Convex Hull Alignment

• Convex Hull Alignment is a method that constructs successive convex relaxations (theta bodies) to approximate the convex hull of algebraic sets defined by polynomial equations.
• It utilizes sums-of-squares certificates and moment matrix duality to convert polynomial nonnegativity conditions into semidefinite programming constraints.
• The approach generalizes Lovász theta body and Lasserre’s hierarchy, achieving finite or asymptotic convergence for optimization problems in combinatorial and algebraic geometry.

Convex hull alignment refers to the systematic construction of successive convex approximations (“theta bodies”) to the convex hull of an algebraic set—i.e., the real variety defined by a system of polynomial equations in $\mathbb{R}^n$. This framework, which leverages sums-of-squares (sos) certificates and the dual moment matrix methodology, yields a convergent or even finitely convergent hierarchy of SDP-representable convex relaxations that are tightly aligned with the true convex hull. The approach generalizes the classical theta body construction for graphs (Lovász theta body) and is closely related to Lasserre’s hierarchy for global polynomial optimization. When the defining ideal is real radical, powerful convergence and exactness results are obtained, making the method particularly effective for both finite and infinite algebraic varieties.

1. Algebraic Varieties, Polynomial Ideals, and Problem Setup

Let $I \subseteq \mathbb{R}[x_1, ..., x_n]$ be a polynomial ideal. The real variety

$$V_{\mathbb{R}}(I) = \{ x \in \mathbb{R}^n : f(x) = 0, \forall f \in I \}$$

serves as the (possibly discrete or infinite) set for which the convex hull $\mathrm{cl}(\mathrm{conv}(V_{\mathbb{R}}(I)))$ is sought. Instead of direct computation, the alignment method constructs a sequence of outer convex relaxations—theta bodies—defined in terms of the ideal $I$ and the sos structure of polynomials over $I$.

Key steps:

• Selection of a $\theta$-basis for $\mathbb{R}[x]/I$ to enable computations modulo the ideal.
• Focus on linear polynomials $l(x)$ nonnegative on $V_{\mathbb{R}}(I)$. If $l(x)$ is congruent, mod $I$, to a sum of squares of degree-$k$ polynomials, then for all $x \in V_{\mathbb{R}}(I): l(x) \geq 0$.
• Definition of the $k$th theta body:

$$\mathrm{TH}_k(I) = \left\{ x \in \mathbb{R}^n : l(x) \geq 0 ~ \forall~ l \text{ s.t. } l \equiv \sum_i g_i^2 \pmod{I},~ \deg(g_i) \leq k \right\}$$

This process operationalizes convex hull alignment as the intersection of all linear inequalities “certified” by degree-$k$ sos modulo $I$.

2. Sums-of-Squares Certificates and Moment Matrix Duality

The approach’s core is the equivalence between algebraic nonnegativity certificates (modulo $I$) and semidefinite representations:

• For $l(x)$, check if $l(x) \equiv \sum_i g_i(x)^2 \pmod{I}$ with degree constraint.
• In the dual moment matrix framework, moments indexed by multi-indices of basis monomials (up to degree $k$) are gathered into the vector $y$, and the truncated moment matrix $M_k(y)$ is formed:

$$M_k(y) = \left[ y_{\alpha+\beta} \right]_{\alpha, \beta \in \mathcal{A}_k} \succeq 0$$

where $\mathcal{A}_k$ is the index set of all degree-$\leq k$ basis elements.

• Membership $x \in \mathrm{TH}_k(I)$ is thus encoded by existence of a “moment sequence” $y$ (with $y_0=1$ and $\pi(y) = x$) such that $M_k(y)$ is psd.
• Concrete example for the graph stable set problem:

$$\mathrm{TH}_1(I_G) = \left\{ x \in \mathbb{R}^n : \begin{array}{l} \exists~ M \succeq 0~ \text{with } M_{00} = 1,\ M_{0i} = M_{ii} = x_i,\ M_{ij} = 0~ \forall~ \{i,j\} \text{ an edge} \end{array} \right\}$$

This relaxation is directly derived from the nature of the sos certificates.

3. Hierarchies: Theta Bodies, Lasserre’s Sequence, and Convergence

The sequence of theta bodies,

$$\mathrm{TH}_1(I) \supseteq \mathrm{TH}_2(I) \supseteq \cdots \supseteq \mathrm{cl}(\mathrm{conv}(V_{\mathbb{R}}(I))),$$

serves as a hierarchy of convex relaxations similar to the Lasserre hierarchy in global polynomial optimization. Both hierarchies apply sos and moment matrix constraints to approximate (and, under suitable conditions, exactly recover) the convex hull of the variety or the feasible set.

• Alignment Mechanism: As $k$ increases, $\mathrm{TH}_k(I)$ “tightens” around the true convex hull. For finite varieties, the process terminates after finitely many steps; for infinite varieties or varieties with certain singularities, convergence is asymptotic.
• Lasserre alignment analogy: For equality-constrained cases, the Lasserre hierarchy’s moment matrix relaxations coincide with the succession of theta bodies.

A plausible implication is that, in practical optimization, this structure allows highly tractable outer approximations that fall back to exactness after finitely many steps in combinatorial settings.

4. Role of Real Radical Ideals

Real radicality of $I$—$$I = \{f \in \mathbb{R}[x] : f(x)=0~\forall x \in V_{\mathbb{R}}(I)\}$$—is central. Under real radicality:

• Every nonnegative linear function on $V_{\mathbb{R}}(I)$ admits a $k$-sos certificate for some $k$.
• The moment matrix and direct sos definitions of theta bodies agree up to closure.
• Hierarchy convergence becomes more tractable: for finite $V_{\mathbb{R}}(I)$, finite $k$ exists such that $$\mathrm{TH}_k(I) = \mathrm{conv}(V_{\mathbb{R}}(I))$$.

This tightness does not generally hold for non-real-radical ideals; the alignment may then be only asymptotic and require consideration of closure and infinite steps.

5. Illustrative Examples and Convergence Phenomena

Selected examples from the paper clarify both the reach and practical computation of convex hull alignment:

• Stable set polytope (Lovász theta body): For a graph $G$, $I_G$ comprises $x_i^2 - x_i$ (0/1 constraints) and $x_i x_j$ (for edges), yielding $\mathrm{TH}_1(I_G)$. For perfect graphs, this exactly equals the stable set polytope.
• Odd cycles: Cycle inequalities for odd cycles are $2$-sos mod $I_G$, so $\mathrm{TH}_2(I_G)$ recovers the stable set polytope here as well.
• Algebraic plane curves: For an infinite variety (e.g., a cardioid), $\mathrm{TH}_2(I)$ provides a close, though not exact, convex hull approximation in the absence of a 1-sos certificate.

Convergence results:

• For finite varieties, finite convergence: $\exists$ finite $k$ with $$\mathrm{TH}_k(I) = \mathrm{conv}(V_{\mathbb{R}}(I))$$.
• For compact $V_{\mathbb{R}}(I)$, $$\bigcap_k \mathrm{TH}_k(I) = \mathrm{cl}(\mathrm{conv}(V_{\mathbb{R}}(I)))$$.
• Absence of convex-singular points is sufficient for finite convergence.

6. Applications in Optimization and Algebraic Geometry

Applications of convex hull alignment using theta bodies and sos techniques are extensive:

• Combinatorial optimization: Enables approximation and, in many cases, exact representation of polytopes associated with discrete structures (e.g., maximum stable sets, cuts, etc.) via polynomial-time solvable SDPs.
• Integer programming: The approach generalizes the Lovász theta body, extending to general 0/1 polynomial equation systems.
• Real algebraic geometry: Provides semidefinite representations for convex hulls of semialgebraic sets, which impacts both the study of real varieties and the design of polynomial-time algorithms for global polynomial optimization.
• Exactness and strong relaxations: Offers criteria for when convex hull alignment is attained at a specific step, facilitating the design of tailored relaxations for graph-theoretic and other combinatorial problems (e.g., maxcut, automorphism groups).

A summary table of the alignment approach:

Step/Concept	Mathematical Object	Key Property/Outcome
Algebraic set	$V_{\mathbb{R}}(I)$	Real variety: solution set of polynomial equations
Theta body definition	$\mathrm{TH}_k(I)$	Outer convex approximation, k-sos mod $I$ for certifying inequalities
SDP representation	$M_k(y) \succeq 0$	Feasibility yields $x \in \mathrm{TH}_k(I)$
Hierarchy of relaxations	$\mathrm{TH}_1(I) \supseteq ...$	Converges to cl(conv($V_{\mathbb{R}}(I)$)); finite for finite $V_{\mathbb{R}}(I)$
Real radical ideal	$I = \sqrt[\mathbb{R}]{I}$	Tight certificates; easier convergence

References to Lovász Theta Body and Generalizations

This approach is motivated by and generalizes the Lovász theta body for graphs, extending its principles to arbitrary polynomial ideals—thereby encompassing a large class of optimization and geometric problems. The convex hull alignment thus achieved is both algorithmically tractable and theoretically grounded, connecting modern polynomial optimization, real algebraic geometry, and the theory of semidefinite relaxations in a unifying framework.