基于分解策略处理多分类不均衡问题的方法

针对多分类不均衡问题,提出了一种新的基于一对一（one-versus-one,OVO）分解策略的方法。首先基于OVO分解策略将多分类不均衡问题分解成多个二值分类问题;再利用处理不均衡二值分类问题的算法建立二值分类器;接着利用SMOTE过抽样技术处理原始数据集;然后采用基于距离相对竞争力加权方法处理冗余分类器;最后通过加权投票法获得输出结果。在KEEL不均衡数据集上的大量实验结果表明,所提算法比其他经典方法具有显著的优势。     

Bregmanized Domain Decomposition for Image Restoration

Computational problems of large-scale data are gaining attention recently due to better hardware and hence, higher dimensionality of images and data sets acquired in applications. In the last couple of years non-smooth minimization problems such as total variation minimization became increasingly important for the solution of these tasks. While being favorable due to the improved enhancement of images compared to smooth imaging approaches, non-smooth minimization problems typically scale badly with the dimension of the data. Hence, for large imaging problems solved by total variation minimization domain decomposition algorithms have been proposed, aiming to split one large problem into N>1 smaller problems which can be solved on parallel CPUs. The N subproblems constitute constrained minimization problems, where the constraint enforces the support of the minimizer to be the respective subdomain. In this paper we discuss a fast computational algorithm to solve domain decomposition for total variation minimization. In particular, we accelerate the computation of the subproblems by nested Bregman iterations. We propose a Bregmanized Operator Splitting–Split Bregman (BOS-SB) algorithm, which enforces the restriction onto the respective subdomain by a Bregman iteration that is subsequently solved by a Split Bregman strategy. The computational performance of this new approach is discussed for its application to image inpainting and image deblurring. It turns out that the proposed new solution technique is up to three times faster than the iterative algorithm currently used in domain decomposition methods for total variation minimization.     

A new Lagrangian relaxation algorithm for hybrid flowshop scheduling to minimize total weighted completion time

We investigate the problem of scheduling n jobs in s-stage hybrid flowshops with parallel identical machines at each stage. The objective is to find a schedule that minimizes the sum of weighted completion times of the jobs. This problem has been proven to be NP-hard. In this paper, an integer programming formulation is constructed for the problem. A new Lagrangian relaxation algorithm is presented in which precedence constraints are relaxed to the objective function by introducing Lagrangian multipliers, unlike the commonly used method of relaxing capacity constraints. In this way the relaxed problem can be decomposed into machine type subproblems, each of which corresponds to a specific stage. A dynamic programming algorithm is designed for solving parallel identical machine subproblems where jobs may have negative weights. The multipliers are then iteratively updated along a subgradient direction. The new algorithm is computationally compared with the commonly used Lagrangian relaxation algorithms which, after capacity constraints are relaxed, decompose the relaxed problem into job level subproblems and solve the subproblems by using the regular and speed-up dynamic programming algorithms, respectively. Numerical results show that the new Lagrangian relaxation method produces better schedules in much shorter computation time, especially for large-scale problems.     

An augmented Lagrangian decomposition method for quasi-separable problems in MDO

Several decomposition methods have been proposed for the distributed optimal design of quasi-separable problems encountered in Multidisciplinary Design Optimization (MDO). Some of these methods are known to have numerical convergence difficulties that can be explained theoretically. We propose a new decomposition algorithm for quasi-separable MDO problems. In particular, we propose a decomposed problem formulation based on the augmented Lagrangian penalty function and the block coordinate descent algorithm. The proposed solution algorithm consists of inner and outer loops. In the outer loop, the augmented Lagrangian penalty parameters are updated. In the inner loop, our method alternates between solving an optimization master problem and solving disciplinary optimization subproblems. The coordinating master problem can be solved analytically; the disciplinary subproblems can be solved using commonly available gradient-based optimization algorithms. The augmented Lagrangian decomposition method is derived such that existing proofs can be used to show convergence of the decomposition algorithm to Karush–Kuhn–Tucker points of the original problem under mild assumptions. We investigate the numerical performance of the proposed method on two example problems.     

Decomposing the user-preference in multiobjective optimization

Preference information (such as the reference point) of the decision maker (DM) is often used in multiobjective optimization; however, the location of the specified reference point has a detrimental effect on the performance of multiobjective evolutionary algorithms (MOEAs). Inspired by multiobjective evolutionary algorithm-based decomposition (MOEA/D), this paper proposes an MOEA to decompose the preference information of the reference point specified by the DM into a number of scalar optimization subproblems and deals with them simultaneously (called MOEA/D-PRE). This paper presents an approach of iterative weight to map the desired region of the DM, which makes the algorithm easily obtain the desired region. Experimental results have demonstrated that the proposed algorithm outperforms two popular preference-based approaches, g-dominance and r-dominance, on continuous multiobjective optimization problems (MOPs), especially on many-objective optimization problems. Moreover, this study develops distinct models to satisfy different needs of the DM, thus providing a new way to deal with preference-based multiobjective optimization. Additionally, in terms of the shortcoming of MOEA/D-PRE, an improved MOEA/D-PRE that dynamically adjusts the size of the preferred region is proposed and has better performance on some problems.     

An improved hybrid particle swarm optimization algorithm for fuzzy p-hub center problem

The p-hub center problem is useful for the delivery of perishable and time-sensitive system such as express mail service and emergency service. In this paper, we propose a new fuzzy p-hub center problem, in which the travel times are uncertain and characterized by normal fuzzy vectors. The objective of our model is to maximize the credibility of fuzzy travel times not exceeding a predetermined acceptable efficient time point along all paths on a network. Since the proposed hub location problem is too complex to apply conventional optimization algorithms, we adapt an approximation approach (AA) to discretize fuzzy travel times and reformulate the original problem as a mixed-integer programming problem subject to logic constraints. After that, we take advantage of the structural characteristics to develop a parametric decomposition method to divide the approximate p-hub center problem into two mixed-integer programming subproblems. Finally, we design an improved hybrid particle swarm optimization (PSO) algorithm by combining PSO with genetic operators and local search (LS) to update and improve particles for the subproblems. We also evaluate the improved hybrid PSO algorithm against other two solution methods, genetic algorithm (GA) and PSO without LS components. Using a simulated data set of 10 nodes, the computational results show that the improved hybrid PSO algorithm achieves the better performance than GA and PSO without LS in terms of runtime and solution quality.     

面向复杂超多目标优化问题的自适应增强学习进化算法

在解决超多目标优化问题中,基于分解的进化算法是一种较为有效的方法.传统的分解方法依赖于一组均匀分布的参考向量,它借助聚合函数将多目标优化问题分解为一组单目标子问题,然后对这些子问题同时进行优化.然而,由于参考向量分布和Pareto前沿形状的不一致性,导致这些预定义的参考向量在解决复杂超多目标优化问题时表现较差.对此,提出一种基于自适应增强学习的超多目标进化算法(MaOEA-ABL).该算法主要分为两个阶段:第1阶段,采用一种自适应增强学习算法对预定义的参考向量进行调整,在学习过程中删除无用向量,增加新的向量;第2阶段,设计一种对Pareto形状无偏好的分解方法.为验证所提出算法的有效性,选取具有复杂Pareto前沿的MaF系列测试函数进行仿真研究,结果显示, MaOEA-ABL算法的IGD (inverted generational distance)均值在67%的测试函数上超过了对比算法,从而表明该算法在复杂超多目标优化问题中表现良好.     

Segmenting a surface mesh into pants using Morse theory

A pair of pants is a genus zero orientable surface with three boundary components. A pants decomposition of a surface is a finite collection of unordered pairwise disjoint simple closed curves embedded in the surface that decompose the surface into pants. In this paper, we present two Morse theory based algorithms for pants decomposition of a surface mesh. Both algorithms operates on a choice of an appropriate Morse function on the surface. The first algorithm uses this Morse function to identify handles that are glued systematically to obtain a pants decomposition. The second algorithm uses the Reeb graph of the Morse function to obtain a pants decomposition. Both algorithms work for surfaces with or without boundaries. Our preliminary implementation of the two algorithms shows that both algorithms run in much less time than an existing state-of-the-art method, and the Reeb graph based algorithm achieves the best time efficiency. Finally, we demonstrate the robustness of our algorithms against noise.     

Limitations of Incremental Dynamic Programming

We consider so-called “incremental” dynamic programming algorithms, and are interested in the number of subproblems produced by them. The classical dynamic programming algorithm for the Knapsack problem is incremental, produces nK subproblems and nK ² relations (wires) between the subproblems, where n is the number of items, and K is the knapsack capacity. We show that any incremental algorithm for this problem must produce about nK subproblems, and that about nKlogK wires (relations between subproblems) are necessary. This holds even for the Subset-Sum problem. We also give upper and lower bounds on the number of subproblems needed to approximate the Knapsack problem. Finally, we show that the Maximum Bipartite Matching problem and the Traveling Salesman problem require exponential number of subproblems. The goal of this paper is to leverage ideas and results of boolean circuit complexity for proving lower bounds on dynamic programming.     

Splitting augmented Lagrangian method for optimization problems with a cardinality constraint and semicontinuous variables

We propose a new splitting augmented Lagrangian method (SALM) for solving a class of optimization problems with both cardinality constraint and semicontinuous variables constraint. The proposed approach, inspired by the penalty decomposition method in [Z.S. Lu and Y. Zhang, Sparse approximation via penalty decomposition methods, SIAM J. Optim. 23(4) (2013), pp. 2448–2478], splits the problem into two subproblems using auxiliary variables. SALM solves two subproblems alternatively. Furthermore, we prove the convergence of SALM, under certain assumptions. Finally, SALM is implemented on the portfolio selection problem and the compressed sensing problem, respectively. Numerical results show that SALM outperforms the well-known tailored approach in CPLEX 12.6 and the penalty decomposition method, respectively.     

On Two Techniques of Combining Branching and Treewidth

Branch & Reduce and dynamic programming on graphs of bounded treewidth are among the most common and powerful techniques used in the design of moderately exponential time exact algorithms for NP hard problems. In this paper we discuss the efficiency of simple algorithms based on combinations of these techniques. The idea behind these algorithms is very natural: If a parameter like the treewidth of a graph is small, algorithms based on dynamic programming perform well. On the other side, if the treewidth is large, then there must be vertices of high degree in the graph, which is good for branching algorithms. We give several examples of possible combinations of branching and programming which provide the fastest known algorithms for a number of NP hard problems. All our algorithms require non-trivial balancing of these two techniques. In the first approach the algorithm either performs fast branching, or if there is an obstacle for fast branching, this obstacle is used for the construction of a path decomposition of small width for the original graph. Using this approach we give the fastest known algorithms for Minimum Maximal Matching and for counting all 3-colorings of a graph. In the second approach the branching occurs until the algorithm reaches a subproblem with a small number of edges (and here the right choice of the size of subproblems is crucial) and then dynamic programming is applied on these subproblems of small width. We exemplify this approach by giving the fastest known algorithm to count all minimum weighted dominating sets of a graph. We also discuss how similar techniques can be used to design faster parameterized algorithms. A preliminary version of this paper appeared as Branching and Treewidth Based Exact Algorithms in the Proceedings of the 17th International Symposium on Algorithms and Computation (ISAAC 2006) [15]. Additional support by the Research Council of Norway.     

Parallel growing and training of neural networks using outputparallelism

In order to find an appropriate architecture for a large-scale real-world application automatically and efficiently, a natural method is to divide the original problem into a set of subproblems. In this paper, we propose a simple neural-network task decomposition method based on output parallelism. By using this method, a problem can be divided flexibly into several subproblems as chosen, each of which is composed of the whole input vector and a fraction of the output vector. Each module (for one subproblem) is responsible for producing a fraction of the output vector of the original problem. The hidden structure for the original problem's output units are decoupled. These modules can be grown and trained in parallel on parallel processing elements. Incorporated with a constructive learning algorithm, our method does not require excessive computation and any prior knowledge concerning decomposition. The feasibility of output parallelism is analyzed and proved. Some benchmarks are implemented to test the validity of this method. Their results show that this method can reduce computational time, increase learning speed and improve generalization accuracy for both classification and regression problems.     

A decomposition approach for the p-median problem on disconnected graphs

The p-median problem seeks for the location of p facilities on the vertices (customers) of a graph to minimize the sum of transportation costs for satisfying the demands of the customers from the facilities. In many real applications of the p-median problem the underlying graph is disconnected. That is the case of p-median problem defined over split administrative regions or regions geographically apart (e.g. archipelagos), and the case of problems coming from industry such as the optimal diversity management problem. In such cases the problem can be decomposed into smaller p-median problems which are solved in each component k for different feasible values of p_k, and the global solution is obtained by finding the best combination of p_k medians. This approach has the advantage that it permits to solve larger instances since only the sizes of the connected components are important and not the size of the whole graph. However, since the optimal number of facilities to select from each component is not known, it is necessary to solve p-median problems for every feasible number of facilities on each component. In this paper we give a decomposition algorithm that uses a procedure to reduce the number of subproblems to solve. Computational tests on real instances of the optimal diversity management problem and on simulated instances are reported showing that the reduction of subproblems is significant, and that optimal solutions were found within reasonable time.     

微电子生产过程调度问题基于指标快速预报的分解算法

微电子生产过程调度问题具有规模大和约束复杂等特点,如菜单、Setup时间和组批约束等,其优化调度具有一定难度.针对以最小化平均流经时间为调度目标的较大规模微电子生产过程调度问题,提出一种基于指标快速预报的分解方法(DM-IFP).首先,通过松弛不可中断约束,设计一种代理方法,即基于机器负载的操作完工时间快速预测方法(CTP-ML);其次,设计基于CTP-ML的问题分解方法,将原问题迭代分解为多个连续交迭的子问题;然后,提出一种基于双信息素的蚁群算法(ACO-D)用于求解分解后的子问题,其全局调度目标采用CTP-ML获取,有效保证了全局优化性能;最后,针对一些不同规模的仿真数据,将所提出方法与一些代表性的算法进行详尽的数值对比,计算结果表明所提出方法在所获解的质量和收敛性上均有改善.     

On convex transformability and the solution of nonconvex problems via the dual of Falk

In structural optimization, most successful sequential approximate optimization (SAO) algorithms solve a sequence of strictly convex subproblems using the dual of Falk. Previously, we have shown that, under certain conditions, a nonconvex nonlinear (sub)problem may also be solved using the Falk dual. In particular, we have demonstrated this for two nonconvex examples of approximate subproblems that arise in popular and important structural optimization problems. The first is used in the SAO solution of the weight minimization problem, while the topology optimization problem that results from volumetric penalization gives rise to the other. In both cases, the nonconvex subproblems arise naturally in the consideration of the physical problems, so it seems counter productive to discard them in favor of using standard, but less well-suited, strictly convex approximations. Though we have not required that strictly convex transformations exist for these problems in order that they may be solved via a dual approach, we have noted that both of these examples can indeed be transformed into strictly convex forms. In this paper we present both the nonconvex weight minimization problem and the nonconvex topology optimization problem with volumetric penalization as instructive numerical examples to help motivate the use of nonconvex approximations as subproblems in SAO. We then explore the link between convex transformability and the salient criteria which make nonconvex problems amenable to solution via the Falk dual, and we assess the effect of the transformation on the dual problem. However, we consider only a restricted class of problems, namely separable problems that are at least C ¹ continuous, and a restricted class of transformations: those in which the functions that represent the mapping are each continuous, monotonic and univariate.     

Inclusion/Exclusion Meets Measure and Conquer

Inclusion/exclusion and measure and conquer are two central techniques from the field of exact exponential-time algorithms that recently received a lot of attention. In this paper, we show that both techniques can be used in a single algorithm. This is done by looking at the principle of inclusion/exclusion as a branching rule. This inclusion/exclusion-based branching rule can be combined in a branch-and-reduce algorithm with traditional branching rules and reduction rules. The resulting algorithms can be analysed using measure and conquer allowing us to obtain good upper bounds on their running times. In this way, we obtain the currently fastest exact exponential-time algorithms for a number of domination problems in graphs. Among these are faster polynomial-space and exponential-space algorithms for #Dominating Set and Minimum Weight Dominating Set (for the case where the set of possible weight sums is polynomially bounded), and a faster polynomial-space algorithm for Domatic Number. This approach is also extended in this paper to the setting where not all requirements in a problem need to be satisfied. This results in faster polynomial-space and exponential-space algorithms for Partial Dominating Set, and faster polynomial-space and exponential-space algorithms for the well-studied parameterised problem k-Set Splitting and its generalisation k-Not-All-Equal Satisfiability.     

Domain Decomposition Methods for Nonlocal Total Variation Image Restoration

Nonlocal total variation (TV) regularization (Gilboa and Osher in Multiscale Model Simulat 7(3): 1005–1028, 2008; Zhou and Schölkopf in Pattern recognition, proceedings of the 27th DAGM symposium. Springer, Berlin, pp 361–368, 2005) has been widely used for the natural image processing, since it is able to preserve repetitive textures and details of images. However, its applications have been limited in practice, due to the high computational cost for large scale problems. In this paper, we apply domain decomposition methods (DDMs) (Xu et al. in Inverse Probl Imag 4(3):523–545, 2010) to the nonlocal TV image restoration. By DDMs, the original problem is decomposed into much smaller subproblems defined on subdomains. Each subproblem can be efficiently solved by the split Bregman algorithm and Bregmanized operator splitting algorithm in Zhang et al. (SIAM J Imaging Sci 3(3):253–276, 2010). Furthermore, by using coloring technique on the domain decomposition, all subproblems defined on subdomains with same colors can be computed in parallel. Our numerical examples demonstrate that the proposed methods can efficiently solve the nonlocal TV based image restoration problems, such as denoising, deblurring and inpainting. It can be computed in parallel with a considerable speedup ratio and speedup efficiency.     

Large-scale convex optimal control problems: time decomposition,incentive coordination, and parallel algorithm

A parallel algorithm based on time decomposition and incentive coordination is developed for long-horizon optimal control problems. This is done by first decomposing the original problem into subproblems with shorter time horizon, and then using the incentive coordination scheme to coordinate the interaction of subproblems. For strictly convex problems it is proved that the decomposed problem with linear incentive coordination is equivalent to the original problem, in the sense that each optimal solution of the decomposed problem produces one global optimal solution of the original problem and vice versa. In other words, linear incentive terms are sufficient in this case and impose no additional computation burden on the subproblems. The high-level parameter optimization problem is shown to be nonconvex, despite the uniqueness of the optimal solution and the convexity of the original problem. Nevertheless, the high-level problem has no local minimum, even though it is nonconvex. A parallel algorithm based on a prediction method is developed, and a numerical example is used to demonstrate the feasibility of the approach     

Planning Robot Navigation among Movable Obstacles (NAMO) through a Recursive Approach

This paper addresses the NP-complete problem of Navigation Among Movable Obstacles (NAMO) in which a robot is required to find a collision-free path toward a goal through manipulating and transferring some movable objects on its way. The robot’s main goal is to optimize a performance criterion such as runtime, length of transit or transfer paths, number of manipulated obstacles, total number of displacements of all objects, etc. We have designed a recursive algorithm capable of solving various NAMO problems, ranging from linear monotone to nonlinear non-monotone, and with convex or concave polygonal obstacles. Through the adopted approach, the original problem is decomposed into recursively-solved subproblems, in each of which only one movable object is manipulated. In each call of the algorithm, first a Visibility Graph determines a path from the robot’s current configuration to an intermediate goal configuration, and then a tentative final configuration for the last object intercepting the path is calculated using the Penetration Depth concept. It is assumed that the objects can be pulled or pushed, but not rotated, in a continuous space, and under such assumptions the method is complete and locally optimal for convex objects, with a worst-case time complexity of O(n⁴3^m) in which m is the number of movable objects and n is the number of all vertices on them. Several computational experiments showed that compared to the existing methods in the literature, the proposed recursive method achieved equal or smaller number of transferred obstacles or the total number of displacements of all objects in majority of the test problems.     

A problem reduction based approach to discrete optimization algorithm design

The paper presents a novel approach to formal algorithm design for a typical class of discrete optimization problems. Using a concise set of program calculation rules, our approach reduces a problem into subproblems with less complexity based on function decompositions, constructs the problem reduction graph that describes the recurrence relations between the problem and subproblems, from which a provably correct algorithm can be mechanically derived. Our approach covers a large variety of algorithms and bridges the relationship between conventional methods for designing efficient algorithms (including dynamic programming and greedy) and some effective methods for coping with intractability (including approximation and parameterization).