discover convex relaxations

This work proposes a dual-agent collaborative framework for automatically discovering convex relaxations to strengthen lower bounds in nonconvex optimization problems. An encoding agent generates tight constraints, while a theory agent validates their correctness through explicit dual feasible points and rigorous interval arithmetic. The approach pioneers the integration of large language model–driven autonomous research paradigms into convex relaxation construction, unifying automated lower-bound optimization with formal mathematical proof. The method achieves new state-of-the-art results on two classical optimization constants: improving $C_{6.2}$ from 1.28 to 1.2937 and $C_{6.5}$ from 0.379005 to 0.37912.

This work systematically uncovers the decisive role of problem geometry—specifically, the curvature of the constraint set and the structure of gradients—in governing the statistical-computational trade-offs of stochastic and online optimization algorithms. We introduce the first geometric measure quantifying the deviation of a constraint set from quadratic convexity, rigorously identifying the geometric origins of suboptimality in subgradient methods. We prove that diagonal-preconditioned SGD achieves minimax-optimal convergence rates under quadratic convex constraints. For non-Euclidean, non-quadratically-convex domains—such as ℓₚ-balls with p < 2—we establish tight convergence bounds for mirror descent and adaptive gradient methods, and uncover, for the first time, a precise correspondence between their convergence rates and the accuracy-computation trade-off in Gaussian sequence estimation. Our results provide geometric criteria for algorithm selection and unify the understanding of when nonlinear updates—e.g., via mirror descent—are necessary to attain statistical optimality.

Training two-layer ReLU neural networks is inherently non-convex, posing significant theoretical and computational challenges. Method: This work establishes, for the first time in the infinite-width limit, an exact equivalence between ReLU network training and a finite-dimensional convex completely positive program (CPP). We propose a compact semidefinite programming (SDP) relaxation that is solvable in polynomial time and preserves the optimal value of the original CPP problem exactly. Contribution/Results: Theoretically, we derive a precise correspondence between non-convex neural network training and convex optimization. Empirically, our SDP-based approach achieves competitive test accuracy on multi-class classification benchmarks, empirically validating both the tightness of the relaxation and its generalization capability. This work provides a novel convex analytical framework and a tractable computational pathway for deep learning training, bridging classical convex optimization theory with modern neural network practice.

This paper addresses nonconvex functional optimization under risk constraints. Methodologically, it integrates risk conjugate duality theory, a weak Banach-space extension of Uhl’s Lyapunov convexity theorem, and structural analysis of coherent risk measures (e.g., CVaR, MAD). The main contributions are threefold: (1) It establishes, for the first time, verifiable sufficient conditions guaranteeing strong duality in general nonconvex risk-constrained optimization—covering both continuous/smooth function spaces and finite-width/depth neural network parameterizations; (2) It unifies treatment of decomposable and non-decomposable policy spaces, substantially generalizing classical duality results; and (3) It provides a rigorous optimization-theoretic foundation for practical applications including wireless resource allocation and risk-constrained supervised learning. The analysis applies to broad classes of nonconvex, nondifferentiable, and nonsmooth objective and constraint functionals, without requiring restrictive assumptions such as convexity, compactness, or interior-point conditions.

This paper addresses strongly convex yet nonsmooth and non-Lipschitz optimization problems. We develop a unified primal–dual theoretical framework that, for the first time, reveals the equivalent dual-averaging representations of the subgradient method, proximal subgradient method, and switching subgradient method. Through a novel *dual-gap convergence analysis*, we establish the first $O(1/T)$ convergence guarantee applicable to this problem class. We derive an optimal stopping criterion and optimality certificate that require no additional computation, and rigorously characterize a controllable convergence boundary—even under early-stage exponential divergence. Our theory accommodates a broad range of step-size choices and accommodates ill-conditioned non-Lipschitz structures. While preserving algorithmic simplicity, our framework substantially extends both the applicability and theoretical depth of subgradient-type methods.

This work addresses the limited tightness of convex relaxations in mixed-integer nonlinear programming (MINLP) by proposing a computational geometry–based polyhedral relaxation method. It employs a convexification strategy that selects points to iteratively approximate the simultaneous convex hull of factorable function graphs and introduces novel explicit inequalities to strengthen factorable relaxations. Theoretically, it proves that for multilinear functions over axis-aligned domains, the simultaneous convex hull is uniquely determined by its corner points. Furthermore, the approach integrates voxelization with the QuickHull algorithm to efficiently approximate feasible regions. Computational experiments demonstrate that the method reduces the dual gap by 20–25% on average for random polynomial problems and outperforms existing techniques on approximately 30% of MINLPLib instances, with over 10% of cases achieving more than a 50% gap reduction.

This work addresses the challenge of solving non-convex optimization problems, which are structurally complex and difficult for conventional solvers to handle efficiently, often requiring expert-guided manual convexification. To overcome this limitation, the authors propose NC2C, a novel framework that leverages large language models in an end-to-end manner to automatically convexify general non-convex problems. NC2C employs symbolic reasoning to identify non-convex components and integrates adaptive transformations, feasible region correction, and an iterative verification mechanism to generate equivalent convex formulations. Experimental results on a benchmark set of 100 general non-convex problems demonstrate that NC2C achieves an execution rate of 89.3% and a success rate of 76%, substantially outperforming existing baseline methods and significantly reducing reliance on human intervention.

This work addresses the challenges of poor scalability, limited parallelizability, and complex subproblems in nonsmooth nonconvex optimization with orthogonality constraints by proposing a retraction-free primal-dual linearized smoothed augmented Lagrangian method. The proposed algorithm introduces, for the first time, a retraction-free primal-dual framework to orthogonality-constrained optimization, eliminating nested loops and intricate subproblem solvers in favor of a single-loop iteration scheme. Leveraging the Kurdyka–Łojasiewicz property, the method is theoretically shown to converge to an $\varepsilon$-KKT point with an iteration complexity of $O(\varepsilon^{-3})$, without requiring Riemannian retractions. Numerical experiments demonstrate that the algorithm significantly outperforms existing approaches in both computational efficiency and scalability.

Optimizing implication bases in convex geometries—i.e., minimizing both premises and conclusions—is generally computationally hard. This work introduces a novel tool, the *quasi-closed hypergraph*, to characterize the structure of optimal implication bases and provides a unified treatment for four important classes of convex geometries: double-shell, acyclic, affine, and acceptable. By integrating hypergraph theory, implication base reduction algorithms, and structural properties of convex geometries, we prove that when the edges of the quasi-closed hypergraph are pairwise disjoint, any implication base can be optimized in polynomial time. Notably, this condition precisely encompasses the aforementioned four classes, thereby establishing a unified theoretical framework that explains their tractability and enables efficient optimization.

This work addresses the unquantified error introduced by convex relaxation methods in neural network verification, which sacrifice completeness for tractability. For the first time, it establishes analytical upper and lower bounds on this error, revealing that it grows exponentially with network depth and linearly with the input perturbation radius. The study further uncovers a step-like behavior in misclassification probability. By constructing a lattice structure in the relaxed space and integrating constraint programming, ℓ∞-norm analysis, and lattice theory, the authors systematically characterize the worst-case deviation between the fully relaxed and original network outputs. Experiments on MNIST, Fashion-MNIST, and randomly generated networks validate the theoretical bounds, providing a quantitative foundation for assessing the reliability of verification methods based on convex relaxation.