arXiv subjects

Ruoyu Diao

Publications and source records attributed to Ruoyu Diao.

5 recordsLinked to original sources

arXiv2026

A Multiscale Primal-Dual Interior-Point Relaxation Method for Large-Scale Optimal Transport Problems

Large-scale optimal transport (OT) problems involve a vast number of transport variables, leading to prohibitive memory and computational costs. To address these challenges, we propose a multiscale primal-dual interior-point relaxation method (MSIPRM). The multiscale outer framework constructs a hierarchy of standard OT problems at progressively finer levels. At each level, the OT problem is solved over a sequence of adaptively refined active sets initialized based on the solution support at the previous level. This yields a sequence of closely related sparse subproblems, thereby substantially reducing memory requirements. The primal-dual interior-point relaxation method (IPRM) serves as the inner solver for each sparse subproblem. Since IPRM does not require strictly interior iterates, it can readily use the solution of the previous subproblem as a warm start. To efficiently obtain the Newton direction, we solve a reduced Schur complement system derived from the normal equations. Furthermore, we develop an effective support-identification strategy based on the approximate solutions obtained by IPRM. We establish condition number estimates for the Schur complement matrices and analyze the global and local convergence properties of the algorithm. Numerical experiments on large-scale test problems demonstrate the computational efficiency and scalability of MSIPRM and show that it compares favorably with existing solvers. In particular, MSIPRM can handle instances whose full formulations contain trillions of transport variables.

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arXiv2026

Polynomial iteration complexity of a path-following smoothing Newton method for symmetric cone programming

It has long remained open whether smoothing Newton methods (SNMs) for symmetric cone programming (SCP) admit polynomial iteration complexity. A key difficulty lies in the lack of an analogue of the self-concordant convex framework underlying interior-point methods (IPMs). In this paper, inspired by Nemirovski's self-concordant convex-concave theory, we address this open problem by introducing a reduced barrier augmented Lagrangian (BAL) function. We prove that the reduced BAL function is self-concordant convex-concave and establish that the parameterized smooth system arising in SNMs coincides with the first-order optimality conditions of an associated minimax problem. Motivated by this equivalence, we propose a path-following smoothing Newton method (PFSNM). The reduced BAL function induces a central path and an associated neighborhood, which provide estimates for the Newton decrement needed for the path-following analysis. As a result, the method achieves an iteration complexity of $\mathcal{O}(\sqrt{\nu}\ln(1/\varepsilon))$, matching the best-known short-step complexity for IPMs. Numerical results on standard benchmarks show that PFSNM is competitive with several well-known interior-point solvers, and the observed performance is consistent with the theoretical development.

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arXiv2026

A sequential linear complementarity problem method for generalized Nash equilibrium problems

Generalized Nash equilibrium problems (GNEPs) arise in various applications where multiple players minimize individual cost functions subject to coupled constraints. A relatively unexplored approach to solving such problems is via a sequence of (mixed) linear complementarity problems (LCPs). Compared with the nonlinear equilibrium subproblems arising in recently popular penalty-based methods such as augmented Lagrangian methods, these LCPs are often substantially easier to solve. However, the existing literature on this approach is very limited, largely because of the difficulty of assessing the search directions generated by the subproblems and establishing a principled step-length acceptance criterion. This paper proposes a sequential linear complementarity problem (SLCP) method with a comprehensive convergence analysis. To assess the search directions, we introduce a novel merit function analogous to the $\ell_1$ penalty function in sequential quadratic programming. The merit function is shown to decrease along the search directions generated by the subproblems under suitable assumptions, thereby guaranteeing the global convergence of the SLCP method. We further establish local quadratic convergence and analyze the solvability of the subproblems. Preliminary numerical results demonstrate the effectiveness and competitiveness of the proposed method relative to existing approaches.

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arXiv2025

Stability for Nash Equilibrium Problems

This paper is devoted to studying the stability properties of the Karush-Kuhn-Tucker (KKT) solution mapping $S_{\rm KKT}$ for Nash equilibrium problems (NEPs) with canonical perturbations. Firstly, we obtain an exact characterization of the strong regularity of $S_{\rm KKT}$ and a sufficient condition that is easy to verify. Secondly, we propose equivalent conditions for the continuously differentiable single-valued localization of $S_{\rm KKT}$. Thirdly, the isolated calmness of $S_{\rm KKT}$ is studied based on two conditions: Property A and Property B, and Property B proves to be sufficient for the robustness of both $E(p)$ and $S_{\rm KKT}$ under the convex assumptions, where $E(p)$ denotes the Nash equilibria at perturbation $p$. Furthermore, we establish that studying the stability properties of the NEP with canonical perturbations is equivalent to studying those of the NEP with only tilt perturbations based on the prior discussions. Finally, we provide detailed characterizations of stability for NEPs whose each individual player solves a quadratic programming (QP) problem.

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arXiv2025

A Newton Augmented Lagrangian Method for Symmetric Cone Programming with Complexity Analysis

Symmetric cone programming covers a broad class of convex optimization problems, including linear programming, second-order cone programming, and semidefinite programming. Although the augmented Lagrangian method (ALM) is well-suited for large-scale problems, its subproblems are often not twice continuously differentiable, preventing the direct use of classical Newton methods. To address this issue, we observe that barrier functions used in interior-point methods (IPMs) naturally serve as effective smoothing terms to alleviate such nonsmoothness. By combining the strengths of ALM and IPMs, we construct a novel augmented Lagrangian function and subsequently develop a Newton augmented Lagrangian (NAL) method. By leveraging the self-concordance property of the barrier function, the proposed method is shown to achieve an $\mathcal{O}(1/{\epsilon})$ complexity bound. In addition, a spectral analysis reveals that the condition numbers of the Schur complement matrices arising in the NAL method are of order $\mathcal{O}(1/{\mu})$, which is better than the $\mathcal{O}(1/{\mu^2})$ order of classical IPMs. This improvement is further illustrated by a heatmap of condition numbers. Numerical experiments conducted on standard benchmarks indicate that the NAL method exhibits significant performance improvements compared to several existing methods.

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