semidefinite programming formulation

Verifying feasibility of degenerate semidefinite programs (SDPs) remains challenging when exact feasible solutions involve irrational numbers, as rational-arithmetic solvers only yield approximate solutions and cannot rigorously certify feasibility. Method: We propose a symbolic–numerical hybrid approach that does not assume the existence of a rational feasible solution. By constructing an isolated real solution system of polynomial equations corresponding to a maximum-rank exact feasible solution, we reduce feasibility certification to a real algebraic geometry problem, then refine approximate numerical solutions using numerical algebraic geometry techniques for exact certification. Contribution/Results: This work establishes, for the first time, an algebraic–geometric framework for SDP feasibility verification under irrationality—without requiring rational feasibility. It successfully certifies several degenerate SDP instances on which purely symbolic methods fail, significantly expanding both the scope and robustness of rigorously verifiable SDPs.

This paper addresses exact community recovery in the asymmetric binary Stochastic Block Model (SBM). While existing semidefinite programming (SDP) methods achieve the information-theoretic limit under symmetry, their failure mechanism in the asymmetric setting remains unclear, and no principled SDP design exists for this case. First, the authors uncover the geometric root of symmetric SDP failure: it provably fails to recover communities exactly in certain information-theoretically solvable regimes, exposing a fundamental limitation of standard SDP formulations. Second, they propose a novel adaptive SDP framework that explicitly incorporates asymmetry via prior constraints on community sizes and a weighted objective function reflecting heterogeneous edge probabilities. Theoretically, the new SDP achieves exact recovery over a strictly broader parameter regime and attains the tight information-theoretic threshold. This work extends the applicability boundary of SDP in statistical inference and establishes a new paradigm for learning from structurally imbalanced graphs.

This work addresses the Max-Cut problem by investigating whether the Goemans–Williamson approximation ratio α_GW ≈ 0.87856 can be surpassed under the assumption that the optimal solution of the standard semidefinite programming (SDP) relaxation lies in a fixed low-dimensional space and satisfies triangle inequalities. To this end, we propose a novel randomized rounding algorithm based on the signs of low-dimensional Gaussian projections and establish a corresponding geometric anti-concentration lemma. Our approach yields, for any fixed dimension d, a polynomial-time algorithm achieving an approximation ratio strictly better than α_GW, with the expected cut value at least (α_GW + 2^{-O(d)}) times the SDP optimum—demonstrating significant performance gains in small dimensions.

This work addresses the optimization landscape analysis of the Burer–Monteiro (BM) low-rank factorization method for MaxCut-type semidefinite programs (SDPs). We identify the condition number of the associated Laplacian matrix as a key spectral criterion: when this condition number falls below a tight, explicitly characterized threshold, every second-order critical point of the nonconvex BM formulation is guaranteed to be a global optimum, and no spurious local minima exist. This constitutes the first tight sufficient condition ensuring global convergence of the BM method for MaxCut-type SDPs, markedly improving the theoretical solvability boundary for canonical problems such as ℤ₂-synchronization. Technically, we integrate tools from Riemannian optimization, spectral graph theory, and second-order critical point analysis to establish a precise quantitative relationship between the Laplacian condition number and the benignness of the optimization landscape. Our results provide a more rigorous theoretical foundation for solving nonconvex low-rank SDPs.

Understanding the interplay between association schemes and Lovász–Schrijver lift-and-project relaxations—particularly SDP-based ones—in combinatorial optimization, especially for stable set polytopes and hypergraph matching. Method: We introduce the notion of *deeply vertex-transitive graphs*, construct *hypermatching pseudo-schemes* (noncommutative association structures), and develop a systematic contraction method to derive commutative sub-schemes. Our approach integrates algebraic combinatorics, spectral graph theory, and SDP analysis. Contribution/Results: We derive Delsarte–Hoffman-type tight bounds on clique and stability numbers; precisely characterize the lift-and-project rank of hypergraph matching relaxations; and establish a general contraction framework from noncommutative pseudo-schemes to commutative sub-schemes. This yields a unified algebraic–convex optimization paradigm for symmetric combinatorial optimization problems.

本文解决了最大割问题中SDP精确性识别的复杂性问题，通过构造具有多项式界整数权重的平方和对偶证书证明了加权图和简单无权图的强NP难性。

Standard graph neural networks (GNNs) struggle to effectively solve large-scale linear semidefinite programming (SDP) problems. This work systematically analyzes the representational limitations of GNNs in SDP solving and introduces a novel, highly expressive GNN architecture capable of precisely emulating the iterative update dynamics of first-order optimization solvers while explicitly modeling key structural properties of SDPs. Evaluated on both synthetic datasets and the SDPLib benchmark suite, the proposed method significantly reduces prediction error and optimality gap, and achieves up to 80% acceleration in solution time under warm-start settings. These results demonstrate the effectiveness and promise of integrating machine learning with convex optimization for scalable SDP solving.

This work presents the first experimental study that simultaneously approximates solutions to both the Max-Cut problem and the Weighted Fractional Cut Cover problem. Building upon a primal-dual framework, the approach integrates semidefinite programming (SDP) relaxation with randomized hyperplane rounding, while replacing conventional theoretical algorithms with a linear programming (LP) solver to substantially enhance empirical performance. Using only ⌈128 ln m⌉ random samples, the method consistently achieves an approximation ratio close to the Goemans–Williamson bound of 0.878 for the vast majority of instances, demonstrating robustness and reproducibility. Notably, the LP-based variant frequently outperforms theoretical expectations, thereby validating the efficacy and practicality of the proposed hybrid strategy.

This study addresses the problem of reconstructing optimal phylogenetic trees under the balanced minimum evolution (BME) criterion in distance-based phylogenetics. For the first time, it introduces semidefinite programming (SDP) to this domain by formulating a tight convex relaxation of the BME problem and coupling it with a tailored iterative rounding strategy to efficiently convert continuous solutions into valid tree topologies. The proposed approach not only establishes a novel optimization framework for BME but also demonstrates potential for extension to other phylogenetic inference problems. Experimental results on both simulated and real datasets show that the method accurately reconstructs phylogenetic trees, confirming its effectiveness and generalizability.

This work reformulates the problem of solving semidefinite programs (SDPs) as the search for optimal strategies in zero-sum semidefinite games, particularly targeting instances that challenge existing methods. Under natural constraint qualifications, it establishes a constructive and complete equivalence between primal-dual SDPs and zero-sum semidefinite games: the game value is zero if and only if a strong optimal solution exists; otherwise, the framework yields an infeasibility certificate for either the primal or the dual problem. By integrating SDP duality theory, a semidefinite generalization of von Stengel’s construction, techniques for handling generalized duality phenomena, and explicit bounds on solution coordinates, this approach extends applicability to a broader class of SDPs and overcomes limitations of prior methods.