An inexact continuation accelerated proximal gradient algorithm for low n-rank tensor recovery

The low n-rank tensor recovery problem is an interesting extension of the compressed sensing. This problem consists of finding a tensor of minimum n-rank subject to linear equality constraints and has been proposed in many areas such as data mining, machine learning and computer vision. In this paper, operator splitting technique and convex relaxation technique are adapted to transform the low n-rank tensor recovery problem into a convex, unconstrained optimization problem, in which the objective function is the sum of a convex smooth function with Lipschitz continuous gradient and a convex function on a set of matrices. Furthermore, in order to solve the unconstrained nonsmooth convex optimization problem, an accelerated proximal gradient algorithm is proposed. Then, some computational techniques are used to improve the algorithm. At the end of this paper, some preliminary numerical results demonstrate the potential value and application of the tensor as well as the efficiency of the proposed algorithm.     

基于变分的图像恢复算法及收敛性

提出了一种保持边缘的正则化图像恢复算法,该方法可有效地用于求解线性逆问题的 非凸优化过程.通过对正则化函数及相应泛函性质的理论分析,得出了使泛函达到最小的正则 化函数表达式;引入一个与原非凸泛函相应的二元泛函,将非凸优化问题转化为本质上的凸优 化问题,采用松弛迭代算法获得非凸优化问题的局部极小解;证明了所提出的算法是全局收敛 的.通过实验验证了算法的有效性.     

Distributed stochastic mirror descent algorithm for resource allocation problem

In this paper, we consider a distributed resource allocation problem of minimizing a global convex function formed by a sum of local convex functions with coupling constraints. Based on neighbor communication and stochastic gradient, a distributed stochastic mirror descent algorithm is designed for the distributed resource allocation problem. Sublinear convergence to an optimal solution of the proposed algorithm is given when the second moments of the gradient noises are summable. A numerical example is also given to illustrate the eff ectiveness of the proposed algorithm.     

Numerical algorithm for solving the problem of synthesis of impulse controls under uncertainty

Feedback control problem for a linear system under an unknown but bounded disturbance is considered. A numerical algorithm for the synthesis of controls in the problem of impulse control based on the representation of the value function using methods of convex analysis in terms of the conjugate function and the subsequent approximation of the conjugate function by piecewise affine convex functions is described.     

一类非线性极小极大问题的粒子群-邻近点算法

针对每个分量函数都是凸函数的离散型非线性极小极大问题,提出一种全局收敛的粒子群-邻近点混合算法。该算法利用极大熵函数将极小极大问题转化为一个光滑函数的无约束凸优化问题;利用邻近点算法为外层算法,内层算法采用粒子群算法来优化此问题;数值结果表明,该算法数值稳定性好、收敛快,是求解此类非线性极小极大问题的一种有效算法。     

一种加权剖分简单多边形为三角形和凸四边形子域的算法

针对计算几何与有限元网格自动剖分中多边形子域剖分问题，给出了一种适用于有限元网格子域单元（即大单元）剖分的标准，并提出了一种通过在可视点对之间引进适当的多边形剖分和根据子域单元的形状质量判定因子来引导剖分的算法。由于建立的权函数和凹角（凸角）本身有关，因此对同属于凹角（凸角）的权函数也可以加以权值上的区分。该算法通过分步进行剖分，即先将简单多边形剖分为凸多边形，然后再将凸多边形剖分为凸六边形和凸五边形，最后将凸六边形和凸五边形剖分为三角形和凸四边形，以得到满足要求的剖分结果。在以上的每个剖分过程中，都引进了权重来引导剖分，使得剖分结果更加优化、合理。     

二阶智能体的分布式非光滑资源分配算法

分布式资源分配问题旨在满足局部约束下完成一定量资源分配的同时使全局成本函数最小.首先,针对无向连通网络下二阶积分器型线性智能体系统,结合Karush-Kuhn-Tucker条件,提出一种初始值任意的分布式优化算法,其中,全局等式约束对偶变量实现比例积分控制,局部凸函数不等式约束对偶变量实现自动获取.当全局成本函数为非光滑凸函数时,借助集值LaSalle不变性原理理论证明所提出算法渐近收敛到全局最优解.其次,将所提出算法推广至无向连通网络下参数未知的Euler-Lagrange多智能体系统.当全局成本函数为非光滑凸函数时,借助Barbalat引理理论证明所提出算法渐近收敛到全局最优解.最后,通过数值仿真验证了所提算法的有效性.     

Two stochastic optimization algorithms for convex optimization with fixed point constraints

Two optimization algorithms are proposed for solving a stochastic programming problem for which the objective function is given in the form of the expectation of convex functions and the constraint set is defined by the intersection of fixed point sets of nonexpansive mappings in a real Hilbert space. This setting of fixed point constraints enables consideration of the case in which the projection onto each of the constraint sets cannot be computed efficiently. Both algorithms use a convex function and a nonexpansive mapping determined by a certain probabilistic process at each iteration. One algorithm blends a stochastic gradient method with the Halpern fixed point algorithm. The other is based on a stochastic proximal point algorithm and the Halpern fixed point algorithm; it can be applied to nonsmooth convex optimization. Convergence analysis showed that, under certain assumptions, any weak sequential cluster point of the sequence generated by either algorithm almost surely belongs to the solution set of the problem. Convergence rate analysis illustrated their efficiency, and the numerical results of convex optimization over fixed point sets demonstrated their effectiveness.     

罚函数凸优化迭代算法及在无人机路径规划中的应用

针对无人机路径规划问题,建立了具有定常非线性系统、非仿射等式约束、非凸不等式约束的非凸控制问题模型,并对该模型进行了算法设计和求解。基于迭代寻优的求解思路,提出了凸优化迭代求解方法和罚函数优化策略。前者利用凹凸过程(CCCP)和泰勒公式对模型进行凸化处理,后者将经处理项作为惩罚项施加到目标函数中以解决初始点可行性限制。经证明该方法严格收敛到原问题的Karush-Kuhn-Tucker(KKT)点。仿真实验验证了罚函数凸优化迭代算法的可行性和优越性,表明该算法能够为无人机规划出一条满足条件的飞行路径。     

A Cut-Based Algorithm for the Nonlinear Dual of the Minimum Cost Network Flow Problem

We consider a convex, or nonlinear, separable minimization problem with constraints that are dual to the minimum cost network flow problem. We show how to reduce this problem to a polynomial number of minimum s,t-cut problems. The solution of the reduced problem utilizes the technique for solving integer programs on monotone inequalities in three variables, and a so-called proximity-scaling technique that reduces a convex problem to its linear objective counterpart. The problem is solved in this case in a logarithmic number of calls, O(log U), to a minimum cut procedure, where U is the range of the variables. For a convex problem on n variables the minimum cut is solved on a graph with O(n²) nodes. Among the consequences of this result is a new cut-based scaling algorithm for the minimum cost network flow problem. When the objective function is an arbitrary nonlinear function we demonstrate that this constrained problem is solved in pseudopolynomial time by applying a minimum cut procedure to a graph on O(nU) nodes.     

Bayes optimality in linear discriminant analysis

We present an algorithm which provides the one-dimensional subspace where the Bayes error is minimized for the C class problem with homoscedastic Gaussian distributions. Our main result shows that the set of possible one-dimensional spaces v, for which the order of the projected class means is identical, defines a convex region with associated convex Bayes error function g(v). This allows for the minimization of the error function using standard convex optimization algorithms. Our algorithm is then extended to the minimization of the Bayes error in the more general case of heteroscedastic distributions. This is done by means of an appropriate kernel mapping function. This result is further extended to obtain the d-dimensional solution for any given d, by iteratively applying our algorithm to the null space of the (d - 1)-dimensional solution. We also show how this result can be used to improve up on the outcomes provided by existing algorithms, and derive a low-computational cost, linear approximation. Extensive experimental validations are provided to demonstrate the use of these algorithms in classification, data analysis and visualization.     

Nonsmooth DC programming approach to clusterwise linear regression: optimality conditions and algorithms

The clusterwise linear regression problem is formulated as a nonsmooth nonconvex optimization problem using the squared regression error function. The objective function in this problem is represented as a difference of convex functions. Optimality conditions are derived, and an algorithm is designed based on such a representation. An incremental approach is proposed to generate starting solutions. The algorithm is tested on small to large data sets.     

An optimization algorithm for checking feasibility of robust H ∞-control problem for linear time-varying uncertain systems

An algorithm for checking feasibility of the robust H _∞-control problem for systems with time-varying norm bounded uncertainty is suggested. This algorithm is an iterative procedure on each step of which an optimization problem for a linear function under convex constraints determined by LMIs is solved. The effectiveness of the proposed algorithm is demonstrated on the numerical example of a parametrically disturbed pendulum.     

Maximizing for the sum of ratios of two convex functions over a convex set

This paper presents an algorithm for globally maximizing a sum of convex–convex ratios problem with a convex feasible region, which does not require involving all the functions to be differentiable and requires that their sub-gradients can be calculated efficiently. To our knowledge, little progress has been made for globally solving this problem so far. The algorithm uses a branch and bound scheme in which the main computational effort involves solving a sequence of linear programming subproblems. Because of these properties, the algorithm offers a potentially attractive means for globally solving the sum of convex–convex ratios problem over a convex feasible region. It has been proved that the algorithm possesses global convergence. Finally, the numerical experiments are given to show the feasibility of the proposed algorithm.     

An exact penalty function algorithm for control problems with state and control constraints

This paper describes an exact penalty function algorithm for solving control problems with state, control, and terminal constraints and establishes its convergence properties. A convex optimal control problem is defined whose solution yields a search direction which satisfies the control constraints and reduces a first-order estimate of the exact penalty function. Step length is determined using an Armijo-like procedure. An adaptive procedure for adjusting the penalty parameter completes the algorithm.     

Self-organizing maps by difference of convex functions optimization

We offer an efficient approach based on difference of convex functions (DC) optimization for self-organizing maps (SOM). We consider SOM as an optimization problem with a nonsmooth, nonconvex energy function and investigated DC programming and DC algorithm (DCA), an innovative approach in nonconvex optimization framework to effectively solve this problem. Furthermore an appropriate training version of this algorithm is proposed. The numerical results on many real-world datasets show the efficiency of the proposed DCA based algorithms on both quality of solutions and topographic maps.     

The solution of an affine problem and its application in control

An optimization problem with an affine feasible set is studied. By converting the affine set to a closed convex set which contains the solution, the Meyer-Polak algorithm can be used. The selection of the constraints is the key issue. A theorem has been derived for the selection of the constraints for the affine problem where the cost function is of p-norm form. Its application in a control problem is demonstrated     

基于鞍点法的自适应分布式资源分配算法

研究一类带有不等式约束为凸函数的多智能体系统分布式资源分配问题.在资源分配问题中,各智能体拥有仅自身可知的局部成本函数和局部凸不等式约束.分布式资源分配旨在如何利用智能体间的信息交互设计一种分布式优化算法,完成定量资源分配的同时还保证最小化全局成本函数.针对该问题,基于卡罗需-库恩-塔克条件和比例积分控制思想,首先提出一种自适应分布式优化算法,其中凸不等式约束的对偶变量可实现自适应获取;然后,为了降低系统的通信资源消耗,设计一种动态事件触发控制策略以实现离散时间通信的分布式资源分配算法;最后,通过数值仿真验证所设计算法的有效性.     

基于广义凸下界估计的多模态差分进化算法

针对多模态优化问题,提出了基于广义凸下界估计模型的改进差分进化算法。首先,基于模型变换方法将原优化问题转变为单位单纯形约束条件下的严格递增射线凸优化问题;其次,基于广义凸理论,利用差分进化算法中更新个体的适应度知识,建立原优化问题广义凸下界估计模型,设计实现了基于 N-叉树的估计模型快速计算方法;进而,综合考虑原问题目标值与其估计值之间的差异,提出一种基于有偏采样的小生境指标,并设计区域进化树更新策略来保证算法的局部搜索能力。数值实验结果表明,提出的算法能够有效地发现并维持一定数量的满意解模态,动态地实现全局模态搜索到模态内局部增强的自适应平滑过渡。对于给出的测试问题,能够发现所有的全局最优解以及一些较好的局部极值解。     

一类DEDS最优调度问题的解法

本文提出了带存储器生产线的一类新的最优调度问题,给出了最优调度目标函数的 具体形式,指出它不是凸函数;在一个变量时给出了最优调度的公式解,在多个变量时得到了 一个迭代寻优的算法.