A new smoothing technique for mathematical programs with equilibrium constraints

A kind of mathematical programs with equilibrium constraints （MPEC） is studied. By using the idea of successive approximation, a smoothing nonlinear programming, which is equivalent to the MPEC problem, is proposed. Thereby, it is ensured that some classical optimization methods can be applied for the MPEC problem. In the end, two algorithm models are proposed with the detail analysis of the global convergence.

Wolfe-Type Duality for Mathematical Programs with Equilibrium Constraints

This paper considers the mathematical programs with equilibrium constraints(MPEC).It is well-known that,due to the existence of equilibrium constraints,the Mangasarian-Fromovitz constraint qualification does not hold at any feasible point of MPEC and hence,in general,the developed numerical algorithms for standard nonlinear programming problems can not be applied to solve MPEC directly.During the past two decades,much research has been done to develop numerical algorithms and study optimality,stability,and sensitivity for MPEC.However,there are very few results on duality for MPEC in the literature.In this paper,we present a Wolfe-type duality for MPEC and,under some suitable conditions,we establish various duality theorems such as the weak duality,direct duality,converse duality,and strict converse duality theorems.We further show that a linear MPEC is equivalent to a linear programming problem in some sense.

An Algorithm for Traffic Equilibrium Flow with Capacity Constraints of Arcs

In the traffic equilibrium problem, we introduce capacity constraints of arcs, extend Beckmann’s formula to include these constraints, and give an algorithm for traffic equilibrium flows with capacity constraints on arcs. Using an example, we illustrate the application of the algorithm and show that Beckmann’s formula is a sufficient condition only, not a necessary condition, for traffic equilibrium with capacity constraints of arcs.

Continuity for vector optimization problems with equilibrium constraints

The concept of vector optimization problems with equilibrium constraints (VOPEC) is introduced. By using the continuity results of the approximate solution set to the equilibrium problem, we obtain the same results of the marginal map and the approximate value in VOPEC (ε) for vector-valued mapping.

非线性均衡约束下的光纤数据抗干扰通信方法

为了更加精准地解决非线性扰动变量对光纤通信造成的影响,消除扰动噪声对通信速率及通信质量的负面影响,结合扰动参量的非线性特征,对其约束均衡参量进行逐级优化,通过对其常规状态下光纤数据通信扰动模型的建立与抗扰条件分析、扰动信号的非线性分布范围确定与非线性均衡约束下扰动信号优化,实现扰动参量的彻底消除,同时提升原有光纤数据传输能力.通过创建的仿真数据对比证明提出方法的抗扰效果真实有效,其优化效果能够满足现阶段全场景应用需要,切实解决光纤数据通信扰动问题.

“双碳”目标下考虑各方努力水平的生鲜供应链网络均衡研究

为探讨不同竞争与合作模式下生鲜供应链成员最优努力水平、消费者低碳偏好和原材料产能约束对生鲜供应链利润的影响,本文结合生鲜产品易腐性、季节性的特点,以多个供应商、制造商、零售商及需求市场组成的生鲜供应链为研究对象,利用网络均衡的方法构建供应链网络模型,并设计基于对数二次型逼近的算法求解。研究结果表明:当所有生鲜供应链成员协同努力时,一定程度的努力投入能够使相关成员及供应链整体利润达到最优;当生鲜供应链各层级之间协同努力时,供应商和零售商努力水平的提高更有利于整体利润最大化;当生鲜供应链成员各自努力时,决策者需密切关注竞争对手或其余层级成员的努力水平;消费者低碳偏好的增加有利于提升制造商利润,但对供应商与零售商利润的改善有限;原材料产能约束能够使供应商利润得到较为显著提升,但并不利于改善其余供应链成员利润。

基于BIM技术的建筑群施工造价偏差预警

常规的建筑群施工造价偏差预警方法以竣工结算阶段的造价偏差预警为主,忽略了累积造价偏差,导致造价偏差预警存在明显差异。因此,设计了基于BIM技术的建筑群施工造价偏差预警方法。动态测算建筑群施工阶段工程造价偏差,将不确定性问题的造价价值信息提取出来,为造价偏差预警行为提供正确描述。基于BIM技术构建建筑群施工造价偏差预警模型,利用IFC描述BIM造价偏差预警的标准格式,为建筑群施工各阶段的造价偏差提供预警决策。约束建筑群施工造价阶段性预警变量,针对不同的施工阶段进行预警条件优化,将各预警变量进行均衡约束,确保施工造价偏差预警效果。采用实例分析,验证了该预警方法的效果更佳,能够应用于实际生活中。

求解一类含均衡约束的均衡问题

本文研究一类含均衡约束的均衡问题(EPEC),求解其共用约束乘子成比例的正则稳定点,将此类EPEC转化为一个标准的含均衡约束的数学规划问题(MPEC)进行求解.并分析相似产品市场竞争中存在的此类博弈模型,将其按上述方法进行有效求解.

An SQP algorithm for mathematical programs with nonlinear complementarity constraints

In this paper, we describe a successive approximation and smooth sequential quadratic programming （SQP） method for mathematical programs with nonlinear complementarity constraints （MPCC）. We introduce a class of smooth programs to approximate the MPCC. Using an 11 penalty function, the line search assures global convergence, while the superlinear convergence rate is shown under the strictly complementary and second-order sufficient conditions. Moreover, we prove that the current iterated point is an exact stationary point of the mathematical programs with equilibrium constraints （MPEC） when the algorithm terminates finitely.

An Implicit Smooth Conjugate Projection Gradient Algorithm for Optimization with Nonlinear Complementarity Constraints

This paper discusses a special class of mathematical programs with equilibrium constraints. At first, by using a generalized complementarity function, the discussed problem is transformed into a family of general nonlinear optimization problems containing additional variable μ. Furthermore, combining the idea of penalty function, an auxiliary problem with inequality constraints is presented. And then, by providing explicit searching direction, we establish a new conjugate projection gradient method for optimization with nonlinear complementarity constraints. Under some suitable conditions, the proposed method is proved to possess global and superlinear convergence rate.

A Novel Stackelberg-Game-Based Energy Storage Sharing Scheme Under Demand Charge

Demand response(DR)using shared energy storage systems(ESSs)is an appealing method to save electricity bills for users under demand charge and time-of-use(TOU)price.A novel Stackelberg-game-based ESS sharing scheme is proposed and analyzed in this study.In this scheme,the interactions between selfish users and an operator are characterized as a Stackelberg game.Operator holds a large-scale ESS that is shared among users in the form of energy transactions.It sells energy to users and sets the selling price first.It maximizes its profit through optimal pricing and ESS dispatching.Users purchase some energy from operator for the reduction of their demand charges after operator's selling price is announced.This game-theoretic ESS sharing scheme is characterized and analyzed by formulating and solving a bi-level optimization model.The upper-level optimization maximizes operator's profit and the lower-level optimization minimizes users'costs.The bi-level model is transformed and linearized into a mixed-integer linear programming(MILP)model using the mathematical programming with equilibrium constraints(MPEC)method and model linearizing techniques.Case studies with actual data are carried out to explore the economic performances of the proposed ESS sharing scheme.

Coherent Application of a Contact Structure to Formulate Classical Non-Equilibrium Thermodynamics

This contribution presents an outline of a new mathematical formulation for Classical Non-Equilibrium Thermodynamics (CNET) based on a contact structure in differential geometry. First a non-equilibrium state space is introduced as the third key element besides the first and second law of thermodynamics. This state space provides the mathematical structure to generalize the Gibbs fundamental relation to non-equilibrium thermodynamics. A unique formulation for the second law of thermodynamics is postulated and it showed how the complying concept for non-equilibrium entropy is retrieved. The foundation of this formulation is a physical quantity, which is in non-equilibrium thermodynamics nowhere equal to zero. This is another perspective compared to the inequality, which is used in most other formulations in the literature. Based on this mathematical framework, it is proven that the thermodynamic potential is defined by the Gibbs free energy. The set of conjugated coordinates in the mathematical structure for the Gibbs fundamental relation will be identified for single component, closed systems. Only in the final section of this contribution will the equilibrium constraint be introduced and applied to obtain some familiar formulations for classical (equilibrium) thermodynamics.

输配协同下的输配能量市场双层出清模式

在传统输配网分离决策的电能市场出清模式中,会出现节点电价过高和风电资源消纳能力不足等问题,为协调输配电网资源与信息不均衡问题,分析输配协同决策下的电力市场特点,构建了输配协同下输电运营商(Transmission System Operator, TSO)与配电运营商(Distribution System Operator, DSO)电能市场双层出清模式,下层DSO决策利用KKT条件将其转化为均衡约束,经二阶锥的对偶规划与下层KKT的线性化处理,其转化为带均衡约束的数学规划(Mathematical Program with Equilibrium Constraints, MPEC)单层决策模型。实验结果表明,该模型成功调动输配侧资源,降低了市场出清价格,提高了系统经济性,为电力系统运行和市场协调带来积极影响。

能耗均衡约束下的多源异质传感器数据动态汇聚算法

多源异质传感器数据在无线传感网络中汇聚时,易受到无线传感网络冗余数据的干扰,导致其通信开销大、精度低。为此,提出能耗均衡约束下多源异质传感器数据动态汇聚算法。构建能耗均衡约束模型,保证多源异质数据节点在汇聚过程中所消耗的能量趋于平均值,采用监督判别投影算法对数据进行降维处理,消除数据中的冗余度,通过模糊分簇阈值筛选机制获取汇聚区域的簇头阈值,将簇区域内数据汇聚到传感器中,完成多源异质传感器数据的动态汇聚。仿真结果表明,所提算法的隐私泄露率在25%左右,通信开销在9 MB~46 MB之间,汇聚数据正确率高于90%,证明所提算法可有效降低通信开销,提高数据汇聚精度。

考虑劳动力流动限制的供应链网络均衡模型

新冠疫情引起的劳动力不足对劳动密集型企业造成了巨大的产能压力,“共享员工”是疫情期间许多企业缓解生产压力的一种做法。针对由多个相互竞争的异构供应链组成的供应链网络系统,利用变分不等式的方法,建立考虑劳动力流动限制的供应链网络均衡模型。利用欧拉算法求解该问题,分析了劳动力资源对供应链网络均衡决策的影响,并进一步分析“共享员工”措施对供应链运营效率的影响。研究表明,劳动力资源不足会降低供应链的产量及利润;“共享员工”可以帮助供应链根据自身产业结构及需求更好地分配有限的劳动力;“共享员工”在某些情况下会损害供应链的运行效率。

VaR约束下两个相互竞争保险公司的最优再保险投资策略

本文研究了VaR约束下两个竞争保险公司的最优再保险-投资策略。我们假设保险公司的动态盈余过程用经典的Cramer-Lundberg(C-L)风险模型来描述,该模型中的保费由损失相依保费原则确定。此外,保险公司可以购买比例再保险并投资于一个由一个无风险资产和一个风险资产组成的金融市场,其中风险资产的价格过程由几何布朗运动描述。首先,我们以保险公司相对终端财富的期望效用最大化为目标,建立了VaR约束下的优化问题。接下来,我们利用最优控制理论和动态规划原理解决了相应的约束优化问题。特别地,我们在指数效用下得到了三种不同情形下的纳什均衡策略。最后,通过具体的数值分析,阐述了一些参数对最优再保险策略和最优投资策略的影响,并得到了一些有意义的结论。

碳交易机制下考虑公平关切的低碳供应链网络均衡模型

文章以实现供应链网络的低碳可持续发展为目标,基于碳交易机制与制造商公平关切特征,利用变分不等式理论构建了库存能力约束下的网络均衡模型。研究结果表明,制造商有利的不公平关切行为不利于自身利润的增加,而不利不公平关切行为却能使其获得更多收益,并且零售商的利润也表现出相同的变化趋势。而当零售商存在库存能力限制时,制造商有利的不公平关切行为会进一步增加能力约束对低碳网络定价均衡决策的影响程度,而不利不公平关切行为会进一步削弱能力约束对低碳网络定价均衡决策的影响程度。另外,碳交易机制虽然提高了产品的价格,但对激励企业积极投资减排技术具有促进作用。

面向多产品拆解的拆解序列计划问题的模型与算法

再制造系统拆解车间在一个计划期内通常需要拆解多个产品。为实现多个产品拆解的整体最优,研究了面向多产品拆解的拆解序列计划问题,并建立了以拆解利润最大化为目标和以拆解能耗为约束的整数规划模型。结合问题特点提出了一种改进平衡优化算法对所研究问题进行求解。该算法构造了解的编码与解码方法,以提高算法的适用性;设计了邻域搜索方法,以加强算法的局部搜索能力,从而改善解的质量。通过对5个不同产品数量的算例进行求解,并与标准的平衡优化算法、改进的重力搜索算法以及遗传算法进行对比分析,验证了所提算法在解决该问题上的有效性。

Dimension-down iterative algorithm for the mixed transportation network design problem

An optimal dimension-down iterative algorithm （DDIA） is proposed for solving a mixed （continuous/ discrete） transportation network design problem （MNDP）, which is generally expressed as a mathematical programming with equilibrium constraints （MPEC）. The upper level of the MNDP aims to optimize the network performance via both the expansion of existing links and the addition of new candidate links, whereas the lower level is a traditional Wardrop user equilibrium （UE） model. The idea of the proposed DDIA is to reduce the dimensions of the problem. A group of variables （discrete/continuous） are fixed to altemately optimize another group of variables （continuous/discrete）. Some continuous network design problems （CNDPs） and discrete network design problems （DNDPs） are solved repeatedly until the optimal solution is obtained. A numerical example is given to demonstrate the efficiency of the proposed algorithm.

关于核函数选取的方法

在支持向量机技术中,核函数选取的好坏直接影响支持向量机的性能.目前关于核函数的研究在理论和应用两方面均取得了一定的成果,但还未深入到足以指导核函数的选取.本文从混合核函数着手研究,建立若干选取规则,得到关于核函数选取的方法.采用平衡约束规划(MPEC)模型来优化选取参数,解决了参数的选取问题.