Two new predictor-corrector algorithms for second-order cone programming

Based on the ideas of infeasible interior-point methods and predictor-corrector algorithms,two interior-point predictor-corrector algorithms for the second-order cone programming(SOCP) are presented.The two algorithms use the Newton direction and the Euler direction as the predictor directions,respectively.The corrector directions belong to the category of the Alizadeh-Haeberly-Overton(AHO) directions.These algorithms are suitable to the cases of feasible and infeasible interior iterative points.A simpler neighborhood of the central path for the SOCP is proposed,which is the pivotal difference from other interior-point predictor-corrector algorithms.Under some assumptions,the algorithms possess the global,linear,and quadratic convergence.The complexity bound O(rln(ε0/ε)) is obtained,where r denotes the number of the second-order cones in the SOCP problem.The numerical results show that the proposed algorithms are effective.

SMOOTHING NEWTON ALGORITHM FOR THE CIRCULAR CONE PROGRAMMING WITH A NONMONOTONE LINE SEARCH

In this paper, we present a nonmonotone smoothing Newton algorithm for solving the circular cone programming(CCP) problem in which a linear function is minimized or maximized over the intersection of an affine space with the circular cone. Based on the relationship between the circular cone and the second-order cone(SOC), we reformulate the CCP problem as the second-order cone problem(SOCP). By extending the nonmonotone line search for unconstrained optimization to the CCP, a nonmonotone smoothing Newton method is proposed for solving the CCP. Under suitable assumptions, the proposed algorithm is shown to be globally and locally quadratically convergent. Some preliminary numerical results indicate the effectiveness of the proposed algorithm for solving the CCP.

Day-ahead Optimization Schedule for Gas-electric Integrated Energy System Based on Second-order Cone Programming

This paper proposes an optimal day-ahead opti-mization schedule for gas-electric integrated energy system(IES)considering the bi-directional energy flow.The hourly topology of electric power system(EPS),natural gas system(NGS),energy hubs(EH)integrated power to gas(P2G)unit,are modeled to minimize the day-ahead operation cost of IES.Then,a second-order cone programming(SOCP)method is utilized to solve the optimization problem,which is actually a mixed integer nonconvex and nonlinear programming issue.Besides,cutting planes are added to ensure the exactness of the global optimal solution.Finally,simulation results demonstrate that the proposed optimization schedule can provide a safe,effective and economical day-ahead scheduling scheme for gas-electric IES.

Nonsingularity in second-order cone programming via the smoothing metric projector

Based on the differential properties of the smoothing metric projector onto the second-order cone,we prove that,for a locally optimal solution to a nonlinear second-order cone programming problem,the nonsingularity of the Clarke's generalized Jacobian of the smoothing Karush-Kuhn-Tucker system,constructed by the smoothing metric projector,is equivalent to the strong second-order sufficient condition and constraint nondegeneracy,which is in turn equivalent to the strong regularity of the Karush-Kuhn-Tucker point.Moreover,this nonsingularity property guarantees the quadratic convergence of the corresponding smoothing Newton method for solving a Karush-Kuhn-Tucker point.Interestingly,the analysis does not need the strict complementarity condition.

Exact Computable Representation of Some Second-Order Cone Constrained Quadratic Programming Problems

Solving the quadratically constrained quadratic programming(QCQP)problem is in general NP-hard.Only a few subclasses of the QCQP problem are known to be polynomial-time solvable.Recently,the QCQP problem with a nonconvex quadratic objective function over one ball and two parallel linear constraints is proven to have an exact computable representation,which reformulates the original problem as a linear semidefinite program with additional linear and second-order cone constraints.In this paper,we provide exact computable representations for some more subclasses of the QCQP problem,in particular,the subclass with one secondorder cone constraint and two special linear constraints.

基于SOCP的采样矩阵求逆算法分析

为克服自适应波束形成技术中采样矩阵求逆(SMI)算法存在计算繁琐、运算量庞大等缺点,提出一种基于二阶锥规划(SOCP)算法的采样矩阵求逆改进算法,即针对SMI算法中数据块生成权向量的最小二乘模型,将其转化为二阶锥规划(SOCP)形式,然后使用SOCP优化方法能够快速准确地求出最优解.实际仿真结果表明:基于SOCP的SMI改进算法(SOCP-SMI)具有计算精确高、运算量小,且对波束特性的改善效果明显.

Unified convergence analysis of a second-order method of multipliers for nonlinear conic programming

In this paper,we accomplish the unified convergence analysis of a second-order method of multipliers(i.e.,a second-order augmented Lagrangian method)for solving the conventional nonlinear conic optimization problems.Specifically,the algorithm that we investigate incorporates a specially designed nonsmooth(generalized)Newton step to furnish a second-order update rule for the multipliers.We first show in a unified fashion that under a few abstract assumptions,the proposed method is locally convergent and possesses a(nonasymptotic)superlinear convergence rate,even though the penalty parameter is fixed and/or the strict complementarity fails.Subsequently,we demonstrate that for the three typical scenarios,i.e.,the classic nonlinear programming,the nonlinear second-order cone programming and the nonlinear semidefinite programming,these abstract assumptions are nothing but exactly the implications of the iconic sufficient conditions that are assumed for establishing the Q-linear convergence rates of the method of multipliers without assuming the strict complementarity.

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。

基于数据驱动与物理模型的主动配电网双时间尺度协调优化

高比例间歇性分布式电源与电动汽车接入配电网时,容易导致功率与电压频繁、快速、剧烈波动。文中结合数据驱动与物理建模方法,提出了一种配电网双时间尺度有功无功协调优化策略。针对短时间尺度(分钟级或秒级)的功率波动,以静止无功补偿器、分布式电源无功功率为决策变量,以网损最小为目标函数,计及物理约束,针对平衡与不平衡配电网分别构建了二阶锥与二次规划模型。针对长时间尺度(小时级)的优化,以有载调压变压器分接头变比、可投切电容电抗器挡位、储能系统充放电功率为动作,以网损为代价,计及节点电压越限惩罚,构建了马尔可夫决策过程。为克服连续-离散动作空间维数灾,采用一种基于松弛-预报-校正的深度确定性策略梯度强化学习求解算法。通过IEEE 33节点与IEEE 123节点配电系统验证了所提方法的有效性。

Strain localization of Mohr-Coulomb soils with non-associated plasticity based on micropolar continuum theory

To address the problems of strain localization, the exact Mohr-Coulomb (MC) model is used based on second-order cone programming (mpcFEM-SOCP) in the framework of micropolar continuum finite element method. Using the uniaxial compression test, we focused on the earth pressure problem of rigid wall segment involving non-associated plasticity. The numerical results reveal that when mpcFEM-SOCP is applied, the problems of mesh dependency can be effectively addressed. For geotechnical strain localization analysis involving non-associated MC plasticity, mpcFEM-SOCP in conjunction with the pseudo-time discrete scheme can improve the numerical stability and avoid the unreasonable softening issue in the pressure-displacement curves, which may be encountered in the conventional FEM. It also shows that the pressure-displacement responses calculated by mpcFEM-SOCP with the pseudo-time discrete scheme are higher than those calculated by mpcFEM-SOCP with the Davis scheme. The inclination angle of shear band predicted by mpcFEM-SOCP with the pseudo-time discrete scheme agrees well with the theoretical solution of non-associated MC plasticity.

Quadratic Optimization over a Second-Order Cone with Linear Equality Constraints

This paper studies the nonhomogeneous quadratic programming problem over a second-order cone with linear equality constraints.When the feasible region is bounded,we show that an optimal solution of the problem can be found in polynomial time.When the feasible region is unbounded,a semidefinite programming(SDP)reformulation is constructed to find the optimal objective value of the original problem in polynomial time.In addition,we provide two sufficient conditions,under which,if the optimal objective value is finite,we show the optimal solution of SDP reformulation can be decomposed into the original space to generate an optimal solution of the original problem in polynomial time.Otherwise,a recession direction can be identified in polynomial time.Numerical examples are included to illustrate the effectiveness of the proposed approach.

基于混合整数二阶锥规划的主动配电网有功–无功协调多时段优化运行

通过日前多时段优化,可以为主动配电网中的分布式电源、无功补偿装置和储能装置安排合理高效的生产计划,以达到调节电压水平、提高能源资源利用率、节能降损的目的。首先建立基于三相Distflow潮流的辐射状配电网有功–无功协调动态优化模型。该模型考虑分布式电源出力、储能装置充放电功率、电容器阻投切等连续和离散决策变量,是一个典型的混合整数非凸非线性规划。该类模型缺乏严格高效的求解方法。为此,采用二阶锥松弛技术将其中的三相潮流方程进行凸化松弛,使得优化问题转化为可有效求解的混合整数二阶锥规划模型。最后,采用扩展的IEEE 33节点三相测试系统仿真计算,利用MOSEK等算法包求得电网中各设备动作时刻和投运容量,验证了所提方法寻优稳定、松弛精确、计算高效等特性。

基于鲁棒均值–方差优化的发电自调度算法及鲁棒代价分析

解除管制电力市场背景下,发电厂商作为价格的接受者,需向电力交易中心提供发电交易策略来最大化自身的收益,从而形成发电自调度的优化模型。然而,当考虑电价不确定性时,发电商一方面希望最大化收益,另一方面需要最小化不确定性带来的风险。为此,该文建立了一种鲁棒均值方差优化模型,以收益最大化和风险最小化为多目标,进而获得多目标优化的Pareto前沿。通过等价转化发现,鲁棒均值方差模型与非鲁棒均值方差模型具有相同的数学形式,均为一个二阶锥优化。进一步分析了鲁棒模型对收益、风险以及Pareto前沿的代价。最后采用30节点系统对鲁棒均值方差优化的发电厂自调度模型以及鲁棒代价进行详细的分析和对比,结果证明提出方法和分析的正确性。

计及风电相关性的二阶锥动态随机最优潮流

可再生能源的大规模接入增加了系统运行调度中的不确定性。同时,实际运行中风电功率具有较强相关性,忽略这些因素将会带来较大的计算误差。目前二阶锥规划多应用于单个时间断面下的最优潮流,但只考虑单时段的最优潮流无法有效计及风电功率的不确定性以及相关性。而现有动态随机最优潮流缺乏对多维风电功率的准确建模,求解计算效率偏低,无法保证收敛性。针对以上问题,文中提出了计及风电相关性的二阶锥动态随机最优潮流模型。基于Pair Copula函数对多维风电功率进行建模,并通过凸松弛将非线性动态随机最优潮流模型转化为二阶锥规划模型,利用商业软件Gurobi结合改进三点估计法对模型进行求解。通过与传统模型以及不计及风电相关性的方案进行对比,验证该模型的有效性和实用性。

基于TDOA/AOA混合的高精度室内可见光定位算法

为了提高室内三维空间的定位精度,提出了一种基于联合到达时间差与到达角度(time difference of arrival/angle of arrival,TDOA/AOA)信息的混合定位算法。由于构建的目标函数具有非凸性,采用传统定位算法在目标函数求解过程中会出现局部最优解的问题。因此,针对该问题,将目标函数转成二次约束二次规划问题,通过引入半定松弛(semi-definite relaxation,SDR)方法将目标函数转换为二阶锥规划(second order cone programming,SOCP)问题,寻找全局最优解。其次,针对SOCP无法对凸包外的目标进行有效定位的问题,在该算法的基础上引入了惩罚项,使松弛后的约束条件进一步逼近原始约束条件,解决了定位过程中的凸包问题。数值仿真结果表明:在10m×10m×3m的三维定位空间内,选取40×40个测试点,平均定位误差为1.39cm,可实现室内三维空间高精度定位。与传统的混合定位算法相比,均能够获得较高的定位精度。

基于二阶锥规划的宽带波束形成器设计

宽带波束形成器有两种典型的实现方式,分别基于FIR滤波器和基于长方形阵列,将这两种波束形成器的波束响应表达为统一的形式,并根据参考波束的选择情况,为该类波束优化问题的求解建立了数学模型,该数学模型属于二阶锥约束问题,可以用二阶锥规划方法进行求解。仿真结果表明,提出的求解宽带恒定束宽响应波束形成器权系数的数学模型具有通用性,并且把二阶锥规划方法运用到宽带波束形成器优化设计中,其约束控制灵活,问题求解方便,设计结果精确。

基于迭代二阶锥的唯相位波束形成

唯相位自适应波束形成技术对普通相控阵雷达自适应干扰抑制是非常重要的,在数字阵列雷达中,唯相位技术则可以充分利用各阵元发射模块微波功率从而提高雷达的威力。该文提出了一种新的唯相位波束形成方法,假定初始唯相位权重矢量有个较小的相位扰动,将唯相位模型的目标函数和约束函数在该相位扰动区域内分别用泰勒一阶展开式来近似,则可以将原来的非凸问题转化为凸优化问题,通过二阶锥规划方法(SOCP)求得使当前目标函数最小的扰动矢量,然后更新得到新的权重矢量并代替原来的权重矢量,重复上述迭代过程直到满足停止条件,可以得到满足要求的唯相位权重。计算机仿真结果验证了该方法的正确性和有效性。

求解大规模机组组合问题的二阶锥规划方法

基于混合整数二阶锥规划(mixed integer second-order cone programming,MI-SOCP)提出一种求解电力系统计及爬坡约束机组组合问题(unit commitment,UC)的新方法。利用UC问题的混合整数二次规划(mixed integer quadratic programming,MI-QP)模型和一个简单混合整数集合的凸包表示,产生UC问题一个更紧的MI-SOCP模型。将最小覆盖不等式作为割平面,应用内点割平面法求解MI-SOCP以获得不计爬坡约束UC问题的机组启停状态。为满足爬坡约束,提出一种简单易行的机组启停状态修正方法。100机组96时段等多个系统的仿真结果表明,利用内点割平面法求解2种模型时,MI-SOCP能比MI-QP获得质量更好的次优解,所提方法能有效处理爬坡约束,适用于大规模的UC问题。

改进的宽带阵元延迟线恒定束宽波束形成算法

针对时域宽带恒定束宽波束形成计算复杂度高、硬件实现复杂等问题,提出基于离散空间响应变化(SRV)约束的宽带阵元延迟线(SDL)恒定束宽波束形成算法。该算法首先建立SDL宽带阵列模型,然后分别在有无强干扰的情况下,采用约束优化的方式,在保证期望来向增益趋于单位响应的同时,对期望方向逼近精度、旁瓣增益、主瓣宽度、SRV等进行约束,使得可能的最大旁瓣均值最小,最后将其转化为二阶锥规划问题,并利用内点法进行求解。实验表明,该算法的频率不变性较好,收敛速度较快,计算复杂度较低。

基于发射波束域-平行因子分析的MIMO雷达收发角度估计

该文提出了基于发射波束域-平行因子分析的双基地MIMO雷达收发角度估计方法。针对传统MIMO雷达发射功率分散的问题,将发射功率聚集范围与期望发射方向矢量相结合,提出了发射波束加权矩阵的优化准则,并转化为二阶锥规划形式,通过内点法求出其数值解,进而利用平行因子分析(PARAFAC)算法估计出收发角度。仿真结果表明:离线设计得到的波束加权矩阵使发射功率聚集于感兴趣的目标空域,且近似服从均匀分布,因此,在发射功率和角度估计算法相同时,基于发射波束域的角度估计精度优于全向等功率发射时的角度估计精度。