Convex Relaxation

Convex relaxations make nonconvex optimization tractable by replacing hard nonconvex problems with convex surrogates that give valid bounds. Yet their design is still largely manual: a human expert must decide where to relax, which relaxation to use, and how much formulation growth is worth paying. We show that this design process can be learned. The method proposed in this paper, AutoConvexRelax, represents the symbolic formulation of a nonconvex problem as a tripartite graph linking variables, symbolic terms, and their objective or constraint positions. It then learns a policy that composes symbolic rewrites into problem-specific relaxation trajectories as the formulation evolves. We prove that finite action sequences generated by the learned policy preserve relaxation validity under explicit domain, occurrence-context, and representation-consistency conditions. We further establish computable error bounds for mixed relaxations and sufficient conditions for feasible recovery and global-optimality certification. As a first instantiation, we apply the method to nonconvex quadratic problems and fractional problems. On 241 benchmark quadratic instances, the learned policy strengthens McCormick lower bounds by 13.52%. Its mean lower bound is 3.70% below strengthened semidefinite relaxations on the normalized comparison metric, while it uses substantially smaller formulations. The learned policy retains this trade-off on fractional instances and reaches near-semidefinite bounds on beamforming and sensor-network localization instances. These results suggest that convex relaxation design can move from a fixed handcrafted choice to an automated learning process that adapts to problem structures. Code is available at https://github.com/auto4opt/AutoConvexRelax.

Certified robustness is critical for deploying neural networks in safety-sensitive settings, but computing exact certificates typically requires solving nonconvex optimization problems. Practical verifiers use convex relaxations: linear programming (LP) relaxations are scalable but often too conservative, whereas semidefinite programming (SDP) relaxations can be substantially tighter but are computationally expensive. We introduce a hierarchy of second-order cone programming (SOCP) relaxations that provide stronger certificates than LP while avoiding the cost of global positive-semidefinite constraints. Our construction combines convex-hull relaxations of ReLU activations, selective McCormick lifting over certified variable bounds, and second-order cone constraints induced by 2-by-2 principal minors. Adding these constraints progressively yields a monotone hierarchy of increasingly tight bounds with a controllable computation-tightness trade-off. On fully connected and convolutional networks on MNIST and CIFAR-10, SOCP-based verification improves certified robust accuracy over LP-based verification and is competitive with alpha,beta-CROWN, while remaining orders of magnitude faster than SDP-based verification. These results position selective second-order constraints as a practical middle ground between scalable LP-based and tight SDP-based certification. Finally, we introduce class-optimized robust evaluation (CORE), which applies tighter relaxations only to target classes not certified by cheaper ones. On the fully connected MNIST benchmark, CORE variants match SDP-cert's certified robust accuracy while reducing average verification time by up to 86.30%.

Embedding trained neural networks as surrogates within optimisation problems is an established practice in operations research. The prevailing approach uses feedforward neural networks (FNNs) with ReLU activations, whose piecewise-linear structure admits an exact but computationally intensive mixed-integer programming (MIP) reformulation as the networks grow. We advocate input convex neural networks (ICNNs) as structurally superior surrogates when the underlying response is approximately convex or concave. The convex architecture offers two computational advantages. First, the ICNN-MIP formulation tends to yield a tighter linear programming (LP) relaxation than its FNN-MIP counterpart, with no integrality gap in favourable instances. Second, ICNNs uniquely admit an LP-based reformulation via epigraph representations of ReLU activations, though this embedding is not always exact. When it is not, we exploit the properties of ICNNs to construct the strongest continuous relaxation over box domains, namely, the convex hull of the ICNN's graph, bounded below by the epigraph and above by the concave envelope; this construction is tractable under input convexity but hard for general ReLU networks. On this basis, we develop a branch-and-bound algorithm that builds this relaxation at each node, branches directly on input variables rather than intermediate variables as in MIP reformulations, and terminates at the root node whenever the epigraph embedding is valid. Case studies on humanitarian food aid, oil well routing, and wine blending show that ICNN surrogates match FNN accuracy and deliver gains in solve time and scalability, supporting ICNN as the default surrogate when the underlying function is convex, concave, or well-approximated as such.

Recent work shows that LLM agents can improve sharp-constant inequalities by searching for extremal constructions, which yield upper bounds. We address the complementary side: a lower bound holds for every admissible function and follows from a convex relaxation of the nonconvex problem, with tighter relaxations giving stronger bounds. We instantiate the autoresearch paradigm to discover such relaxations: a coding agent proposes valid tightening constraints, a theory agent verifies each one and searches for counterexamples, and every reported bound is certified by an explicit dual-feasible point checked in rigorous interval arithmetic. On two optimization constants studied by \citet{tao2025alphaevolve} - the first autocorrelation inequality ($C_{6.2}$) and the Erdős minimum-overlap constant ($C_{6.5}$) - we improve the certified lower bounds from $1.28$ to $1.2937$ and from $0.379005$ to $0.37912$, respectively.

We study the graph alignment problem for correlated Gaussian Orthogonal Ensemble (GOE) matrices, where the goal is to recover a hidden vertex permutation given two correlated symmetric Gaussian matrices $(A, B)$ with correlation $1/\sqrt{1+σ^2}$. While the maximum likelihood estimator is information-theoretically optimal, its computation, which reduces to a quadratic assignment problem, is intractable. Motivated by this, we analyze convex relaxations based on minimizing $\|AX - XB\|_F$ over the set of doubly stochastic matrices and the unit hypercube. We show that when the correlation parameter satisfies $σ= o(n^{-1/2}/\log^4 n)$, the solution of either relaxation $(X^\star)$ concentrates around the ground-truth permutation matrix $(Π^\star)$, i.e., $\|X^\star-Π^\star\|_F^2 = o(n)$, implying recovery of all but a vanishing fraction of vertices after simple post-processing. Combined with existing lower bounds, our results precisely characterize that $\|X^\star-Π^\star\|_F^2$ transitions from $o(n)$ for $σ= \tilde{o}(n^{-1/2})$ to $Ω(n)$ for $σ= \tildeΩ(n^{-1/2})$. In doing so, our analysis significantly tightens prior results and extends them beyond doubly stochastic relaxations.

Many learning tasks in science and engineering are characterized by sparse datasets, which limits the effectiveness of purely data-driven approaches. At the same time, these problems are often accompanied by rich domain knowledge derived from physical laws, operational requirements, and expert heuristics. Such knowledge is frequently expressed as rules involving logical propositions and linear inequalities. Existing neuro-symbolic methods typically enforce these rules approximately through soft penalties, assume input-independent rules when designing specialized architectures, or rely on non-differentiable post-processing at inference time to achieve hard constraint satisfaction. While recent advances in differentiable optimization layers enable end-to-end feasibility enforcement within neural networks, extending these approaches to logical or mixed-integer rules remains challenging due to inherent nonconvexity. In this work, we propose a unified end-to-end framework for enforcing hard, input-dependent mixed integer linear constraints within neural networks. Our approach represents rules as disjunctive constraints and applies hierarchical convex relaxations to obtain convex hull formulations. These relaxations yield tractable linear constraints that can be embedded as differentiable optimization layers while enabling exact rule satisfaction. We demonstrate the effectiveness of the proposed framework on real-world datasets, achieving perfect rule satisfaction and strong predictive performance.

Exact solution of hard combinatorial optimization problems often relies on strong convex relaxations, but solving these relaxations repeatedly inside a branch-and-bound algorithm can be prohibitively expensive. Hence, we consider this challenge for Max-Cut, where branch and bound commonly uses semidefinite programming (SDP) relaxations to bound subproblems. We propose a Max-Cut-specific graph neural network that serves as a principled, lightweight neural proxy for these SDP solvers and can be plugged directly into an exact branch-and-bound framework. The proposed architecture has update steps of complexity $\mathcal{O}(n^2 + ne)$, and predicts both primal- and dual-feasible SDP solutions. The primal SDP solutions yield feasible Max-Cut solutions via the Goemans--Williamson algorithm. In addition, it is trained in a self-supervised fashion without requiring solved SDP relaxations as labels. Empirically, we show that our architecture can substantially reduce the cost of bounding in exact Max-Cut solving by up to $10.6 \times$ compared with using the state-of-the-art SDP solver Mosek. Our work highlights the potential of learned, validity-preserving surrogates for accelerating exact optimization over structured convex relaxations.

We study the problem of denoising observations $Y_i=X_i+Z_i$, where the latent variables $X_i$ are sampled from a low-dimensional manifold in $\mathbb{R}^n$ and the noise variables $Z_i$ are isotropic Gaussian. We propose a convex-relaxation estimator that first reduces dimension by principal component analysis and then projects the observations onto the convex hull of the projected latent manifold. We construct a statistical oracle that estimates its supporting hyperplanes from empirical Gaussian tail probabilities of the noisy sample. Under a lower-mass condition on the latent distribution, we prove finite-sample guarantees for the oracle and derive error bounds for the resulting denoiser. The analysis combines risk bounds for least-squares projection under convex constraints with entropy bounds for convex hulls. We also verify the assumptions of the framework for a Cryo-Electron Microscopy observation model by establishing suitable covering number and Lipschitz estimates for the associated group action and imaging operators.

Many neural network (NN) verification systems represent the network's input-output relation as a constraint program. Sound and complete, representations involve integer constraints, for simulating the activations. Recent works convexly relax the integer constraints, improving performance, at the cost of soundness. Convex relaxations consider outputs that are unreachable by the original network. We study the worst case divergence between the original network and its convex relaxations; both qualitatively and quantitatively. The relaxations' space forms a lattice, where the top element corresponds to a full relaxation, with every neuron linearized. The bottom element corresponds to the original network. We provide analytical upper and lower bounds for the $\ell_\infty$-distance between the fully relaxed and original outputs. This distance grows exponentially, w.r.t. the network's depth, and linearly w.r.t. the input's radius. The misclassification probability exhibits a step-like behavior, w.r.t. input radius. Our results are supported by experiments on MNIST, Fashion MNIST and random networks.

Global solvers have emerged as a powerful paradigm for 3D vision, offering certifiable solutions to nonconvex geometric optimization problems traditionally addressed by local or heuristic methods. This survey presents the first systematic review of global solvers in geometric vision, unifying the field through a comprehensive taxonomy of three core paradigms: Branch-and-Bound (BnB), Convex Relaxation (CR), and Graduated Non-Convexity (GNC). We present their theoretical foundations, algorithmic designs, and practical enhancements for robustness and scalability, examining how each addresses the fundamental nonconvexity of geometric estimation problems. Our analysis spans ten core vision tasks, from Wahba problem to bundle adjustment, revealing the optimality-robustness-scalability trade-offs that govern solver selection. We identify critical future directions: scaling algorithms while maintaining guarantees, integrating data-driven priors with certifiable optimization, establishing standardized benchmarks, and addressing societal implications for safety-critical deployment. By consolidating theoretical foundations, practical advances, and broader impacts, this survey provides a unified perspective and roadmap toward certifiable, trustworthy perception for real-world applications. A continuously-updated literature summary and companion code tutorials are available at https://github.com/ericzzj1989/Awesome-Global-Solvers-for-3D-Vision.

We consider the densest submatrix problem, which seeks the submatrix of fixed size of a given binary matrix that contains the most nonzero entries. This problem is a natural generalization of fundamental problems in combinatorial optimization, e.g., the densest subgraph, maximum clique, and maximum edge biclique problems, and has wide application the study of complex networks. Much recent research has focused on the development of sufficient conditions for exact solution of the densest submatrix problem via convex relaxation. The vast majority of these sufficient conditions establish identification of the densest submatrix within a graph containing exactly one large dense submatrix hidden by noise. The assumptions of these underlying models are not observed in real-world networks, where the data may correspond to a matrix containing many dense submatrices of varying sizes. We extend and generalize these results to the more realistic setting where the input matrix may contain many large dense subgraphs. Specifically, we establish sufficient conditions under which we can expect to solve the densest submatrix problem in polynomial time for random input matrices sampled from a generalization of the stochastic block model. Moreover, we also provide sufficient conditions for perfect recovery under a deterministic adversarial. Numerical experiments involving randomly generated problem instances and real-world collaboration and communication networks are used empirically to verify the theoretical phase-transitions to perfect recovery given by these sufficient conditions.

In this paper, we propose a novel locally statistical active contour model (LACM) based on Aubert-Aujol (AA) denoising model and variational level set method, which can be used for SAR images segmentation with intensity inhomogeneity. Then we transform the proposed model into a global optimization model by using convex relaxation technique. Firstly, we apply the Split Bregman technique to transform the global optimization model into two alternating optimization processes of Shrink operator and Laplace operator, which is called SB_LACM model. Moreover, we propose two fast models to solve the global optimization model , which are more efficient than the SB_LACM model. The first model is: we add the proximal function to transform the global optimization model to a general ROF model[29], which can be solved by a fast denoising algorithm proposed by R.-Q.Jia, and H.Zhao; [29] was submitted on 29-Aug-2013, and our early edition was ever submitted to TGRS on 12-Jun-2012, Venkatakrishnan et al. [30] proposed their PnP algorithm on 29-May-2013, so Venkatakrishnan and we proposed the PnP algorithm almost simultaneously. Thus we obtain a fast segmentation algorithm with global optimization solver that does not involve partial differential equations or difference equation, and only need simple difference computation. The second model is: we use a different splitting approach than one model to transform the global optimization model into a differentiable term and a general ROF model term, which can be solved by the same technique as the first model.