THE APPROXIMATION PROBLEM ON THE CLOSED CONVEX CONE AND ITS NUMERICAL SOLUTION

In this paper, we study the approximation problem on the closed convex cone, and prove that there exists a unique solution of the approximation problem, then give the algorithm to compute the unique solution.

SUPER EFFICIENCY IN THE NEARLY CONE-SUBCONVEXLIKE VECTOR OPTIMIZATION WITH SET-VALUED FUNCTIONS

Some properties for convex cones are discussed, which are used to obtain an equivalent condition and another important property for nearly cone-subconvexlike set-valued functions. Under the nearly cone-subconvexlikeness, some characterizations of the super efficiency are given in terms of scalarization and Lagrangian multipliers. Related results are generalized.

BENSON PROPER EFFICIENCY IN THE NEARLY CONE-SUBCONVEXLIKE VECTOR OPTIMIZATION WITH SET-VALUED FUNCTIONS

Some properties of convex cones are obtained and are used to derive several equivalent conditions as well as another important property for nearly cone-subconvexlike set-valued functions. Under the assumption of nearly cone-subconvexlikeness,a Lagrangian multiplier theorem on Benson proper efficiency is presented. Related results are generalized.

On Iteration Complexity of a First-Order Primal-Dual Method for Nonlinear Convex Cone Programming

Nonlinear convex cone programming(NCCP)models have found many practical applications.In this paper,we introduce a flexible first-order primal-dual algorithm,called the variant auxiliary problem principle(VAPP),for solving NCCP problems when the objective function and constraints are convex but may be nonsmooth.At each iteration,VAPP generates a nonlinear approximation of the primal augmented Lagrangian model.The approximation incorporates both linearization and a distance-like proximal term,and then the iterations of VAPP are shown to possess a decomposition property for NCCP.Motivated by recent applications in big data analytics,there has been a growing interest in the convergence rate analysis of algorithms with parallel computing capabilities for large scale optimization problems.We establish O(1/t)convergence rate towards primal optimality,feasibility and dual optimality.By adaptively setting parameters at different iterations,we show an O(1/t2)rate for the strongly convex case.Finally,we discuss some issues in the implementation of VAPP.

Common Fixed Points for a Countable Family of Quasi-Contractive Mappings on a Cone Metric Space with the Convex Structure

In this paper, we consider a countable family of surjective mappings {Tn}n∈N satisfying certain quasi-contractive conditions. We also construct a convergent sequence { Xn } n c∈Nby the quasi-contractive conditions of { Tn } n ∈N and the boundary condition of a given complete and closed subset of a cone metric space X with convex structure, and then prove that the unique limit x＂ of {xn}n∈N is the unique common fixed point of {Tn}n∈N. Finally, we will give more generalized common fixed point theorem for mappings {Ti,j}i,j∈N. The main theorems in this paper generalize and improve many known common fixed point theorems for a finite or countable family of mappings with quasi-contractive conditions.

Common Fixed Points for a Countable Family of Non-self Mappings in Cone Metric Spaces with the Convex Prop erty

A new common fixed point result for a countable family of non-self mappings defined on a closed subset of a cone metric space with the convex property is obtained, and from which, a more general result is given. Our main results improve and generalize many known common fixed point theorems.

A Different Approach to Cone-Convex Optimization

In classical convex optimization theory, the Karush-Kuhn-Tucker (KKT) optimality conditions are necessary and sufficient for optimality if the objective as well as the constraint functions involved is convex. Recently, Lassere [1] considered a scalar programming problem and showed that if the convexity of the constraint functions is replaced by the convexity of the feasible set, this crucial feature of convex programming can still be preserved. In this paper, we generalize his results by making them applicable to vector optimization problems (VOP) over cones. We consider the minimization of a cone-convex function over a convex feasible set described by cone constraints that are not necessarily cone-convex. We show that if a Slater-type cone constraint qualification holds, then every weak minimizer of (VOP) is a KKT point and conversely every KKT point is a weak minimizer. Further a Mond-Weir type dual is formulated in the modified situation and various duality results are established.

Higher-Order Minimizers and Generalized (F,<i>ρ</i>)-Convexity in Nonsmooth Vector Optimization over Cones

In this paper, we introduce the concept of a (weak) minimizer of order k for a nonsmooth vector optimization problem over cones. Generalized classes of higher-order cone-nonsmooth (F, ρ)-convex functions are introduced and sufficient optimality results are proved involving these classes. Also, a unified dual is associated with the considered primal problem, and weak and strong duality results are established.

K-凸函数的性质

借助集值映射的K-回收映射和K-雅可比映射得到了K-凸函数的两个性质。

图-凸集值映射的性质

借助回收映射和余切导数得到了图-凸集值映射的两个性质.

RSS和AOA融合的单站低空无线电辐射源定位算法

提出一种在低空场景下基于接收信号强度(Rcecived Signal Strength,RSS)与到达角度(Angle of Arrival,AOA)信息融合的单站无源定位算法。该算法采用单架无人机设备虚拟多站设备接收无线电辐射源信号,融合RSS估计的距离信息与AOA方向角信息,依据最小二乘准则(LS)构造算法的优化目标函数,采用凸松弛技术将目标函数等价为二阶锥规划(SOCP)问题并通过内点法求解。实验结果表明,该算法的定位精度在2 km范围内可达20 m,其定位性能优于单站无源定位算法,且由于采用单架无人机采集信号,其设备复杂度相较于多站无源定位较低。

计及压缩机运行域的动态最优电-气能流凸化算法

压缩机是电-气互联系统的关键元件,其运行工作点是否位于运行域内关系到系统的安全稳定运行,同时其模型非凸性大幅增加了最优电-气能流的求解难度。为此,该文在压缩机准确建模的基础上,针对电力和天然气系统迥异的传输特性,提出一种计及压缩机运行域的动态最优电-气能流凸化算法。首先,根据压缩机运行参数与状态变量间的映射关系,构建压缩机三维运行域模型,并利用超平面实现运行域的凸包络松弛。其次,考虑差异化传输特性建立动态最优电-气能流模型,实现模型的混合整数二阶锥凸重构,并提出一种基于动态线性化与紧缩凸包络的凸化算法。最后,算例验证所提模型的有效性,表明所提算法在计算精度和效率上均优于连续锥规划算法。

基于凸包理论的含风电电力系统负荷恢复方案优化

大停电事故后,合理的负荷恢复策略对加快电力系统恢复、减少停电损失具有重要意义。针对标准交流潮流方程的非凸性导致负荷恢复模型求解效率低的问题,提出了基于凸包理论的含风电电力系统负荷恢复方案优化方法。首先,简要介绍了凸包理论以及非线性函数的凸包构建方法。其次,针对风电出力的不确定性,采用条件风险价值理论处理风电波动带来的风险,进而建立了直角坐标系下基于凸包理论的电力系统负荷恢复二阶锥优化模型,其中,交流潮流方程中的非线性项采用凸包表达。针对无法精确确定凸包参数的问题,提出了基于迭代更新的参数优化方法,减少因凸包引入的松弛误差。所提出的模型为混合整数二阶锥规划模型,可采用商业求解器高效求解。最后,采用IEEE 39节点系统和改进的广东电力系统验证了所提出模型的有效性。

CHARACTERIZATION OF SOLID CONES

The author gives a dual characterization of solid cones in locally convex spaces. From this the author obtains some criteria for judging convex cones to be solid in various kinds of locally convex spaces. Using a general expression of the interior of a solid cone, the author obtains a number of necessary and sufficient conditions for convex cones to be solid in the framework of Banach spaces. In particular, the author gives a dual relationship between solid cones and generalized sharp cones. The related known results are improved and extended.

半拟可微拟凸规划的KKT型最优性条件

最优性条件在优化问题中起着重要的作用,它为优化算法的研究提供了重要的理论依据。众所周知,凸规划方面最优性条件已比较完善。然而,由于拟凸函数性质的特殊性,对于拟凸规划问题解的Karush-Kuhn-Tucker(KKT)类型最优性条件的研究相对较少。本文利用半拟可微刻画了拟凸规划的最优性条件,同时研究了可行集法锥与带半拟可微性质的约束函数之间的关系,并证明了上述两个结果与Greenberg-Pierskalla次微分的关系。

Multicriterion Clustering Model with Cone Dominated Structure

A multicriterion clustering model with cone dominated structure is established and the concept of cone-efficient clustering is put forward. The properties of cone-efficient clustering are disscussed, and a method to solve multicriterion clustering is put forward.The model of Ref.[3] is a special case of that proposed in this paper.

THE EXISTENCE THEOREM OF THE CONE-WEAK SUBDIFFERENTIAL OF SET-VALUED MAPPING

In this paper, the existence theorem of the cone weak subdifferential of set valued mapping in locally convex topological vector space is proved. Received March 30,1998. 1991 MR Subject Classification: 47H17,90C29.

考虑智能软开关接入的主动配电网扩展规划方法

柔性互联装置的广泛应用给主动配电网(active distribution network,ADN)规划带来巨大挑战。该文提出一种考虑智能软开关(soft open point,SOP)接入的ADN扩展规划方法,对变电站新建及扩容,线路新建,智能软开关、分布式电源、储能系统以及无功补偿等设备的选址定容进行协同规划。首先,考虑分布式电源出力和负荷功率不确定性,采用基于改进高斯混合模型的聚类方法构建典型日场景。在此基础上,以年综合费用最小为目标函数,建立了考虑SOP接入的ADN扩展规划模型。然后,通过线性化和二阶锥松弛技术,将原始非凸非线性规划模型转化为混合整数二阶锥规划(mixed-integer second-order cone programming,MISOCP)模型,并提出逐次收缩凸松弛算法以获得凸松弛间隙足够小的原问题最优解。最后,在54节点主动配电网算例上验证了所提规划模型和求解算法的可行性与有效性。

参数自适应凸优化下的月面着陆最优轨迹规划

针对月面高精度下降着陆问题,提出了一种基于最优观测器的参数自适应凸优化月面着轨迹规划算法。首先,针对下降着陆主减速段需要尽可能减少燃料消耗的问题,考虑约束及动力学模型的非凸性,采用无损凸化技术将月面下降着陆问题转化为二阶锥规划问题;其次,为减少下降着陆过程中质量、比冲等参数不确定性影响,设计了基于Riccati方程的最优观测器;再根据标称参数凸优化所产生的最优轨迹进行飞行,利用该过程中加速度计实时测量信息,结合推力器输出信息,实现对着陆器参数的在线实时估计;在参数估计实现收敛后,通过在线求解二阶锥规划问题实现最优轨迹的在线规划。仿真结果表明,该观测器能够实现在线参数实时估计;该算法相较其他定点着陆算法,仅需要发动机具有离散的推力域,且具有精度更高的优势。

基于丰度反演及光谱变换的书画霉斑虚拟修复

古书画在保存过程中容易滋生霉斑病害,霉斑不仅影响书画的外观,还会对纸质文物造成不可逆转的永久性破坏。为了给霉斑清洗提供直观参考,提出基于丰度反演及光谱变换的书画霉斑提取与虚拟修复方法。选取与霉斑区域光谱曲线存在明显差异的450~600 nm之间244个波段作为特征波段,通过连续最大角凸锥端元提取与灰度分割,提取霉斑区域。再对高光谱影像进行主成分变换,选取包含信息量最大的前三主成分合成影像,利用Criminisi算法对霉斑区域进行修复,再将修复后的影像进行主成分逆变换,完成高光谱影像的虚拟修复。以中国近现代画家倪田的《捕鱼图》为例进行了研究,发现修复后的霉斑区域与画体融入性较好,边界自然平滑,均方根误差值普遍变小。研究结果可为书画霉斑清洗提供直观的修复依据,具有较强的实用性。