本文研究了带不等式约束的DC型切向凸优化问题的混合型对偶。首先利用约束规范条件建立了混合型对偶模型。其次,利用伪凸函数的性质建立了带不等式约束的DC型切向凸优化问题的弱对偶定理、强对偶定理和逆对偶定理。并且推广了前人已有的结论。In this paper, we study mixed-type duality for DC-type tangentially convex optimization problems with inequality constraints. Firstly, we introduce constraint qualification conditions to establish a mixed-type dual model. Subsequently, leveraging properties of pseudoconvex functions, we derive weak duality, strong duality, and converse duality theorems for the proposed optimization problem. Furthermore, our results generalize and extend existing conclusions from prior studies.
本文基于混合耗散不等式约束优化提出了一种面向离散时间脉冲系统的分布式故障检测方法,充分考虑了系统单元间的相互作用特性。首先针对脉冲作用下的故障检测观测器设计,确保其满足故障混合敏感性和干扰混合鲁棒性要求。通过引入混合矢量耗散性概念,系统地处理脉冲效应、故障混合敏感性和干扰混合鲁棒性条件,确保脉冲误差动态相对于混合供给率是混合矢量耗散的。接着,推导出保证脉冲估计误差动态矢量耗散的充分条件。最后,通过求解混合耗散不等式约束的优化问题离线获得观测器参数,并实现在线故障检测。在数值仿真中,通过与现有方法的比较,验证了本文提出方法的有效性和优越性。This paper presents a novel distributed fault detection approach for discrete-time impulse systems, leveraging an optimization framework constrained by a mixed dissipativity inequality. The interaction characteristics between system units are considered. First, the research emphasizes the design of a fault detection observer that operates under impulse actions, ensuring compliance with fault mixed sensitivity and disturbance mixed robustness requirements. By introducing the concept of mixed vector dissipativity, the proposed approach systematically addresses the effects of impulse actions, fault sensitivity conditions, and disturbance robustness conditions, ensuring that the impulse error dynamics are mixed vector dissipativity with respect to the mixed supply rate. Then, sufficient conditions are derived to guarantee the dynamic vector dissipativity of the impulse estimation error. Finally, the parameters of the fault detection observer are obtained offline by solving the optimization problem constrained by mixed dissipativity inequalities, and online fault detection is realized. In numerical simulation, the effectiveness and superiority of the proposed method are verified by comparing with existing method.
针对只含有不等式约束的优化问题,本文首先给出了其Karush-Kuhn-Tucker (KKT)条件,并利用光滑互补函数将KKT系统转化为一类光滑的方程组问题;其次,将光滑方程组问题转化为无约束优化问题;最后,本文提出一类二阶微分方程系统求解无约束优化问题,并讨论了二阶微分方程系统的解的稳定性及收敛速度。For optimization problems with only inequality constraints, this paper first presents their Karush-Kuhn-Tucker (KKT) conditions, and uses smooth complementarity functions to transform the KKT system into a class of smooth system of equations problems. Secondly, this article transforms the problem of smooth equation systems into an unconstrained optimization problem. Finally, this article proposes a class of second-order differential equation systems for solving unconstrained optimization problems, and discusses the stability and convergence speed of the solutions of second-order differential equation systems.
