semidefinite relaxation

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.

Standard semidefinite programming (SDP) solvers suffer from poor scalability, while the conventional Burer–Monteiro factorization (BMF) introduces nonconvexity, hindering reliable optimization. Method: We propose a biconvex optimization framework based on the bilinear decomposition $Z = XY^ op$, augmented with a structural penalty term $|X - Y|_F^2$ to ensure tractable optimization. Contribution/Results: This work is the first to explicitly formulate SDPs as biconvex problems. We derive a theoretically grounded upper bound on the penalty parameter $gamma$, guaranteeing equivalence to low-rank BMF solutions at stationarity. Our approach establishes a novel biconvex surrogate paradigm for SDPs. Integrated with alternating minimization, it achieves state-of-the-art performance on matrix completion and Max-Cut—two canonical SDP tasks—while significantly improving both efficiency and accuracy for large-scale instances.

This paper addresses nonconvex quadratically constrained quadratic programming (QCQP) problems by proposing an iterative tightening framework based on doubly nonnegative (DNN) relaxation and linear semidefinite programming (LSDP) cutting planes. Methodologically, it integrates DNN relaxation, LSDP cut separation, and relaxation strengthening to progressively tighten bound constraints. The key contribution is the first theoretical characterization linking Karush–Kuhn–Tucker (KKT) points to LSDP cutting planes, enabling a finite-step-convergent local search strategy that improves lower bounds efficiently—without branch-and-bound. On a benchmark set of 140 instances with 100 variables, the method reduces the relative optimality gap to within 0.01% for 138 problems within one hour, significantly outperforming state-of-the-art commercial and academic solvers in both efficiency and robustness.

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。

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 challenge of lacking a general computable convex relaxation for quadratic matrix optimization problems with rank constraints by proposing a lifted semidefinite relaxation framework that does not rely on spectral structure assumptions. By analyzing block redundancies in moment matrices, the authors derive a compact equivalent formulation involving only two small-scale semidefinite constraints. They further design projection-based cutting planes that exploit rank inheritance under linear mappings to strengthen the low-rank constraint. This approach significantly enhances scalability and enables efficient solutions for problems such as matrix completion and reduced-rank regression, handling instances with dimensions up to \(n + m\).

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 addresses the non-convex problem of planning high-order smooth (e.g., minimum-snap), collision-free trajectories for point robots amidst spherical obstacles. The authors formulate the task as a non-convex optimization over polynomial trajectories and present, for the first time, a theoretical analysis of its semidefinite programming (SDP) relaxation. Key contributions include establishing necessary and sufficient conditions for relaxation tightness, revealing the equivalence between relaxed solutions and globally optimal trajectories in an augmented space, and leveraging symmetry to reduce the SDP dimensionality so that it scales linearly with the polynomial degree—irrespective of the ambient environment dimension. Integrated into an RRT framework as a convex steering function, the method achieves 10–100× speedups over SNOPT/IPOPT in quadrotor C⁴-continuous minimum-snap planning, significantly reduces solution time variance, and reliably yields high-quality locally optimal trajectories.