Matrix Optimization Reformulation

• Matrix optimization reformulation is a set of methods that recast matrix-constrained problems using factorization, conic, and manifold techniques to enhance tractability.
• It transforms problems through squared-variable, spectral, and coordinate frameworks, enabling advanced nonlinear programming and revealing structural insights.
• These reformulations yield efficient algorithms—with demonstrated speedups and near-optimal results—across diverse applications such as SDP, QCQP, and power flow.

Matrix optimization reformulation encompasses a broad spectrum of methodologies that recast matrix-constrained or matrix-variable problems into alternative forms—often involving factorizations, variable transformations, conic or manifold constraints—allowing the application of advanced nonlinear programming, conic optimization, or manifold-based algorithms. Reformulation can facilitate tractable solution strategies, improve scalability, enable algorithmic compatibility, and reveal structural insights unavailable in the original representation. This article surveys the principal reformulation paradigms, mathematical underpinnings, and computational consequences as documented in recent literature.

1. Matrix Factorization and Squared-Variable Paradigms

The squared-variable reformulation is a fundamental strategy for nonlinear semidefinite programming (NSDP) and matrix-constrained problems. For positive semidefinite constraints $X\in\mathbb{S}^d$, $X\succeq0$, the variable $X$ is replaced by $FF^\top$, where $F\in\mathbb{R}^{d\times d}$ (nonsymmetric case) or $F\in\mathbb{S}^d$ (symmetric case), converting the problem into the so-called squared-variable form. The main reformulation classes are:

• BC/DSS (Unconstrained Matrix): The original PSD-constrained minimization $\min h(X)$, $X\succeq0$ becomes an unconstrained smooth minimization $\min g(F):=h(FF^\top)$ over $F$ (Ding et al., 4 Feb 2025).
• NSDP/SSV (With Matrix Constraints): For general variable $x$ and matrix constraint $C(x)\succeq0$, introduce a slack variable $F$ and enforce equality $C(x)=FF^\top$, yielding an equality-constrained nonlinear program.
• SSV-Sym (Symmetric Squared-Variable): Further restrict $F$ symmetric, $C(x)=F^2$, which introduces additional spectral/eigenvalue consistency conditions.

While squared-variable reformulations enable the use of standard NLP techniques, first-order (KKT) correspondences can fail due to the non-injective nature of $F\mapsto FF^\top$ (e.g., $FQ$ yields the same $X$ for any orthogonal $Q$), which precludes unique dual variable recovery and impacts dual feasibility unless rank-deficiency or spectral nondegeneracy holds. However, second-order necessary conditions and local minimizers correspond precisely: any $X$ satisfying 2NC in the original PSD-formulation generates $F$ satisfying 2NC in the squared-variable reformulation, and vice versa (with eigenvalue conditions for symmetric squares). Strict local minimizers generally do not exist except at $F=0$ due to rotational symmetry of $F\to FQ$ (Ding et al., 4 Feb 2025).

2. Coordinate and Spectral Reformulation Frameworks

General constrained matrix optimization extends the reformulation logic by decomposing $X$ through spectral or SVD factorizations, separating coordinate and spectral constraints:

• Spectral Decomposition Model: Given $X\in S^n$ with eigenvalues $\lambda(X)$, one rewrites $X=Q\,\operatorname{Diag}(\lambda)\,Q^\top$, $Q\in O(n)$, $\lambda\in\mathbb{R}^n_\downarrow$, translating the original problem into one over manifold and spectral variables (Garner et al., 2024).
• Rectangular Matrix Case: For non-symmetric $X\in\mathbb{R}^{m\times n}$, SVD parameterization $X=U \operatorname{Diag}(\sigma)V^\top$, $U,V$ Stiefel matrices, decouples singular value and coordinate constraints.

This reformulated space is treated as a product manifold $M=O(n)\times\mathbb{R}^n$, and constraints are classified into manifold equality (e.g. $G(X)$) and spectral inequality (e.g. $g(\lambda(X))\leq0$). The staged block-coordinate algorithm alternates between spectral updates (over $\lambda$), coordinate (orthogonal manifold) updates (over $Q$), and joint KKT steps, ensuring descent and feasibility on $M$. Global convergence to $(\epsilon,\epsilon)$-approximate KKT points in $\mathcal{O}(1/\epsilon^2)$ iterations is established under standard regularity conditions. Nonconvex spectral constraints—for instance, rank-1 enforcing $λ_1(X)\geq\delta, λ_2(X),…,λ_n(X)\leq\delta$—can be addressed directly in this framework, often outperforming classical convex relaxations plus randomization in QCQP and generalized SDP settings (Garner et al., 2024).

3. Reformulation to Conic and Matrix-Sparse Models

For quadratic and semidefinite programs, canonical and sparse conic reformulations enable efficient solution and tighter approximation hierarchies:

• SOCP Reformulation for GTRS: By simultaneous block diagonalization (Uhlig canonical form), the quadratically constrained quadratic problem becomes a direct sum of $1\times1$ and $2\times2$ blocks. The reformulated problem uses rotated second-order cone constraints, yielding a pure SOCP of size $O(n)$ without large-scale SDP variables. This enables fast solution, direct back-substitution recovery of primal solutions, and simplified forms of the S-lemma (for inequalities, equalities, and interval constraints) (Jiang et al., 2016).
• Matrix Minor Reformulation (Branch-and-Cut for AC-OPF): SDP rank constraints are equivalently enforced via zeroing all $2\times2$ minors, classified into principal, 3-cycle, and 4-cycle types. These translate into conic, bilinear, or McCormick envelope (linearized) constraints, capturing key physical invariants (cycle angle sums) in power flow. SOCP relaxation with systematic strengthening cuts (edge cuts, cycle cuts, arctan envelopes) yields fast, strong global bound closure far outpacing traditional SDP approaches (Kocuk et al., 2017).
• Sparse Copositive Reformulation for QCQP: For structured scenario-based QCQPs, variables are lifted into a lower-dimensional set-completely positive cone $CMP(\bar{\mathcal C})$ tailored to the problem block structure, leveraging arrow-head and clique decompositions. Inner and outer approximation hierarchies (e.g., $CPS(\bar{\mathcal{C}})$, $CPI(\bar{\mathcal{C}})$) achieve near-exactness under mild assumptions, and facial reduction techniques confirm that added constraints often carve out faces of the sparse cones, closing the completability gap without re-introducing full Burer-lift dimensions. Computational experiments show two orders of magnitude speedup and vanishing optimality gaps compared to full SDP or global MIP solvers (Gabl, 2021).

4. Manifold-Based and Matrix-Free Reformulation Techniques

Matrix optimization reformulation increasingly leverages manifold and matrix-free structures for scalability and efficiency:

• Riemannian Manifold Formulations: Nearest $Ω$-stable matrix problems become smooth optimization on $U(n)$ or $O(n)$, with objectives involving the Schur parametrization and direct handling of spectral constraints via triangular matrices with prescribed diagonals. The Riemannian trust-region algorithm exploits intrinsic manifold geometry, allowing robust solution and eigenvalue multiplicity handling, often outperforming classical eigen-decomposition methods (Noferini et al., 2020).
• Matrix-Free Jacobian Chaining: The chain-product of Jacobian matrices in large-scale simulation can be reformulated in terms of tangent (matrix–Jacobian) and adjoint (Jacobian–matrix) vector kernels, subject to DAG memory (“tape”) constraints. A dynamic-programming recursion over evaluation strategies optimally partitions tangent, adjoint, and explicit multiplication, dramatically reducing the total computational cost under memory constraints (Naumann, 2024).

5. Structure-Exploiting Matrix Reformulation in Application Domains

Domain-specific matrix reformulations capitalize on structural constraints (diagonal, constant modulus, block, symmetry):

• Complex Matrix Derivatives Under Structure Constraints (Wireless/MIMO): Problems with diagonal and constant modulus constraints admit closed-form solutions via specialized complex matrix-derivative identities, reducing capacity maximization and MSE minimization to efficient solution procedures (water-filling, quadratic equations) or fast alternating optimization (AO) algorithms. Under entrywise or block-diagonal decoupled constraints, phase-derivative classification enables O(1) per-update complexity, eschewing matrix inversion or SVD in fully-passive IRS and hybrid analog/digital MIMO designs (Ju et al., 2023).
• Matrix Multiplication Optimization with Mixed Discrete/Continuous Variables: Nonlinear matrix multiplication chains in material science and biology can be converted into compact MILP or MIQCQP formulations exploiting bilinear reformulation, auxiliary variables, and scenario coupling. These models solve to optimality for moderate $N$ and demonstrate superiority over heuristics or brute-force enumeration prevalent in the literature (Kocuk, 2020).

6. Limitations and Practical Implications

Nearly all reformulation paradigms carry inherent caveats:

• First-Order Condition Mismatch: Squared-variable reformulation first-order points do not guarantee dual feasibility or correct positive semidefiniteness unless extra spectral or rank constraints hold (Ding et al., 4 Feb 2025).
• Rotational Symmetry: Factorization-based representations often yield orbits of equivalent solutions (e.g., $F\to FQ$), and strict local minima are rare except at singular points.
• Nonconvexity and Scalability: While spectral, conic, and manifold reformulations are tractable for moderate problem sizes, scalability to very large $n$ may require custom preconditioning, Hessian approximation, or parallel splitting strategies (Noferini et al., 2020Naumann, 2024).
• Structure Constraints: Some structural constraints (sparsity, Toeplitz, symmetry, constant modulus) may not be compatible with all reformulation approaches (e.g., the Schur manifold reparametrization), necessitating specialized algorithms or derivative identities (Ju et al., 2023).

A plausible implication is that the choice of matrix optimization reformulation is tightly problem-dependent, sensitive to variable structure, scale, constraint type, and computational objectives.

7. Computational Performance and Comparative Results

Recent computational studies show that reformulation approaches deliver dramatic improvements in tractability, optimality gap closure, and scalability compared to classical convex or enumeration methods:

Reformulation Class	Relative Gap/Accuracy	Computational Time/Scale	Key Reference
Sparse Copositive (QCQP)	≤0.1% (inner approx.)	2–100× faster than full SDP	(Gabl, 2021)
SOCP for GTRS	Provably tight, O(n) cones	Lower time constant than SDP	(Jiang et al., 2016)
Matrix Minor SOCP (AC-OPF)	Avg. 0.53% gap (root), global	Under 300s (root), <720s (global)	(Kocuk et al., 2017)
Riemannian Nearest $Ω$-stable	Robust to multiplicity	Multiple seconds for $n=100$	(Noferini et al., 2020)
Squared-Variable NSDP	2NC and local min equivalence	Standard NLP solvers	(Ding et al., 4 Feb 2025)
Structure-Exploiting AO	Equal performance, $\sim$40% time reduction	$N_t=30$	(Ju et al., 2023)

Empirical findings suggest reformulated models—when structure is well aligned to the reformulation technique—often outperform state-of-the-art convex relaxation or brute-force approaches by an order of magnitude or more.

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Matrix optimization reformulation continues to evolve, integrating advances in factorization, conic relaxation, manifold optimization, and structural exploitation. Current research demonstrates that careful reformulation tailored to the matrix variable and constraint structure is essential for enabling scalable, reliable, and highly accurate solution of complex matrix optimization problems.