Semidefinite Program Formulations

• Semidefinite Program Formulations are optimization models in which symmetric matrix variables are optimized using linear objectives and constraints under the requirement of positive semidefiniteness.
• They generalize linear programming by incorporating lifting methods and symmetric primal–dual integrality constraints to exactly capture the integer hulls in combinatorial problems.
• Key applications include stable set relaxations via the Lovász theta function, MaxCut problems, and extended formulations, linking convex geometry to discrete optimization.

Semidefinite program (SDP) formulations are optimization models wherein a symmetric matrix variable is optimized with respect to linear objectives and linear constraints, subject to positive semidefiniteness. SDPs extend linear programming by allowing matrix variables and exploiting the cone structure of positive semidefinite matrices, which is critical across combinatorial optimization, quantum information, control theory, and machine learning. Recent research has explored the interplay between convexity, integrality, and duality in SDP formulations for discrete optimization and combinatorial problems, seeking to generalize concepts such as total dual integrality (TDI) and provide frameworks to extend the min–max theorems of polyhedral combinatorics.

1. Total Dual Integrality: Definitions and Generalization

In classical polyhedral combinatorics, a system “Ax ≤ b” is called totally dual integral (TDI) if for every integral objective, the dual linear program achieves its optimum at an integral solution. This property is central to equating polyhedral relaxations and integer hulls and underpins strong min–max relations.

Transitioning to the semidefinite setting, no natural integrality is associated with matrices in the positive semidefinite cone. The generalization introduced in (Silva et al., 2018) defines TDI for SDPs “through a lifting” $L$: an SDP system is totally dual integral if for all integral objective vectors $c$, the dual SDP—obtained after lifting $c$ via $L$ and incorporating a primal–dual symmetric integrality constraint—admits an optimal dual solution that is “integral” in a new, algebraically meaningful sense. This aims to ensure that semidefinite relaxations exactly characterize the integer hull for a class of problems and extends the reach of polyhedral min–max theory to nonlinear, non-polyhedral settings.

2. Primal–Dual Symmetric Integrality Constraints

A central technical development is the introduction of integrality constraints that are symmetric in the primal and dual SDPs. These constraints go beyond simply requiring rank–one solutions and instead rely on a decomposition principle:

• Primal integrality: A feasible matrix $X̂$ in the SDP’s lifted variable space is called integral if it can be represented as a nonnegative integral combination of “atomic” rank–one matrices corresponding to characteristic vectors of subsets of the variable set. Explicitly,

$$X̂ = \sum_{A \subseteq V} m_A\, \begin{bmatrix} 1 & x_A^T \ x_A & x_A x_A^T \end{bmatrix}, \quad m_A \in \mathbb{Z}_+$$

with $x_A$ the incidence vector.

• Dual integrality: The dual slack matrix $Ŝ$ is declared integral if it can be written as

$$Ŝ = \sum_{A \subseteq V} m_A\, \begin{bmatrix} 1 & -\chi_A^T \ -\chi_A & \chi_A \chi_A^T \end{bmatrix}, \quad m_A \in \mathbb{Z}_+$$

with sign reversal creating the primal–dual symmetry.

Ranking 1 in the SDP variable is not a sufficiently general integrality concept for SDPs; in key combinatorial cases like MaxCut or the Lovász theta function, rank–one dual solutions are either over-restrictive or correspond to trivial (complete) combinatorial structures. The symmetric integrality condition remedies this gap, aligning the SDP dual relaxation with combinatorial structures.

3. Theoretical Underpinnings: Support Functions and Convex Geometry

The extension of TDI to SDPs is grounded in a generalization of Hoffman’s theorem—from polytopes to convex sets. For a compact convex set $\mathcal{C}$, if its support function $$h_\mathcal{C}(w) = \sup_{x \in \mathcal{C}} \langle w, x \rangle$$ is integral for every integral $w$, then $\mathcal{C}$ is the convex hull of its integer points. This links the algebraic integrality of dual solutions to the geometric property that the convex relaxation coincides with the integer hull, which is preserved under the new integrality constraints on SDPs. The arguments extend to equivalences involving face contains and optimality over rational data.

By moving the focus from extreme points in polyhedra to faces of arbitrary compact convex sets, the SDP context provides a geometric foundation for asserting that TDI, realized through integrality-constrained duals, captures the integer hull of lifted combinatorial objects.

4. Canonical Applications: Stable Sets and the Lovász Theta Function

A principal application is the semidefinite relaxation of the stable set problem via the Lovász theta function. In SDP relaxations, incidence vectors of stable sets are lifted to rank–one matrices. Imposing the proposed primal–dual symmetric integrality constraints on the dual SDP formulation, (Silva et al., 2018) proves that dual integrality is achievable if and only if the underlying graph is perfect.

This equivalence encapsulates the case where the SDP relaxation is tight—matching the integral polytope—and directly connects to the classical result that a perfect graph’s stable set polytope equals its theta body relaxation. The dual optimal integer solutions correspond to clique covers, and the resulting min–max relations align with combinatorial optimality certificates for perfect graphs.

5. MaxCut SDP and Dual Integrality

For the MaxCut problem, the standard semidefinite programming relaxation is tight only for bipartite graphs. When the dual slack matrix is required to be “integrally decomposable” (according to the symmetric integrality constraint), only special cases (e.g., complete graphs or instances with factorable weights) allow feasible dual solutions, indicating that the MaxCut SDP’s dual TDI property is exceedingly restrictive in general.

When edge weights are nonnegative, the only optimal integer dual solution corresponds to the “trivial” certifying cut. This feature exposes the sharp contrast between the rich combinatorial structure in the stable set scenario and the algebraic barriers in MaxCut, where TDI characterizes tightness only for special classes of graphs (e.g., bipartite).

6. Extended Formulations and Lifted Integrality

Extended formulations—modeling combintorial problems in higher-dimensional lifted spaces—are ubiquitous in modern optimization. TDI in the context of SDPs requires careful alignment between lifting maps and integrality constraints:

• Given a lifting $L : \mathbb{R}^k \to \mathbb{S}^{n+1}$, the SDP is TDI through $L$ if for every integral $c \in \mathbb{Z}^k$, the dual (with integrality-constrained slack) has integer optimal solutions.
• With the canonical diagonal embedding $w \mapsto \operatorname{Diag}(0 \oplus w)$, the notion reduces to the classical LP setting.
• Alternative liftings (e.g., Laplacian for MaxCut) enable TDI analysis for structured SDPs where the projection onto original variables remains integral under the symmetric dual constraint.

If the lifting is appropriately designed, the TDI property can be inherited by the semidefinite relaxation, resulting in exact convexification of the initial integer program and enabling combinatorial min–max results in the lifted SDP framework.

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Table: TDI in LPs versus SDPs

LP Formulation	SDP Formulation	Integrality Notion
Ax ≤ b TDI if dual integral	AX = b, X ⪰ 0 TDI if dual integral	Symmetric primal–dual constraints
Polyhedral integer hull	Convex lift equals integer hull	Exactness of convex relaxation
Min–max for perfect graphs	Min–max for perfect graphs (theta)	Only when dual integrality holds

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Summary

Total dual integrality in semidefinite program formulations generalizes the polyhedral notion by using a primal–dual symmetric integrality constraint in lifted matrix space. This approach, grounded in convex geometric theory, links integrality of dual solutions to the tightness of SDPs for key combinatorial optimizations. The methodology yields powerful insight for perfect graphs (Lovász theta), reveals structural limits in MaxCut SDPs, and extends naturally to broad classes of extended formulations, opening new directions in min–max optimization and convexification of discrete problems.