duality theory

This paper establishes the theoretical foundations of infinite-horizon linear dynamic optimization models. Addressing core issues—including existence of solutions, sufficiency of optimality conditions, and validity of the dynamic programming equation—the study employs convex analysis, infinite-dimensional optimization, and the theory of upper semicontinuous set-valued mappings. It provides the first rigorous proof that the transversality condition unconditionally ensures optimality in such models; reveals that optimal decision rules must be upper semicontinuous correspondences—not necessarily single-valued functions—in linear settings; introduces the novel “two-stage linear cake-eating problem” paradigm and derives necessary conditions for its solution; and, under convex bi-periodic constraints, establishes concavity, continuity, and monotonicity of the value function, along with conditional monotonicity of the policy correspondence. Collectively, these results unify and extend the applicability of the Euler equation, transversality condition, and dynamic programming principle to linear dynamic optimization.

This study addresses the non-smooth, constrained user utility maximization problem inherent in Perturbed Utility Route Choice (PURC) models by introducing a unified convex duality framework. The proposed approach transforms the original problem into an unconstrained, differentiable concave maximization task, enabling efficient gradient-based optimization. By leveraging the convex conjugate of link-specific perturbation functions, the method uniquely recovers optimal route flows link-by-link. This work establishes, for the first time, a rigorous convex duality theory for PURC models, revealing a structural analogy to electrical current flows. The framework facilitates rapid sensitivity analysis and scalable computation, significantly enhancing both efficiency and applicability for real-time solution and parameter sensitivity evaluation in large-scale, complex transportation networks.

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 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 paper addresses the real-time tracking of solutions to time-varying variational inequalities (VIs), relaxing the restrictive assumptions—namely strong monotonicity/strong convexity and sublinear growth of solution trajectories—prevalent in prior work. We consider two novel settings: (1) non-monotone time-varying VIs with sublinearly growing solution paths, and (2) periodically time-varying VIs whose solution paths need not be sublinear. Our method employs a unified analytical framework grounded in dynamical systems modeling, Lyapunov stability analysis, and nonlinear stability theory. Contributions include: (i) the first universal upper bound on tracking error for non-monotone time-varying VIs; (ii) rigorous proof that discrete-time dynamical systems for periodic time-varying VIs exhibit coexisting chaotic and asymptotically convergent behaviors; and (iii) numerical simulations validating both the tightness of the theoretical error bound and the accuracy of predicted dynamical behaviors.

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 reformulates the problem of solving semidefinite programs (SDPs) as the search for optimal strategies in zero-sum semidefinite games, particularly targeting instances that challenge existing methods. Under natural constraint qualifications, it establishes a constructive and complete equivalence between primal-dual SDPs and zero-sum semidefinite games: the game value is zero if and only if a strong optimal solution exists; otherwise, the framework yields an infeasibility certificate for either the primal or the dual problem. By integrating SDP duality theory, a semidefinite generalization of von Stengel’s construction, techniques for handling generalized duality phenomena, and explicit bounds on solution coordinates, this approach extends applicability to a broader class of SDPs and overcomes limitations of prior methods.

This work addresses the instability of classical mirror descent in monotone variational inequality problems, where the method may diverge or cycle due to the absence of a stabilizing convergence mechanism. To overcome this limitation, the paper introduces the Targeted Mirror Descent (TMD) framework, which incorporates a target-point correction mechanism into the dual update to effectively stabilize the optimization dynamics. TMD unifies classical algorithms such as the proximal point method and extragradient method, revealing their underlying convergence principles. It also rectifies the equilibrium misalignment issue present in discounted mirror descent and enables geometric ensembling, allowing multiple heterogeneous mirror maps to operate collaboratively in parallel. The framework guarantees convergence for monotone variational inequalities and facilitates the construction of novel mirror maps, substantially enhancing algorithmic robustness and applicability.

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 proposes a novel paradigm termed “Optimized Natural Physics” to investigate whether optimization algorithms adhere to natural laws of motion induced by the objective function. By establishing equivalence between optimal control problems and generalized KKT conditions, the authors construct a natural vector field governed by non-Newtonian dynamics. Leveraging Pontryagin’s minimum principle, Hamilton–Jacobi inequalities, and energy dissipation mechanisms, they design control strategies possessing inverse optimality. This framework not only unifies the interpretation of diverse existing optimization algorithms but also enables the systematic derivation of new ones. The approach demonstrates that global optimization can be achieved through deliberate modulation of jumps and dissipation, thereby providing a physically intuitive and mathematically unified foundation for optimization theory.