Liwei Zhang

This paper investigates minimax quadratic programming problems with coupled inequality constraints. By leveraging a duality theorem, we develop a dual algorithm that extends the dual active set method to the minimax setting, transforming the original inequality constrained problem into a sequence of equality constrained subproblems. Under a suitable assumption, we prove that the associated S-pairs do not repeat and that the algorithm terminates in a finite number of iterations, guaranteed by the monotonic decrease of the objective function value. To ensure numerical stability and efficiency, the algorithm is implemented using Cholesky factorization and Givens rotations. Numerical experiments on both randomly generated minimax quadratic programs and illustrative applications demonstrate the accuracy, stability, and computational effectiveness of the proposed algorithm.

Second-order optimality conditions are essential for nonsmooth optimization, where both the objective and constraint functions are Lipschitz continuous and second-order directionally differentiable. This paper provides no-gap second-order necessary and sufficient optimality conditions for such problems without requiring convexity assumptions on the constraint set. We introduce the concept of second-order gph-regularity for constraint functions, which ensures the outer second-order regularity of the feasible region and enables the formulation of comprehensive optimality conditions through the parabolic curve approach. An important application of our results is bilevel optimization, where we derive second-order necessary and sufficient optimality conditions for bi-local optimal solutions, which are based on the local solutions of the lower-level problem. By leveraging the Mangasarian-Fromovitz constraint qualification (MFCQ), strong second-order sufficient condition (SSOSC) and constant rank constraint qualification (CRCQ) of lower-level problem, these second-order conditions are derived without requiring the uniqueness of the lower-level multipliers. In addition, if the linear independence constraint qualification (LICQ) holds, these conditions are expressed solely in terms of the second-order derivatives of the functions defining the bilevel problem, without relying on the second-order information from the solution mapping, which would introduce implicit complexities.

Consider the stability properties of the Karush-Kuhn-Tucker (KKT) solution mapping [ ] for Nash equilibrium problems (NEPs) with canonical perturbations. Firstly, we obtain an exact characterization of the strong regularity of [ ] as well as an easily verified sufficient condition. Secondly, we propose equivalent conditions for the continuously differentiable single-valued localization of [ ]. Thirdly, the isolated calmness of [ ] is studied based on the I-property. The P-property is proposed as a sufficient condition for the robust isolated calmness of [ ] under convex assumptions. 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 the NEPs in which each individual player solves a quadratic programming problem. Funding: Financial support from the Strategic Priority Research Program of the Chinese Academy of Sciences [Grant XDA27010101] is gratefully acknowledged. Y.-H. Dai received financial support from the Natural Science Foundation of China [Grants 11991021, 11991020, and 12021001]. L. Zhang received financial support from the Major Program of the National Natural Science Foundation of China [Grants 72192830 and 72192831], the National Natural Science Foundation of China [Grant 12371298], and the 111 Project [Grant B16009].

This paper focuses on a class of inclusion problems of maximal monotone operators over directed multi-agent networks, where each agent is characterized by a splittable local operator. By integrating the ideas from correction, inexact proximal methods, and push-sum protocol, we proposed a distributed proximal-correction algorithm (referred to as DPCA- $\epsilon _{k}^{i}$ ) to solve this inclusion problem of operators, where $\epsilon _{k}^{i}$ is an agent predetermined accuracy for approximately computing proximal mappings. Under mild assumptions on problem setup and agents specified precision $\epsilon _{k}^{i}$ , we proved the convergence and sublinear rate $o(1/k)$ of DPCA- $\epsilon _{k}^{i}$ in terms of the first-order optimality residual. Furthermore, under the metric regularity of local operators of agents, we established its linear convergence rate. Numerical simulations and comparisons with other distributed algorithms illustrate the effectiveness of the proposed algorithm.