Semidefinite Programming and Matrix Convexity

• Semidefinite programming is a convex optimization framework using symmetric matrix variables and linear matrix inequalities to generalize linear programming.
• Dimension-free SDP extends classical formulations by allowing matrix variables of arbitrary size, enabling versatile applications in control theory and quantum information.
• Non-commutative settings in SDP enforce matrix convexity and guarantee LMI representability, simplifying NP-hard problems through robust operator algebra techniques.

Semidefinite programming (SDP) is a central paradigm in convex optimization where the variables are symmetric matrices subject to linear and positive semidefinite (PSD) constraints. A semidefinite programming problem typically seeks to optimize a linear function over the intersection of an affine subspace and the cone of PSD matrices. SDP generalizes linear programming by replacing scalar inequalities with linear matrix inequalities and is foundational across control theory, combinatorial optimization, quantum information, and systems engineering. The notion of dimension-free SDP further generalizes classical formulations by considering unknowns as matrices of arbitrary size, yielding profound connections with non-commutative real algebraic geometry and operator theory.

1. Algebraic Formulation and Dimension-Free SDPs

In the canonical form, an SDP is given as:

$$\begin{aligned} & \text{minimize} \quad & \langle C, X \rangle \ & \text{subject to} \quad & \langle A_i, X \rangle = b_i \quad (i=1,\dots,m), \ & & X \succeq 0, \end{aligned}$$

where $X$ is an $n \times n$ real symmetric matrix variable, $C$ and all $A_i$ are symmetric matrices, $b_i \in \mathbb{R}$, and $X \succeq 0$ denotes that $X$ is positive semidefinite.

In dimension-free SDP, as formalized in (Helton et al., 2011), the variables $x_j$ of the LMI

$L(x) = I + \sum_{j=1}^g A_j x_j > 0$

are elevated to symmetric $n \times n$ matrices $X_j$, and the inequality becomes

$$L(X) = I_n \otimes I + \sum_{j=1}^g A_j \otimes X_j > 0$$

where the same pencil $(A_1, \dots, A_g)$ defines the LMI independently of the dimension $n$, enabling simultaneous consideration of all matrix sizes. The associated solution set at level $n$ is denoted $\mathcal{D}_L(n)$.

2. Non-Commutative Real Algebraic Geometry and Positivstellensatz

Non-commutative (nc) real algebraic geometry (RAG) investigates polynomials in non-commuting variables—appropriate for matrix-valued unknowns and operator-theoretic formulations. Seminal results in nc RAG, surveyed in (Helton et al., 2011), include:

• Helton’s Theorem: Every matrix-positive, non-commutative polynomial is a sum of squares. This is in sharp contrast to the commutative field, where such representations are generally much subtler.
• Non-commutative Positivstellensätze: These guarantee that, if an nc polynomial is strictly positive on a “free” LMI domain (i.e., a level set of a matrix-valued LMI), it can be written as a sum-of-squares plus linear matrix polynomials, reflecting “quadratic module” membership or a commutator expression.

Formally, using nc polynomials as

$p(x) = \sum_{w} p_w w$

with $w$ a word in non-commuting variables, positivity on all matrix substitutions is characterized algebraically in terms of sums-of-squares and LMI certificates.

3. Convexity in the Non-Commutative Setting

Convexity on free (matrix) variables differs sharply from the commutative polyhedral case. For a set

$S = \{ X : p(X) > 0 \}$

convexity requires that each level $S(n)$ is convex. The critical rigidity result (Theorem 4.1, (Helton et al., 2011)) states:

• Degree Two Rigidity: Every convex and symmetric nc polynomial must have degree at most two. There is no commutative analogue to this constraint.

For example:

• $p(x) = x^2$ is matrix convex: for symmetric matrices $X, Y$,

$$tX^2 + (1-t)Y^2 - [tX + (1-t)Y]^2 = t(1-t)(X-Y)^2 \geq 0$$

• $p(x) = x^4$ is not matrix convex, as demonstrated by explicit matrix substitutions.

Moreover, every bounded convex nc semi-algebraic set can be represented by an LMI (Theorem 3.3), establishing a structural equivalence: in the non-commutative setting, bounded convexity is equivalent to LMI representability.

4. Matrix Variables, Complete Positivity, and Algebraic Structure

Transitioning from scalars ($\mathbb{R}^g$) to tuples of matrices ($S\mathbb{R}^{n \times n}$) significantly enriches the underlying structure. Unique features arising from this quantization include:

• Complete Positivity: For the inclusion $\mathcal{D}_{L_1} \subseteq \mathcal{D}_{L_2}$, there exists a map $T$ between operator systems such that $T$ is completely positive if and only if the inclusion holds. This enables powerful operator-algebraic tools for analysis.
• Unitary Equivalence (Linear Gleichstellensatz): Two LMIs define identical feasible sets (at all levels $n$) if and only if their minimal defining pencils are unitarily equivalent.
• Intrinsic Robustness: Matrix variable formulations are robust under direct sums and unitary conjugation, which aligns with well-studied operations in operator algebras and control.
• NP-Hardness Relaxation: Problems that are NP-hard in the commutative case may become efficiently tractable (via SDP) after quantization to matrix variables, provided convexity (i.e., degree-two representability) is preserved.

5. Fundamental Formulas and Theorems

Key mathematical statements mapping non-commutative convexity and LMI representability include:

• (1) LMI over Scalars: $L(x) = I + \sum_{j=1}^g A_j x_j > 0$
• (2) NC Polynomial Expansion: $p(x) = \sum_{w} p_w w$
• (3) Matrix Convexity Condition: $p(tX + (1-t)Y) \leq t p(X) + (1-t) p(Y)$

Major theorems (as per (Helton et al., 2011)):

• Linear Positivstellensatz (Thm. 2.1): $D_{L_1} \subseteq D_{L_2}$ if and only if $L_2(x) = \sum V_j^* L_1(x) V_j$ for suitable isometries $V_j$.
• Linear Gleichstellensatz (Thm. 2.2): Two linear pencils generate identical feasible domains iff their minimal pencils are unitarily equivalent.
• Quadratic Rigidity (Thm. 4.1): Non-commutative convex symmetric polynomials must be of degree ≤2.
• LMI Representability (Thm. 3.3): Every bounded convex non-commutative semi-algebraic set has an LMI representation.
• Positivstellensatz (various): Strict positivity over matrix substitutions implies representability as sums-of-squares in a quadratic module, often with much greater rigidity compared to the commutative context.

6. Applications, Computational Tools, and Implications

Control and Systems Engineering: The dimension-free SDP framework is essential in linear systems, signal-flow problems, and control synthesis. Key properties of the system—robust stability, dissipativity, disturbance rejection—are encoded as dimension-free LMIs, with solution sets valid for all dimensions. Computation of the smallest pencil, LMI inclusion, and explicit certificate construction are all enabled by the complete positivity and algebraic characterization results.

Quantum Theory: Non-commutative SDPs underpin quantum information, including entanglement detection, quantum channel estimation, and optimization over operator algebras. These applications exploit Positivstellensätze as algebraic certificates and depend crucially on matrix variable quantization.

Mathematical Software: Availability of symbolic/numerical toolkits (e.g., NCAlgebra in Mathematica, NCSOStools in Matlab) allows practical computer-aided exploitation of non-commutative RAG and dimension-free SDP. These packages implement symbolic algorithms for sum-of-squares decomposition, LMI representation, and basic non-commutative convexity checks.

Philosophical and Structural Conclusions: In engineering-relevant dimension-free SDP problems, the paper demonstrates that convexity and LMI representability are equivalent (under boundedness)—reflecting that non-commutative convexity "hides no extra structure" compared to LMI sets. Matrix-variable relaxations yield more rigid algebraic and geometric structure—simplifying theoretical understanding and enabling robust computational methods.

7. Summary and Structural Insights

Dimension-free semidefinite programming establishes an overview of operator algebra, non-commutative real algebraic geometry, and convex optimization. Matrix variables not only provide more tractable relaxations but also enforce strong rigidity through algebraic structure and complete positivity. Fundamental theorems delineate the landscape: non-commutative convex polynomials are always degree two; every bounded convex semi-algebraic set in the nc setting admits an LMI representation; and, in many cases, NP-hardness is mitigated due to quantization. These insights and methodologies form the foundation for modern non-commutative optimization in systems, quantum information, and beyond.