositive cone $$C = \{A \in \mathbb{S}^n : x^\top A x \geq 0\ \forall x \geq 0\}$$, the outer approximation $C_d$ is defined by requiring the moment (localizing) matrix

$M_d(f_A y) \succeq 0,$

where $f_A(x) = x^\top A x$ and $y_\alpha$ are the moments of a reference measure on $\mathbb{R}_+^n$ (Lasserre, 2010). The duals $C^*_d$ provide explicit inner approximations to the cone of completely positive matrices:

$$C_d^* = \overline{ \left\{ \int_{\mathbb{R}_+^n} xx^\top \sigma(x)\, d\mu(x) : \sigma \in \Sigma[x]_d \right\} }.$$

In the context of robust controller design for polynomial matrix inequalities (PMIs), the method seeks a sequence of polynomials $g_d(x)$ underestimating a robust eigenvalue function, using an LMI formulation:

$$v^\top P(x, u) v - g(x) = r(x,u,v)(1-v^\top v) + \sum_i s_i(x,u,v) a_i(u) + \sum_j t_j(x,u,v) b_j(x),$$

where all multipliers are SOS, leading to the inner superlevel set $G_d = \{x \in B : g_d(x) \geq 0\}$ (Henrion et al., 2011).

For the region of attraction (ROA) in polynomial dynamical systems, the hierarchy is constructed by relaxing an infinite-dimensional LP over occupation measures to a finite SDP parametrized by truncated moment sequences, with the dual given by SOS certificates for the Liouville equation and nonnegativity constraints, yielding polynomial sublevel sets as inner approximations to the ROA (Korda et al., 2012, Oustry et al., 2019).

3. Convergence Guarantees and Theoretical Properties

The haLLMark property of SDP-based hierarchies is asymptotic convergence: at each level, the inner approximation is guaranteed to be contained within the true set, and the sequence fills out the set in the limit. For cones, one has

$$C^* = \operatorname{cl} \left( \bigcup_{d=0}^\infty C_d^* \right),$$

and for feasible sets $P$ of PMI problems,

$$\lim_{d\to\infty} \mathrm{vol}(P \setminus G_d) = 0.$$

In the case of invariant sets for dynamical systems, the Lebesgue measure of the difference between the true set and the union of inner approximations vanishes as the relaxation order increases (Korda et al., 2012, Oustry et al., 2019).

Several papers establish finite convergence under strong conditions. For convex polynomial programs, with a Lagrangian having positive-definite Hessian at a saddle point, the Lasserre hierarchy with an extended quadratic module achieves finite convergence (Jeyakumar et al., 2013).

4. Implementation, Scalability, and Approximability

Each level of the SDP-based hierarchy corresponds to an explicit semidefinite program defined by the moments or Gram matrices associated with the chosen degree. Although theoretically tractable, the size of the SDPs grows quickly with the relaxation order, motivating structured approaches (such as chordal block decompositions (Miller et al., 2019), use of DSOS/SDSOS cones (Kuang et al., 2015), and scalable ADMM algorithms (Sinjorgo et al., 10 Jun 2025)) to manage large instances.

For families of problems where the maximum feasible basis size is limited (for example, in intermediate-level Lasserre relaxations (Sinjorgo et al., 10 Jun 2025)), selection of basis elements (e.g., degree two monomials with the largest support in the optimal theta solution) can provide a practical trade-off between tightness and tractability.

A crucial feature is that all constraints, including SOS conditions, can be formulated as LMIs and thus handled by generic or specialized SDP solvers.

5. Applications and Impact

The SDP-based hierarchy of inner approximations has penetrated a wide range of fields:

• Combinatorial optimization: Bounds on independence number, Max-Cut, and general CSPs via Lasserre/SOS, copositive, or completely positive cone hierarchies (Lasserre, 2010, Sinjorgo et al., 10 Jun 2025, Vargas et al., 5 Sep 2025).
• Control theory: Certified design of robust controllers via inner approximations of nonconvex PMI-defined stability regions (Henrion et al., 2011, Dinh et al., 2012).
• Optimization over nonconvex sets: Quantitative approximations of feasible sets defined by polynomial, set-cover, or semialgebraic constraints (Chlamtac et al., 2012, Guthrie, 2022).
• Polynomial dynamical systems: Certified ROA or invariant set approximation via occupation measures and Lasserre hierarchies (Korda et al., 2012, Oustry et al., 2019).
• Quantum and statistical learning: Moment problems over the hypersphere and links to de Finetti theorems (Doherty et al., 2012).

The hierarchies enable vanishing conservatism and provide practical algorithms for safety verification, controller synthesis, uncertainty quantification, and more.

6. Limitations, Trade-offs, and Innovations

While theoretically powerful, several challenges persist:

• The size of the associated SDP grows combinatorially with both problem dimension and relaxation order.
• In applications such as Set Cover or problems with high integrality gaps, auxiliary techniques (lifting the objective, custom rounding strategies) are essential to exploit the full hierarchy strength (Chlamtac et al., 2012).
• Alternative cones (DSOS, SDSOS, block factor-width) and basis-pursuit-type iterative methods have been introduced to mitigate computational barriers while retaining accuracy (Kuang et al., 2015, Liao et al., 2022, Miller et al., 2019).

Recent developments offer partial facial reduction techniques that generate smaller equivalent SDPs leading to better scalability in practice (Permenter et al., 2014).

7. Connections to Copositive and Completely Positive Cones

A significant thread in the theory is the connection to the copositive and completely positive cones. The hierarchy presented in (Lasserre, 2010) and further explored in (Kuryatnikova et al., 2018, Vargas et al., 5 Sep 2025) uses polynomial moment and sum-of-squares representations to yield convergent sequences of outer (for the copositive cone) and inner (for the completely positive cone) approximations, implemented as spectrahedral sets. The inner approximations of the completely positive cone are especially critical in optimization over quadratic forms and in modeling NP-hard problems as convex programs over tractable subsets of these cones.

Notably, certain graph families, previously requiring nearly as many levels as the independence number to achieve tightness in related hierarchies, can be resolved in a single step with these new SDP-based SOS hierarchies (Vargas et al., 5 Sep 2025). This tightness result precisely delineates the boundary of approximability for these structured cones under the SDP/SOS paradigm.

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This hierarchy of inner approximations thus provides a versatile and unifying framework for tackling nonconvexity in optimization and system analysis, combining rigor, generality, and—through recent computational innovations—increasing practicality across domains.