duality analysis

This paper addresses the challenge of verifying discrete structures—such as state sets, predicates, and ranking functions—in program verification. We propose the first unified primal-dual framework for program verification grounded in Lagrangian duality, modeling verification as generalized Lagrangian optimization over discrete domains and solving it via coordinated iterative refinement of primal and dual variables. Crucially, we formally introduce duality into program verification, revealing a unifying structural principle underlying classical techniques—including abstract interpretation, ranking function synthesis, and inductive proof. Leveraging this insight, we design a novel validity checker that outperforms state-of-the-art tools across multiple benchmarks. Theoretical contributions include monotonicity analysis, fixed-point logical characterization, and synergistic modeling within quantified linear arithmetic—collectively enhancing the systematicity and scalability of verification algorithms.

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.

To address the slow convergence of conventional solvers for large-scale constrained optimization and their inability to meet real-time requirements, this paper proposes a novel machine learning surrogate method that learns dual feasible solutions—thereby overcoming the lack of theoretical guarantees inherent in existing primal-only surrogates. Our approach tightly integrates linear programming structure, convex optimization theory, and dual modeling. Key contributions include: (1) the first smooth, self-supervised dual loss function enabling end-to-end differentiable training; and (2) an implicit-layer-free analytical dual completion strategy that rigorously ensures dual feasibility and enables millisecond-scale inference. Evaluated on large-scale linear optimization benchmarks, our method achieves <1% optimality gap while accelerating solution times by several orders of magnitude over commercial solvers, significantly outperforming unstructured surrogate models.

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 addresses the computation of relative interior points of the dual optimal solution set for the Linear Assignment Problem (LAP). To overcome the inefficiency of existing methods, we propose, for the first time, a linear-time algorithm that constructs a relative interior point directly from any given dual optimal solution and its associated optimal assignment. We further establish a linear-time reduction from incomplete LAP to complete LAP that preserves both optimality and relative interiority. Integrating this algorithm into a dual ascent framework for the Quadratic Assignment Problem (QAP) significantly improves the quality of LP relaxation lower bounds. On standard benchmarks, the resulting lower bounds closely approximate the LP optimum, while runtime remains orders of magnitude lower than commercial LP solvers—particularly advantageous for large-scale and incomplete QAP instances.

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 study addresses the challenge of enhancing both computational efficiency and solution accuracy in primal optimal stopping problems by leveraging dual martingales. We propose a novel approach that integrates high-fidelity approximations of dual martingales with Monte Carlo simulation, effectively reducing the variance of policy estimators. For the first time in numerical experiments, we demonstrate that accurately constructed dual martingales significantly improve solution stability and simultaneously enhance both computational efficiency and estimation accuracy across multiple test cases. Our work underscores the critical role of dual information in numerical methods for optimal stopping and provides a more robust computational framework for applications such as high-dimensional pricing of financial derivatives.

This study investigates whether solutions obtained by reformulating bilevel linear programs into single-level mixed-integer linear programs (MILPs) via the KKT conditions and the Big-M method retain bilevel optimality. We establish, for the first time, that verifying the bilevel optimality of an MILP solution is coNP-complete if even a single Big-M parameter is improperly chosen. Moreover, we show that confirming the global correctness of all Big-M values remains computationally intractable, even when an optimal MILP solution is given. Through complexity-theoretic analysis, we derive two complementary computational lower bounds, demonstrating the inherent intractability of this verification problem. These results apply broadly to uncoupled min-max problems and integer bilevel programs reformulated using strong duality.

This work addresses the ill-conditioning and numerical instability inherent in traditional primal-dual interior-point methods for quadratic programming, which arise from explicitly enforcing complementarity conditions. To overcome this limitation, the authors propose a novel approach that implicitly satisfies the Karush–Kuhn–Tucker (KKT) complementarity conditions. By introducing auxiliary variables, employing a retraction mapping, and replacing the exponential map with the softplus function, the method ensures spectral boundedness of the KKT system, thereby fundamentally mitigating severe ill-conditioning near the solution. Coupled with a linear solver strategy that avoids matrix refactorization at each iteration, the proposed framework not only supports high-accuracy solutions and low-precision arithmetic but also opens new avenues for decomposition-free or indirect solution techniques tailored to large-scale quadratic programming problems.

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.