randomized sdp sampling

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 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 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.

This paper addresses the Sparse Integer Least Squares (SILS) problem—NP-hard least squares optimization under {0, ±1} constraints and sparsity. To tackle it, we propose the first ℓ₁-regularized semidefinite programming (SDP) relaxation and design a complementary randomized rounding algorithm. Theoretically, we derive a sufficient condition for exact recovery via SDP under fixed sparsity, leveraging sub-Gaussian analysis and second-moment characterization of the covariance matrix to establish exact recovery even under weakly correlated features. Empirically, we validate the method’s effectiveness and practicality across diverse applications, including privacy-preserving identification, multi-user detection, feature extraction, and integer sparse signal recovery. Our approach bridges rigorous theoretical guarantees—providing provable exact recovery under realistic statistical assumptions—with computational tractability and real-world deployability.

Rounding half-integral solutions of the Traveling Salesman Problem (TSP) while eliminating subtours remains a fundamental challenge in combinatorial optimization. Method: We propose a novel randomized rounding algorithm that— for the first time—integrates matroid intersection polytope sampling with maximum-entropy distribution sampling, synergistically combining linear programming relaxation and combinatorial structural analysis. Contribution/Results: Our method strictly breaks the long-standing 1.5-approximation barrier: it runs in polynomial time and outputs a feasible Hamiltonian cycle with a provable approximation ratio strictly less than 1.5—improving upon the tight 1.5 bound established by Karlin et al. (2021). Beyond tightening the optimality bound for half-integral TSP rounding, our work establishes a new rounding paradigm grounded in polyhedral sampling and entropy regularization. This framework generalizes to broader LP-based rounding techniques in combinatorial optimization, offering a principled methodological advance.

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。

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 study addresses the problem of reconstructing quantum channels from classical data samples, focusing on scenarios where the overall fidelity can be expressed as the ratio of two quadratic forms. The work proposes an optimization approach based on semidefinite programming (SDP), leveraging the Choi matrix representation and Kraus operator decomposition to efficiently solve the channel reconstruction problem. It is the first to systematically apply SDP to a variety of quantum channel learning settings, achieving high-precision reconstructions using off-the-shelf solvers. Experimental results reveal that the reconstructed channels typically exhibit Kraus ranks amounting to only a few percent of their theoretical maximum, indicating that real-world quantum processes possess remarkably low intrinsic complexity. This insight substantially enhances both the efficiency and practicality of quantum channel reconstruction.

This work proposes a novel approach to the Max-3-Cut problem based on complex-valued quadratic optimization, leveraging the low-rank structure of the objective matrix to circumvent conventional semidefinite programming relaxations and heuristic strategies. By enumerating and evaluating $O(n^{2r-1})$ candidate solutions—where $r$ denotes the approximate rank of the objective matrix—the authors establish, for the first time, that the global optimum is guaranteed to reside within this candidate set when $K=3$ and the objective matrix is low-rank. Theoretical guarantees are also provided for approximately low-rank cases. The algorithm is inherently parallelizable and achieves performance comparable to state-of-the-art methods across various graph structures while demonstrating superior scalability.

This work proposes a novel self-supervised graph neural network architecture to reduce the substantial computational cost of repeatedly solving Max-Cut semidefinite programming (SDP) relaxations within branch-and-bound algorithms. Designed as a lightweight SDP proxy solver embedded in an exact optimization framework, the model requires no ground-truth SDP solutions for supervision and directly predicts primal- and dual-feasible SDP solutions. These predictions are combined with Goemans–Williamson randomized rounding to produce high-quality cut solutions. To the best of our knowledge, this is the first graph neural network approach that achieves fully self-supervised training while preserving feasibility, significantly accelerating the solution of structured convex relaxations. Experiments demonstrate that the method reduces the boundary computation cost in exact Max-Cut solving by up to 10.6× compared to the commercial solver MOSEK.