Linear Matrix Inequality-Based Conditions

• Linear matrix inequality-based conditions are convex, algebraic frameworks using matrix pencils to encode feasibility, domination, and positivity problems.
• They generate explicit certificates through operator factorization and complete positivity, enhancing robust optimization and controller design.
• Their formulations convert NP-hard scalar challenges into tractable semidefinite programs, broadening applications in control theory and signal processing.

Linear matrix inequality–based conditions provide a systematic, algebraic, and convex framework for encoding, analyzing, and certifying a wide variety of feasibility, optimization, and dominance problems in control theory, optimization, signal processing, real algebraic geometry, and beyond. At their core, LMIs express the requirement that a parameter-dependent symmetric (or Hermitian) matrix pencil is positive semidefinite, and many central questions—including whether one LMI region contains another, whether a spectrahedron (the feasible set of an LMI) is nonempty, or whether a polynomial is positive over a spectrahedron—reduce to verifying such conditions or constructing explicit algebraic certificates. The development and relaxation of LMI-based conditions, their connection to operator algebra and Positivstellensätze, and their application to both convex and NP-hard problems are foundational not just theoretically but also for computational practice in systems and optimization research.

1. Linear Matrix Inequalities: Pencils, Spectrahedra, and Domination

A linear matrix inequality (LMI) is determined by a linear matrix pencil of the form

$L(x) = A_0 + \sum_{j=1}^g A_j x_j,$

with $A_j \in \mathbb{S}^d$ (the space of real symmetric $d \times d$ matrices). The associated feasibility region (a "spectrahedron") is

$$D_L(1) = \{ x \in \mathbb{R}^g : L(x) \succeq 0 \}.$$

More generally, under "matricial relaxation," variables $x_j$ are replaced by symmetric matrices $X_j \in \mathbb{S}^n$, and the pencil is evaluated as

$$L(X) = A_0 \otimes I_n + \sum_{j=1}^g A_j \otimes X_j,$$

yielding the graded union $$D_L = \bigcup_n \{ X \in (\mathbb{S}^n)^g : L(X) \succeq 0 \}$$.

The relation of "domination"—when the satisfaction of one LMI implies another ($D_{L_1} \subseteq D_{L_2}$)—is central to relaxation hierarchies, robust analysis, and algebraic certification. This formulation also appears when analyzing the inclusion of feasibility sets in semidefinite programming relaxations of otherwise intractable problems.

2. Matricial (Matrix Variable) Relaxation and Complete Positivity

A crucial conceptual advance is the "matricial relaxation" of LMI containment and domination questions: instead of assessing inclusion at the scalar level ($x_j \in \mathbb{R}$), one requires inclusion at all matrix levels ($X_j \in (\mathbb{S}^n)$ for all $n$). Under mild assumptions (e.g., boundedness of $D_{L_1}$ and existence of a jointly positive definite evaluation), the key results are:

• $D_{L_1} \subseteq D_{L_2}$ if and only if there exists an isometry $V$ (or set of matrices $\{V_j\}_{j=1}^k$) such that

$$L_2(x) = V^* (I_k \otimes L_1(x)) V = \sum_{j=1}^k V_j^* L_1(x) V_j.$$

This factorization generates an explicit algebraic certificate of inclusion, known as a Positivstellensatz for linear pencils, and is intimately connected to the theory of completely positive linear maps in operator algebras. The equivalence is precise: defining a unital linear map $$\tau: \operatorname{span}\{I, A_{1,j}\} \to \mathbb{S}^{d_2}$$ by $\tau(I) = I$ and $\tau(A_{1,j}) = A_{2,j}$, the condition $D_{L_1} \subseteq D_{L_2}$ is equivalent to the complete positivity of $\tau$. The Choi matrix of $\tau$, whose positivity can be tested via semidefinite programming, serves as the test for the domination condition.

3. Positivstellensätze: Algebraic Certificates and Putinar-type Results

A Positivstellensatz provides a representation confirming that a given polynomial is positive over a spectrahedron defined by an LMI. For linear pencils $L$, the aforementioned operator factorization is a linear Positivstellensatz certifying positivity for linear matrix polynomials.

Putinar-type Positivstellensätze generalize this result to potentially nonlinear symmetric polynomials $f$ that satisfy $f(X) \succ 0$ for all $X$ in a bounded matricial LMI set. In this framework, there exist matrix polynomials $A_j$ and $B_k$ such that

$f = \sum_j A_j^* A_j + \sum_k B_k^* L B_k,$

with no need for additional bounding terms due to the automatically archimedean nature of the quadratic module associated with a bounded $D_L$. Thus, positive polynomials on bounded spectrahedra admit explicit sum-of-squares certificates with explicit dependence on the pencil $L$. Nonnegativity certificates (Nichtnegativstellensätze) follow analogously, providing a means to verify the nonnegativity of a polynomial on the feasible region.

4. Algorithmic and Structural Implications

The translation of LMI domination and positivity conditions to complete positivity problems and Positivstellensätze has direct algorithmic consequences. Canonical hard problems such as the matrix cube problem (which asks for the largest box contained in a spectrahedron and is NP-hard in general) can, under the matricial relaxation, be reformulated as feasibility questions for LMIs over matrix variables, which in turn can be checked via semidefinite programs for complete positivity. The Choi matrix formulation is computationally accessible, and factorization conditions yield efficient semidefinite programming approaches for certificate generation.

For equivalence of LMI solution sets, the theory shows that (up to redundancy), two pencils with the same solution set are unitarily equivalent—a strong structural result for the classification problem.

5. Context and Applications

LMI-based conditions have wide application:

• Control theory: Stabilization, optimal control synthesis, and robust performance are frequently formulated via LMIs. Domination conditions are used when comparing different relaxations or controller designs.
• Robust optimization: Certification of worst-case performance and "robust" formulations often require LMI inclusion or contraction arguments.
• Signal processing: Design of filters and arrays can be rephrased as LMI feasibility, dominance, or matrix polytope containment problems.
• NP-hard relaxations: Problems like the (scalar) matrix cube problem, difficult at the scalar level, are amenable to canonical solution at the matricial level via SDPs and operator positivity.

The matricial relaxation, operator factorization, and Positivstellensatz viewpoints unify these applications and provide concrete, testable, and certifiable conditions in terms of explicit matrix factorizations and semidefinite programs.

6. Operator Theory and the Classical Problem Equivalence

The central operator-theoretic realization is that LMI domination (under matricial relaxation) is equivalent to the complete positivity of a unital linear map between operator systems spanned by the pencil coefficients. By Choi's theorem, complete positivity can be certified with a positive semidefinite matrix (the Choi matrix), and when the map is a complete isometry, the pencils' matricial positivity domains exactly coincide.

Typical certificates are of Stinespring type: given an isometry $V$, the pencil $L_2(x)$ is realized as a compression of $L_1(x)$ across a possibly higher-dimensional space. This framework connects classical topics in operator algebras directly to semidefinite optimization and LMI feasibility/containment questions.

7. Summary

Linear matrix inequality–based conditions provide a rigorous algebraic and geometric machinery for certifying positivity, feasibility, and inclusion properties of spectrahedra and enable explicit, tractable certificates and relaxations for problems that are otherwise intractable at the scalar level. The connection with complete positivity, operator system theory, and Positivstellensatz results imparts both structural insights and practical computational tools, undergirding much of modern convex optimization, control, and real algebraic geometry.

The development in "The matricial relaxation of a linear matrix inequality" (Helton et al., 2010) formalizes and clarifies these relationships and demonstrates that the seemingly intractable problem of LMI domination becomes tractable and certifiable via the combination of matricial relaxation, operator factorization, and the machinery of complete positivity and algebraic certificates. These advances underlie efficient semidefinite programming algorithms for relaxation, domination, and positivity of polynomials on spectrahedral domains.