sdp certification

This work addresses the high computational cost in verifying adversarial robustness of multi-class neural networks, which traditionally requires separate optimization for each target class. To overcome this limitation, the authors propose a unified quadratic model that jointly verifies all target classes through a single semidefinite programming (SDP) relaxation. The approach introduces an active neuron pruning strategy to reduce problem dimensionality and accelerate SDP convergence. Notably, this is the first method to handle all output classes simultaneously within a single optimization framework, substantially improving verification efficiency. Experimental results demonstrate that the proposed technique significantly speeds up robustness certification, enabling SDP-based verification to scale to large-scale multi-class datasets.

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 work addresses the excessive relaxation error of linear programming (LP) and semidefinite programming (SDP) formulations in robustness certification of ReLU neural networks. We propose a branch-and-bound framework based on geometric partitioning of the input uncertainty set. Our key contributions are: (i) the first theoretical proof that intelligent partitioning can completely eliminate LP relaxation error for single-layer networks; (ii) a closed-form, layer-wise LP/SDP branching strategy for the NP-hard optimal partitioning problem, extended to an efficient multi-layer heuristic; and (iii) significant reduction of worst-case relaxation gaps, yielding substantial improvements in certified accuracy on MNIST, CIFAR-10, and a breast cancer classifier. We explicitly characterize the applicability boundaries of LP versus SDP branching, and our multi-layer approach achieves state-of-the-art certification performance on large-scale deep networks at the time of publication.

This work investigates the robust satisfiability problem for Promise Constraint Satisfaction Problems (PCSPs), focusing on the expressive power and limitations of semidefinite programming (SDP) algorithms. For symmetric Boolean PCSPs, we establish a complete computational complexity classification. We prove that SDP achieves robust approximation precisely when the PCSP admits either a majority or an alternating threshold polymorphism; moreover, we provide the first algebraic necessary and sufficient condition—formulated via minion homomorphisms—for SDP feasibility to imply exact satisfiability. Innovatively, we introduce spherical Ramsey theory into PCSP analysis, revealing a deep connection between SDP integrality gaps and spherical coloring unsatisfiability. This yields the first algebraic-geometric method for proving SDP hardness gaps. Our results unify and extend robust satisfiability theory to the promise setting, delivering systematic criteria for SDP tractability in PCSPs and proposing a central conjecture governing its scope.

Differentiable optimization in robot visual localization often suffers from local minima and gradient distortion—especially in low-light keypoint detection—compromising robustness and accuracy. Method: This paper proposes a certifiably differentiable framework based on polynomial optimization (POP), which reformulates POP problems into semidefinite programming (SDP) relaxations with certified backward propagation. It integrates implicit differentiation with PyTorch-based end-to-end training, ensuring global optimality guarantees while maintaining computational efficiency. Contribution/Results: To the best of our knowledge, this is the first work to enable certified backpropagation through SDP relaxations of POP problems, theoretically guaranteeing gradient correctness. Experiments demonstrate that the method substantially mitigates failure modes of mainstream differentiable optimizers, significantly improving both keypoint detection robustness and localization accuracy under low-light conditions in robotic visual localization tasks.

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。

针对ReLU神经网络验证中线性规划和半定规划方法产生的保守安全保证问题，提出了一种新的特征值最大化程序$(DNN)^2$，通过搜索非唯一乘数空间来获得全局最优性保证。

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.

本文提出了一种名为CP-Cert的新方法，通过利用中心路径来有效解决在机器人学中由于松弛退化导致难以认证的问题，并应用于位姿注册和点云数据关联。

This work addresses the lack of formal guarantees regarding semantic preservation during problem reformulation and solver correctness in constraint programming. It presents the first end-to-end verified framework implemented in the Lean theorem prover, enabling formal proofs of parameterized equivalence, equisatisfiability, and symmetry-breaking correctness for entire families of problems. The approach combines general, parameterized proofs with instance-level certificate checking, thereby eliminating the need to trust external solvers. Verified certificates are produced via backend transformations, and a single high-level proof suffices for arbitrarily large instances. This methodology achieves dramatic search-space reductions—up to a factor of twenty million—and enables full verification of the largest instances in just a few minutes.