convex optimization

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.

This work addresses the problem of constructing compact extended formulations for multilinear polytopes arising in binary polynomial optimization, with a focus on the intrinsic relationship between hypergraph acyclicity and facial complexity. Methodologically, it integrates hypergraph theory, convex hull characterization, and extended formulation analysis. The contribution is the first complete characterization of polynomial-size extended formulations for the class of acyclic hypergraphs: it precisely identifies the structural properties of acyclic hypergraphs that admit polynomial-time construction of such formulations. This establishes a theoretical bridge between hypergraph topological properties—particularly acyclicity—and the efficiency of optimization modeling. As a result, the paper provides explicit, polynomial-size extended formulations for a broad class of acyclic hypergraphs, substantially reducing the computational complexity of solving the associated binary polynomial optimization problems. The findings advance the scalability and theoretical foundations of nonlinear binary optimization modeling.

This paper resolves two fundamental open problems in convex polynomial programming: (1) existence and boundedness characterization of global optima for unconstrained convex polynomials of degree four or higher; and (2) existence of polynomial-bit-length approximate optimal solutions for convex polynomials of arbitrary degree under polyhedral constraints. Moving beyond conventional approaches reliant on linear optimality characterizations (e.g., KKT conditions), the authors develop a novel, KKT-free framework for analyzing solution boundedness. They establish, for the first time, that ε-approximate optimal solutions admit polynomial bit complexity—positively resolving an open question posed by Nesterov. Integrating the ellipsoid method with a new boundary estimation technique, they design the first polynomial-time approximation algorithm applicable to convex polynomials of arbitrary degree, encompassing both unconstrained and polyhedrally constrained settings. This work bridges a long-standing theoretical and algorithmic gap between linear/convex quadratic programming and semidefinite programming.

Verifying geodesic convexity on Hadamard manifolds—such as the manifold of symmetric positive-definite matrices—is inherently challenging and typically requires case-specific analysis, hindering reliable modeling and optimization in non-Euclidean settings. Method: This paper introduces the *Normalized Geodesic Convex Programming* (DGCP) framework—the first systematic framework for geodesic convexity verification and optimization on Cartan–Hadamard manifolds. It defines geodesic convex atomic functions and composition rules preserving geodesic convexity, grounded in differential geometry and convex analysis, enabling symbolic-level automatic certification. Implemented in Julia as the SymbolicAnalysis.jl library, DGCP seamlessly integrates with manifold optimization solvers for end-to-end modeling and solving. Contribution/Results: DGCP significantly improves modeling reliability and computational efficiency for statistical estimation, matrix learning, and other non-Euclidean optimization tasks, establishing a verifiable convexity foundation for Riemannian optimization.

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 proposes a trajectory-restricted framework for linear convergence analysis that overcomes the conservatism of traditional first-order methods, whose guarantees often rely on global geometric conditions and worst-case constants. Instead of imposing regularity assumptions globally, our approach requires only local geometric properties—such as restricted Polyak–Łojasiewicz inequalities, error bounds, and quadratic growth—on the subset of the space actually traversed by the algorithm. We establish explicit relationships among the associated constants and show that, for piecewise polyhedral composite problems, once iterates enter a well-conditioned active manifold, convergence is governed by the restricted Hoffman constant of that manifold, yielding an improved effective condition number and faster local convergence. The results demonstrate that linear convergence fundamentally depends on the local geometry encountered along the algorithmic trajectory, rather than on global worst-case scenarios.

This work addresses the lack of a unified modular framework for analyzing adaptive optimizers, which hinders a precise characterization of their behavior under constraints on directional reachability, information budgets, and update rules. We propose a geometric–non-geometric decoupled calculus for optimizers: the geometric module, constituted by a family of positive-definite cometrics, captures realizable descent directions, while the non-geometric module governs mechanisms such as information processing, memory, and control. Within this framework, we establish a direction expressivity theorem and a residual theory for constrained cometric families, disentangling directional expressiveness from condition-number complexity and recasting optimizer design as a Pareto optimization problem under modular budgets. Theoretically, we prove that fully positive-definite geometry exactly spans all strictly descending directions; experiments demonstrate that high-information full-metric probes attain numerical precision on deterministic quadratic problems, and a Muon-style implementation preliminarily validates the auditability of matrix-operator updates.

This work addresses adversarial online resource allocation problems without prior information, where conventional linear programming (LP) relaxations often fail under stochastic settings. The paper introduces two dual analysis frameworks: the first leverages regularized convex optimization and KKT conditions within an LP-based approach, while the second establishes a general dual certificate mechanism that does not rely on linear programming. This latter framework unifies the treatment of complex scenarios—including reusable resources, stochastic rewards, and whole-page optimization—and provides a generic template for proving competitive ratios. By offering a unified theoretical foundation for a broad class of online matching and resource allocation problems, the proposed framework not only validates existing algorithms but also facilitates the design of new ones, delivering strong theoretical guarantees across diverse models.