Optimality Conditions for Double-sparsity Constrained Optimization

Sparse optimization has witnessed advancements in recent decades,and the step function finds extensive applications across various machine learning and signal processing domains.This paper integrates zero norm and the step function to formulate a doublesparsity constrained optimization problem,wherein a linear equality constraint is also taken into consideration.By defining aτ-Lagrangian stationary point and a KKT point,we establish the first-order and second-order necessary and sufficient optimality conditions for the problem.Furthermore,we thoroughly elucidate their relationships to local and global optimal solutions.Finally,special cases and examples are presented to illustrate the obtained theorems.

Algorithm Studies on How to Obtain a Conditional Nonlinear Optimal Perturbation (CNOP)

The conditional nonlinear optimal perturbation （CNOP）, which is a nonlinear generalization of the linear singular vector （LSV）, is applied in important problems of atmospheric and oceanic sciences, including ENSO predictability, targeted observations, and ensemble forecast. In this study, we investigate the computational cost of obtaining the CNOP by several methods. Differences and similarities, in terms of the computational error and cost in obtaining the CNOP, are compared among the sequential quadratic programming （SQP） algorithm, the limited memory Broyden-Fletcher-Goldfarb-Shanno （L-BFGS） algorithm, and the spectral projected gradients （SPG2） algorithm. A theoretical grassland ecosystem model and the classical Lorenz model are used as examples. Numerical results demonstrate that the computational error is acceptable with all three algorithms. The computational cost to obtain the CNOP is reduced by using the SQP algorithm. The experimental results also reveal that the L-BFGS algorithm is the most effective algorithm among the three optimization algorithms for obtaining the CNOP. The numerical results suggest a new approach and algorithm for obtaining the CNOP for a large-scale optimization problem.

DISTRIBUTION OF RANDOM ELEMENTS SUBJECTED TO A FLEXIBLE BOUNDARY CONDITION

The probability distribution function of n random elements subjected to the flexible boundary condition is derived. The probability density is a descending curve and converges to a delta function as n tends to infinity. The distribution of the minimum value is discussed in context of ordered statistics.

Convergence analysis of a nonlinear Lagrange algorithm for general nonlinear constrained optimization problems

The convergence analysis of a nonlinear Lagrange algorithm for solving nonlinear constrained optimization problems with both inequality and equality constraints is explored in detail. The estimates for the derivatives of the multiplier mapping and the solution mapping of the proposed algorithm are discussed via the technique of the singular value decomposition of matrix. Based on the estimates, the local convergence results and the rate of convergence of the algorithm are presented when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions. Furthermore, the condition number of the Hessian of the nonlinear Lagrange function with respect to the decision variables is analyzed, which is closely related to efficiency of the algorithm. Finally, the preliminary numericM results for several typical test problems are reported.

A New Kind of Simple Smooth Exact Penalty Function of Constrained Nonlinear Programming

The penalty function method is one basic method for solving constrained nonlinear programming, in which simple smooth exact penalty functions draw much attention for their simpleness and smoothness. This article offers a new kind of simple smooth approximative exact penalty function of general constrained nonlinear programmings and analyzes its properties.

Adaptive Constrained On-board Guidance Technology forPowered Glide Vehicle

To make full use of expanded maneuverability and increased range,adaptive constrained on-board guidance technology is the key capability for a glide vehicle with a double-pulse rocket engine,especially under the requirements of desired target changing and on-line reconfigurable control and guidance.Based on the rapid footprint analysis,whether the new target is within the current footprint area is firstly judged.If not,the rocket engine ignites by the logic obtained from the analysis of optimal flight range by the method of hp-adaptive Gauss pseudospectral method(hp-GPM).Then,an on-board trajectory generation method based on powered quasi-equilibrium glide condition(QEGC)and linear quadratic regulator(LQR)method is used to guide the vehicle to the new target.The effectiveness of the guidance method consisted of powered on-board trajectory generation,LQR trajectory tracking,footprint calculation,and ignition time determination is indicated by some simulation examples.

New smooth gap function for box constrained variational inequalities

A new smooth gap function for the box constrained variational inequality problem （VIP） is proposed based on an integral global optimality condition. The smooth gap function is simple and has some good differentiable properties. The box constrained VIP can be reformulated as a differentiable optimization problem by the proposed smooth gap function. The conditions, under which any stationary point of the optimization problem is the solution to the box constrained VIP, are discussed. A simple frictional contact problem is analyzed to show the applications of the smooth gap function. Finally, the numerical experiments confirm the good theoretical properties of the method.

MERIT FUNCTION AND GLOBAL ALGORITHMFOR BOX CONSTRAINED VARIATIONALINEQUALITIES

The authors consider optimization methods for box constrained variational inequalities. First, the authors study the KKT-conditions problem based on the original problem. A merit function for the KKT-conditions problem is proposed, and some desirable properties of the merit function are obtained. Through the merit function, the original problem is reformulated as minimization with simple constraints. Then, the authors show that any stationary point of the optimization problem is a solution of the original problem. Finally, a descent algorithm is presented for the optimization problem, and global convergence is shown.

A MIXED SUPERLINEARLY CONVERGENT ALGORITHM WITH NONMONOTONE SEARCH FOR CONSTRAINED OPTIMIZATIONS

In the paper, a new mixed algorithm combined with schemes of nonmonotone line search, the systems of linear equations for higher order modification and sequential quadratic programming for constrained optimizations is presented. Under some weaker assumptions,without strict complementary condition, the algorithm is globally and superlinearly convergent.

基于约束区间算法的模糊优化问题的Karush-Kuhn-Tucker条件

主要研究带有不等式约束的模糊优化问题,利用截集构建了与原问题等价的区间值优化问题,基于约束区间算法(CIA)将所得区间值优化问题转为等价的非线性优化问题,从而达到了去模糊化的目的.首先,定义了带有模糊系数函数截集的导数,并利用Zadeh分解定理给出模糊函数的导数概念.其次,在正线性无关约束规范下,建立了模糊优化问题的Karush-Kuhn-Tucker(KKT)条件.最后,利用KKT条件求解具体的模糊优化问题.

饱和约束下交流伺服系统转动惯量的在线辨识

在伺服系统运行过程中,系统饱和会影响观测器参数辨识的准确性。为了解决这个问题,提出一种抗饱和扩展状态观测器(ASESO)。建立单惯量伺服系统的数学模型,针对伺服系统饱和与未饱和2种情况,设计ASESO来在线辨识伺服系统的转动惯量与集总扰动,其中集总扰动用来对系统进行前馈补偿,削弱外部扰动对系统模型线性化的影响。与传统的ESO相比,ASESO在2种系统状态(饱和与未饱和)下有不同的观测器模型。当系统满足限制饱和条件(CSCs)时,观测器处于未饱和状态并实时更新转动惯量;相反,当系统进入饱和状态时,由于系统控制量变化很小,导致并不是所有的CSCs都满足,观测器不再更新转动惯量,保持前一时刻的识别值。最后利用Lyapunov理论验证了所设计方案的稳定性。仿真结果表明:所提方案在2种状态下均保证了转动惯量辨识的准确性。理论分析和仿真结果共同验证了ASESO与传统方法相比的优越性能。

非负组稀疏约束优化问题的最优性条件

基于Bouligand意义下的切锥与法锥和Clarke意义下的切锥与法锥,该文研究了非负组稀疏约束优化问题的最优性理论.该文定义了非负组稀疏约束集的Bouligand切锥与法锥和Clarke切锥与法锥,并给出了它们的等价刻画形式.在目标函数连续可微的条件下,借助于非负组稀疏约束集的切锥和法锥,给出了该优化问题的四类稳定点的定义,并讨论了它们之间的关系.最后,建立了非负组稀疏约束优化问题的一阶和二阶最优性条件.

典型滨海平原城市地下空间地质适宜性评价

地下空间地质适宜性评价是城市地下空间资源开发利用的重要前提。滨海平原城市因其特定的城市类型,需建立特定的地质适宜性评价体系来指导地下空间地质适宜性评价。本文以我国典型滨海平原城市日照市为研究区,深入分析了影响地下空间开发的地质环境因素,提出了基于基础性地质和约束性地质条件的滨海平原城市地质适宜性评价指标体系,进一步采用经验分值法、熵权法和综合评价模型厘定了研究区浅层(0~15m)和次浅层(15~30m)地下空间地质适宜性分区。结果显示,评价指标中水位埋深、断裂带、地面沉降和海水入侵等对地下空间地质适宜性的影响较大。研究区浅层和次浅层地下空间地质适宜性以适宜和较适宜为主,约占研究区面积的90%,不适宜区域约占研究区面积的10%。

虚位移和理想约束系统的不同定义

关于理想的、一般性一阶非完整约束系统,这篇文章认为,虚位移满足的关系式即Chetaev关系式是可以被导出的,但该式中的虚位移应当指的是高斯变分.同样,要从动力学普遍方程出发,借助Chetaev关系式导出系统的带乘子的拉格朗日方程,首先需要判断系统是否为理想约束系统,但理想约束条件式中的虚位移也应当指的是高斯变分.对于理想的、一阶线性非完整约束系统,虚位移所满足的关系式和理想约束条件式中,采用的虚位移既可以是高斯变分,也可以是Jourdain变分.

ON BLOCK PRECONDITIONERS FOR PDE-CONSTRAINED OPTIMIZATION PROBLEMS

Recently, Bal proposed a block-counter-diagonal and a block-counter-triangular precon- ditioning matrices to precondition the GMRES method for solving the structured system of linear equations arising from the Galerkin finite-element discretizations of the distributed control problems in （Computing 91 （2011） 379-395）. He analyzed the spectral properties and derived explicit expressions of the eigenvalues and eigenvectors of the preconditioned matrices. By applying the special structures and properties of the eigenvector matrices of the preconditioned matrices, we derive upper bounds for the 2-norm condition numbers of the eigenvector matrices and give asymptotic convergence factors of the preconditioned GMRES methods with the block-counter-diagonal and the block-counter-triangular pre- conditioners. Experimental results show that the convergence analyses match well with the numerical results.

MATRIX EQUATION AXB=E WITH PX=sXP CONSTRAINT

The matrix equation AXB = E with the constraint PX = sXP is considered, where P is a given Hermitian matrix satisfying P^2 = I and s = ±1. By an eigenvalue decomposition of P, the constrained problem can be equivalently transformed to a well-known unconstrained problem of matrix equation whose coefficient matrices contain the corresponding eigenvector, and hence the constrained problem can be solved in terms of the eigenvectors of P. A simple and eigenvector-free formula of the general solutions to the constrained problem by generalized inverses of the coefficient matrices A and B is presented. Moreover, a similar problem of the matrix equation with generalized constraint is discussed.

The First-Order Necessary Conditions for Sparsity Constrained Optimization

In this paper,we study optimization problems with the sparsity constraints.Firstly we give the expressions of the Mordukhovich(the limiting)normal cone of sparsity constraint and its intersection with a polyhedral set,and then based on these expressions we present the first-order necessary conditions for sparsity constrained optimization.

An improved PSO algorithm for solving nonlinear programing problems with constrained conditions

Engineering optimization problems can be always classified into two main categories including the linear programming(LP)and nonlinear programming(NLP)problems.Each programming problem further involves the unconstrained conditions and constrained conditions for design variables of the optimized system.This paper will focus on the issue about the design problem of NLP with the constrained conditions.The employed method for such NLP problems is a variant of particle swarm optimization(PSO),named improved particle swarm optimization(IPSO).The developed IPSO is to modify the velocity updating formula of the algorithm to enhance the search ability for given optimization problems.In this work,many different kinds of physical engineering optimization problems are examined and solved via the proposed IPSO algorithm.Simulation results compared with various optimization methods reported in the literature will show the effectiveness and feasibility for solving NLP problems with the constrained conditions.

考虑消纳水平的新能源配套储能和输电通道容量协调优化配置

随着以风光为代表的可再生能源发电比例迅速提升,风电场、光伏电站出力的不确定性和波动性给电力实时平衡带来了极大的挑战,配置合理规模的储能可以保障经济性并提高新能源利用率。为此,针对容量已知的新能源电站,提出了一种配套储能和传输线路容量协调优化配置的机会约束规划模型,以储能和传输线路建设成本最小化为目标,以年新能源弃电率不超过规定的指标为机会约束。由于机会约束非凸且缺乏显式表达,基于条件风险价值将机会约束对应的可行域保守转换为线性约束,得到易于求解的线性规划问题,并量化分析了规划方案对新能源场景概率分布的鲁棒性。算例分析结果表明,所提规划模型可以有效地解决考虑新能源消纳能力的储能和传输线路容量配置问题,通过合理配置储能可以降低传输线路的容量从而节约总投资成本。

异质信息网络的复杂条件社区搜索

社区是信息网络的重要属性,社区搜索旨在寻找满足用户给定条件的节点集合,是信息网络分析的重要研究内容.异质信息网络由于包含更加全面、丰富的结构和语义信息,所以异质信息网络的社区搜索近年来受到人们的广泛关注.针对现有异质信息网络的社区搜索方法难以满足复杂条件社区搜索要求的不足,定义了复杂条件社区搜索问题,提出了考虑非对称元路径、受限元路径和禁止节点约束的搜索算法.3种算法分别通过元路径补全策略、调整带标签的批量搜索策略和拆分复杂搜索条件的方式搜索社区,同时针对禁止节点约束的搜索算法设计了基于剪枝策略和近似策略的优化算法以提高搜索效率.在真实数据集上进行了大量实验,实验结果证明了所提算法的有效性和高效性.