compactness arguments

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 investigates generalized linear functions from ℝᵈ to the extended reals ℝ ∪ {±∞}, satisfying linearity axioms under extended arithmetic. Such functions exhibit nontrivial structure not captured by standard d-dimensional linear functions. To characterize them, the authors develop a dimension-d induction-based construction, establishing for the first time their Ω(d²) parametric complexity. The main contributions are threefold: (1) a complete structural characterization of extended-linear and extended-affine functions, proving a bijective correspondence with convex functions whose epigraphs admit vertical supporting hyperplanes; (2) necessary and sufficient conditions for the existence of extended-valued subgradients; and (3) a constructive generalization of the classical functional characterization of proper scoring rules—yielding an explicit equivalence criterion: a scoring rule is proper if and only if it is generated by a given closed proper convex function. These results strengthen foundational connections between convex analysis and scoring rule theory.

This paper investigates the extremal structure of incentive-compatible mechanisms in multidimensional type screening. Focusing on canonical settings—including monopoly pricing, bilateral trade, barter, and delegation versus veto bargaining—it provides a complete characterization of extremal mechanisms. Methodologically, the analysis integrates convex analysis, mechanism design theory, linear utility modeling, and geometric representation of menus. The contributions are threefold: (i) it establishes, for the first time, a formal connection between extremal mechanisms and Gale’s (1954) indecomposable convex bodies; (ii) it proves that any exhaustive mechanism can be approximated arbitrarily closely by extremal mechanisms via generic perturbations; and (iii) it derives a tight upper bound on menu size in the one-dimensional case and shows that, in multidimensional settings, typical exhaustive mechanisms are almost surely extremal with uncountable menus. Collectively, these results unify the boundary characterization of multidimensional incentive-compatible mechanisms, offering novel geometric insights and constructive tools for mechanism design.

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 the lack of an information-theoretic characterization and a point-to-set principle for finite-state dimension (FS dimension). It introduces a precise quantification of the information content of real numbers under finite precision, integrating effective dimension theory, finite-state automaton complexity, and relativized algorithmic information theory. The work establishes, for the first time, a point-to-set principle for FS dimension and thereby defines a robust notion of relative normality. It then rigorously proves the equivalence between relative normality and FS dimension. The main contributions are: (1) an information-theoretic, exact definition of FS dimension; (2) a necessary and sufficient characterization linking relative normality to FS dimension; and (3) a foundational framework enabling future investigation of its equidistribution properties.

This work addresses the computation of strongly stationary solutions in stochastic convex optimization, where strong stationarity is defined by the presence of small elements near zero in the subdifferential. The paper introduces, for the first time, a notion of strong stationarity based on this criterion and employs tools from dimension theory to analyze the structure of subdifferential graphs, revealing how random sampling effectively preserves their essential features. Building on these insights, a proximal-point-type algorithm is developed, leveraging the Moreau envelope and subdifferential analysis to establish rigorous convergence guarantees. The proposed method theoretically overcomes the challenge posed by the lack of uniform convergence of subgradients in neighborhoods of optimal solutions, thereby enabling efficient approximation of strongly stationary points.

This work addresses the challenge in bilevel optimization where non-isolated minima manifolds in the lower-level problem render the upper-level objective nondifferentiable. To overcome this, the authors propose a “select-then-differentiate” framework that, under a local Polyak–Łojasiewicz condition, introduces a unique optimistic selection to ensure hypergradient differentiability and enables explicit hypergradient computation via the pseudoinverse. Theoretically, global uniqueness of the lower-level solution is unnecessary; local smoothness of the upper-level objective is guaranteed as long as the selected solution is nondegenerate on the manifold. The proposed HG-MS method converges to stationary points of the optimistic objective, with complexity governed by the intrinsic dimension of the manifold. Empirically, it significantly outperforms existing approaches on LLM source reweighting tasks, achieving state-of-the-art results on GSM8K and MATH benchmarks and leading performance on MT-Bench.

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 Max-Cut problem by investigating whether the Goemans–Williamson approximation ratio α_GW ≈ 0.87856 can be surpassed under the assumption that the optimal solution of the standard semidefinite programming (SDP) relaxation lies in a fixed low-dimensional space and satisfies triangle inequalities. To this end, we propose a novel randomized rounding algorithm based on the signs of low-dimensional Gaussian projections and establish a corresponding geometric anti-concentration lemma. Our approach yields, for any fixed dimension d, a polynomial-time algorithm achieving an approximation ratio strictly better than α_GW, with the expected cut value at least (α_GW + 2^{-O(d)}) times the SDP optimum—demonstrating significant performance gains in small dimensions.

This study addresses the challenge of eliciting complete preference information from agents in high-dimensional matching markets by proposing a cognitively lightweight yet efficient matching mechanism. The approach leverages agents’ reported preferences over object features, applies singular value decomposition (SVD) to identify dominant directions of variation in the feature space, and reduces the high-dimensional matching problem to a one-dimensional ranking task. By integrating geometric distributionally robust optimization with a one-dimensional sorting algorithm, the mechanism computes a solution in \(O(N \log N)\) time. Theoretical analysis uncovers a novel connection between Nash social welfare and distributionally robust optimization, guaranteeing approximate welfare optimality, distributional truthfulness, and symmetry. Empirical results demonstrate that the method achieves 99% of optimal welfare while running three orders of magnitude faster than direct optimization, making it suitable for applications such as school choice, labor markets, and course allocation.