convex analysis

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 paper studies the student-school assignment problem under capacity constraints and group-level fairness requirements, jointly optimizing individual utilities (e.g., preference rankings), school enrollment caps, and inter-group fairness—such as across ethnicity or geography—formulated either via concave objective functions or explicit group-wise constraints, and supporting arbitrary covering constraints to capture multi-criteria and ordinal optimization needs. We propose, for the first time, a unified algorithmic framework that integrates convex programming modeling with systematic rounding techniques, yielding tunable randomized or deterministic algorithms. These run in polynomial time and provide controlled trade-offs among utility loss, capacity violations, and fairness deviations. Theoretically, our approach achieves provable approximation guarantees and naturally generalizes to covering constraints and ranking-aware settings. It exhibits strong scalability and practical deployability.

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 feasibility and optimization problems over smooth and/or strongly convex constraint sets. To overcome the computational bottleneck of conventional methods—relying on expensive projection or linear optimization oracles—we propose a scalable, accelerated first-order algorithm that requires neither projections nor linear oracles, but only adaptive one-dimensional line searches and normal vector computations. Our key contribution is the first establishment of structural inheritance: the squared Minkowski gauge inherits both smoothness and strong convexity from the underlying constraint set, thereby circumventing analytical difficulties arising from its inherent nonsmoothness and lack of strong convexity. Theoretically, the algorithm achieves an $O(1/T)$ convergence rate under strong convexity, $O(1/T^2)$ under smoothness, and accelerated linear convergence when both properties hold—matching known lower bounds. Moreover, per-iteration complexity is substantially reduced.

This paper studies geometric hitting set and covering problems for families of convex polyhedra parameterized continuously, motivated by modeling two-stage finite-adaptive decisions in robust optimization—particularly nonlinear settings under left-hand-side uncertainty. Methodologically, it establishes a novel paradigm linking continuous parametric hitting set problems to finite-adaptive robust optimization; develops algorithms integrating computational geometry, convex analysis, parametric polyhedral theory, and robust optimization modeling. Contributions include: (i) the first strongly polynomial-time algorithm for the problem when both the polyhedral dimension and the parameter space dimension are constants; (ii) a strongly quadratic-time algorithm for single-parameter families in constant dimension. These results break computational bottlenecks in nonlinear robust optimization and provide the first strongly polynomially solvable framework—with efficient algorithms—for finite-adaptive robust optimization.

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 paper introduces the Graph-based Convex Set (GCS) optimization framework, which generalizes classical combinatorial problems—including shortest path, traveling salesman (TSP), and minimum spanning tree (MST)—into a unified graph optimization model where vertices are convex sets and edges encode coupling constraints and costs. The core methodological innovation is the first systematic use of perspective (homogenization) transformations to automatically reformulate GCS problems as mixed-integer convex programs (MICPs), enabling globally optimal solutions via off-the-shelf MICP solvers. The approach integrates integer linear modeling, perspective lifting, and generic branch-and-bound algorithms, implemented in the open-source Python library GCSOPT. Numerical experiments demonstrate that the framework achieves both strong expressiveness and scalability: it unifies and exactly solves diverse nonconvex and combinatorial–convex hybrid problems while preserving theoretical rigor—thereby significantly expanding the interface between convex and combinatorial optimization.

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.

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.

This work addresses the approximate optimization of weighted Nash Social Welfare (NSW) maximization and two related variants of unrelated machine scheduling. It introduces a novel convex programming relaxation that, for the first time, reduces the exponentially sized configuration linear program to a polynomial-sized compact formulation while preserving approximation guarantees. Combined with the rounding technique of Feng and Li, the approach is efficiently implementable using standard linear programming solvers, achieving an $e^{1/e} \approx 1.445$ approximation ratio for weighted NSW with only an additive loss of $\ln(1+\varepsilon)$. The method successfully extends to unrelated machine scheduling, matching the current best-known approximation performance. Additionally, the paper provides a constructive proof for the unweighted case and simplifies the analysis of the EF1 fairness gap.