Spatial branch-and-bound algorithm for MIQCPs featuring multiparametric disaggregation

Spatial branch-and-bound (B&B) is widely used for the global optimization of non-convex problems. It basically works by iteratively reducing the domain of the variables so that tighter relaxations can be achieved that ultimately converge to the global optimal solution. Recent developments for bilinear problems have brought us piecewise relaxation techniques that can prove optimality for a sufficiently large number of partitions and hence avoid spatial B&B altogether. Of these, normalized multiparametric disaggregation (NMDT) exhibits a good performance due to the logarithmic increase in the number of binary variables with the number of partitions. We now propose to integrate NMDT with spatial B&B for solving mixed-integer quadratically constrained minimization problems. Optimality-based bound tightening is also part of the algorithm so as to compute tight lower bounds in every step of the search and reduce the number of nodes to explore. Through the solution of a set of benchmark problems from the literature, it is shown that the new global optimization algorithm can potentially lead to orders of magnitude reduction in optimality gap when compared to commercial solvers BARON and GloMIQO.     

LP-based bounds for the container and multi-container loading problem

New relaxations are developed in this paper for problems of optimal packing of small (rectangular-shaped) pieces within one or several larger containers. Based on these relaxations tighter bounds for the Container Loading Problem (CLP) and the Multi-Container Loading Problem (MCLP) are obtained.
The new relaxations for the CLP and MCLP lead to linear programming problems. A corresponding solution approach is discussed which is based on a column generation technique. Results of computational tests are also given.     

One-dimensional relaxations and LP bounds for orthogonal packing

We consider the feasibility problem in d -dimensional orthogonal packing ( d 2), called the Orthogonal Packing Problem (OPP): given a set of d -dimensional rectangular items, decide whether all of them can be orthogonally packed in the given rectangular container without rotation. We review two kinds of one-dimensional (1D) relaxations of OPP. The first kind is non-preemptive cumulative-resource scheduling, equivalently 1D contiguous stock cutting. The second kind is simple (preemptive) 1D stock cutting. In three and more dimensions we distinguish the so-called bar and slice preemptive relaxations of OPP. We review some models of these problems and compare the strength of their LP relaxations with regard to a certain OPP instance, theoretically and numerically. Both the theory and computational results in 2D and 3D show the advantage of the bar relaxation. We also compare the LP bounds with the commonly used volume bounds from dual-feasible functions. Moreover, we test the so-called probing (temporary fixing) of intersection variables of OPP with the aim to strengthen the relaxations.     

Semi-infinite relaxation of joint chance constraints in chance-constrained programming Part 1. Zero-order stochastic decision rules†

A new class of semi-infinite deterministic (‘determinizations’) dominants and relaxations of joint chance-constraints in chance-constrained programming is developed and specialized to zero-order stochastic decision rule situations. Tight constraint relaxations are obtained where only the partial information of means and variances is known. The tight non-linear semi-infinite relaxations are related to an accessible finite subsystem. When the chance constraints involve linear inequalities, for a large class, the non-linear tight system is proved equivalent to a linear program. Its solutions for the Prekopa-Szantai reservoir construction examples agree well with theirs     

Algorithmic Aspects of Using Small Instance Relaxations in Parallel Branch-and-Cut

Essential for the success of branch-and-cut algorithms for solving combinatorial optimization problems are the availability of reasonable tight relaxations and effective routines for solving the associated separation problems. In this paper we introduce the concept of small instance relaxations which can be particularly useful for problems with symmetric structure. Small instance relaxations are based on the facets of polytopes associated with small instances of the combinatorial optimization problem to be solved and can be generated automatically by facet enumeration. For a certain class of symmetric problems, we describe a general approach to the separation problem. Algorithmic aspects of using small instance relaxations effectively (parallel separation, facet selection, cutting plane selection) are discussed in detail. Extensive computational results are presented for the linear ordering problem and a certain betweenness problem. Received March 31, 1998; revised November 9, 1998.     

认知OFDM中的抗干扰分段编码设计

认知正交频分复用(OFDM)中主用户突发干扰会造成认知用户数据包丢失。为此,提出一种基于认知OFDM的抗干扰分段编码方案。分段编码基于低冗余度的优化设计方法进行构造。该抗干扰分段编码方案可通过对偶校验分组恢复丢失的数据包,在认知用户通信过程中避免主用户突发干扰。实验结果表明,在低干扰率情况下,该方案比无码率编码方案具有更低的帧差错率和更高的吞吐量性能。     

Input-to-state stability and interconnections of discontinuous dynamical systems

In this paper we will extend the input-to-state stability (ISS) framework to continuous-time discontinuous dynamical systems (DDS) adopting piecewise smooth ISS Lyapunov functions. The main motivation for investigating piecewise smooth ISS Lyapunov functions is the success of piecewise smooth Lyapunov functions in the stability analysis of hybrid systems. This paper proposes an extension of the well-known Filippov’s solution concept, that is appropriate for ‘open’ systems so as to allow interconnections of DDS. It is proven that the existence of a piecewise smooth ISS Lyapunov function for a DDS implies ISS. In addition, a (small gain) ISS interconnection theorem is derived for two DDS that both admit a piecewise smooth ISS Lyapunov function. This result is constructive in the sense that an explicit ISS Lyapunov function for the interconnected system is given. It is shown how these results can be applied to construct piecewise quadratic ISS Lyapunov functions for piecewise linear systems (including sliding motions) via linear matrix inequalities.     

Piecewise nonlinear goal-directed CPPI strategy

Traditional portfolio insurance (PI) strategy, such as constant proportion portfolio insurance (CPPI), only considers the floor constraint but not the goal aspect. This paper proposes a goal-directed (GD) strategy to express an investor’s goal-directed trading behavior and combines this floor-less GD strategy with the goal-less CPPI strategy to form a piecewise linear goal-directed CPPI (GDCPPI) strategy. The piecewise linear GDCPPI strategy shows that there is a wealth position M at the intersection of the GD and CPPI strategies. This M position guides investors to apply the CPPI strategy or the GD strategy depending on whether current wealth is less than or greater than M, respectively. In addition, we extend the piecewise linear GDCPPI strategy to a piecewise nonlinear GDCPPI strategy. This paper applies genetic algorithm (GA) technique to find better piecewise linear GDCPPI strategy parameters than those under the Brownian motion assumption. This paper also applies forest genetic programming (GP) technique to generate the piecewise nonlinear GDCPPI strategy. The statistical tests show that the GP strategy outperforms the GA strategy which in turn outperforms the Brownian strategy.     

Stability analysis of piecewise discrete-time linear systems

Presents a stability analysis method for piecewise discrete-time linear systems based on a piecewise smooth Lyapunov function. It is shown that the stability of the system can be established if a piecewise Lyapunov function can be constructed and, moreover, the function can be obtained by solving a set of linear matrix inequalities (LMIs) that is numerically feasible with commercially available software.     

分段光滑的半监督支持向量分类机

为了解决半监督支持向量分类优化模型中的非凸非光滑问题,基于分段逼近的思想提出了一个分段函数,并以此逼近非凸非光滑的目标函数。给出的分段函数可以根据不同的精度要求选择不同的逼近参数,同时构造出基于上述分段函数的光滑半监督支持向量机模型。采用了LDS(Low Density Separation)算法求解模型,分析了其对对称铰链损失函数的逼进精度。理论分析和数值实验结果都证明分段光滑的半监督支持向量机的分类性能和效率优于以往提出的光滑模型。     

Formulations for the nonbifurcated hop-constrained multicommodity capacitated fixed-charge network design problem

This paper addresses the multicommodity capacitated fixed-charge network design problem with nonbifurcated flows and hop constraints. We present and compare mathematical programming formulations for this problem and we study different relaxations: Lagrangean relaxations, linear programming relaxations, and partial relaxations of the integrality constraints. In particular, we show that the Lagrangean bound obtained by relaxing the flow conservation equations is tighter than the linear programming relaxation bound. We present computational results on a large set of randomly generated instances.     

散乱数据拟合的自适应分片逆尺度空间算法

散乱数据拟合(逼近)是在信号处理、计算机图形学等领域中被广泛研究的问题, 近些年,利用优化方法获得散乱数据的稀疏表示逼近解也成为了优化和曲面重构交叉领域的热 点。基于由B 样条生成的PSI 空间中的散乱点曲面拟合问题和分片稀疏的联系,将分片稀疏性 引入到Bregman 逆尺度空间算法(ISS)中,提出一种自适应的分片逆尺度空间(aP_ISS)算法,处 理散乱数据的曲面拟合问题。通过对逆尺度空间系统分片符号一致性分析,得到了自适应分片 逆尺度空间系统的性能保证定理和避免了aP_ISS 参数的选取。应用到散乱点曲面重构问题上 的数值实验结果表明,该算法不仅可以有效拟合曲面,还能够较好保护分片稀疏性。     

Convergent relaxations of polynomial matrix inequalities and static output feedback

Using a moment interpretation of recent results on sum-of-squares decompositions of nonnegative polynomial matrices, we propose a hierarchy of convex linear matrix inequality (LMI) relaxations to solve nonconvex polynomial matrix inequality (PMI) optimization problems, including bilinear matrix inequality (BMI) problems. This hierarchy of LMI relaxations generates a monotone sequence of lower bounds that converges to the global optimum. Results from the theory of moments are used to detect whether the global optimum is reached at a given LMI relaxation, and if so, to extract global minimizers that satisfy the PMI. The approach is successfully applied to PMIs arising from static output feedback design problems.     

Stability analysis and H infinity controller design of fuzzy large-scale systems based on piecewise Lyapunov functions.

This paper presents a novel approach to stability analysis of a fuzzy large-scale system in which the system is composed of a number of Takagi-Sugeno (T-S) fuzzy subsystems with interconnections. The stability analysis is based on Lyapunov functions that are continuous and piecewise quadratic. It is shown that the stability of the fuzzy large-scale systems can be established if a piecewise Lyapunov function can be constructed, and, moreover, the function can be obtained by solving a set of linear matrix inequalities (LMIs) that are numerically feasible. It is also demonstrated via a numerical example that the stability result based on the piecewise quadratic Lyapunov functions is less conservative than that based on the common quadratic Lyapunov functions. The H infinity controllers can also be designed by solving a set of LMIs based on these powerful piecewise quadratic Lyapunov functions.     

Equivalence of sum of squares convex relaxations for quadratic distance problems

This paper deals with convex relaxations for quadratic distance problems, a class of optimization problems relevant to several important topics in the analysis and synthesis of robust control systems. Some classes of convex relaxations are investigated using the sum of squares paradigm for the representation of positive polynomials. The main contribution is to show that two different relaxations, based respectively on the Positivstellensatz and on properties of homogeneous polynomial forms, are equivalent. Relationships among the considered relaxations are discussed and numerical comparisons are presented, highlighting their degree of conservatism. Copyright © 2012 John Wiley & Sons, Ltd.     

Using relaxations to improve search in distributed constraint optimisation

Densely connected distributed constraint optimisation problems (DisCOP) can be difficult to solve optimally, but finding good lower bounds on constraint costs can help to speed up search. We show how good lower bounds can be found by solving relaxed problems obtained by removing inter-agent constraints. We present modifications to the Adopt DisCOP algorithm that allow an arbitrary number of relaxations to be performed prior to solving the original problem. We identify useful relaxations based on the solving structure used by Adopt, and demonstrate that when these relaxations are incorporated as part of the search it can lead to significant performance improvements. In particular, where agents have significant local constraint costs, we achieve over an order of magnitude reduction in messages exchanged between agents. Finally, we identify cases where such relaxation techniques produce less consistent benefits.     

Stability analysis and H/sub /spl infin// controller design of fuzzy large-scale systems based on piecewise Lyapunov functions

This paper presents a novel approach to stability analysis of a fuzzy large-scale system in which the system is composed of a number of Takagi-Sugeno (T-S) fuzzy subsystems with interconnections. The stability analysis is based on Lyapunov functions that are continuous and piecewise quadratic. It is shown that the stability of the fuzzy large-scale systems can be established if a piecewise Lyapunov function can be constructed, and, moreover, the function can be obtained by solving a set of linear matrix inequalities (LMIs) that are numerically feasible. It is also demonstrated via a numerical example that the stability result based on the piecewise quadratic Lyapunov functions is less conservative than that based on the common quadratic Lyapunov functions. The H/sub /spl infin// controllers can also be designed by solving a set of LMIs based on these powerful piecewise quadratic Lyapunov functions.     

分片支撑矢量机

文中借鉴了分段线性识别的基本思想,提出了分片支撑矢量机模型.该模型首先将特征空间剖分成若干子空间,在每个子空间中基于支撑矢量机构造一个最优分类面,然后,将各个分类面链接起来构成一个分片最优分类面以逼近理论上的最优分类超曲面.同时,文中还从理论上分析探讨了其推广能力的界,为分片支撑矢量机模型提供了坚实的基础.最后,经典双螺旋线数据实验结果表明,相对于传统支撑矢量机,分片支撑矢量机的计算速度、分类能力以及推广能力均有了明显提高.     

单一与非单一鲁棒控制器的设计

针对普通二次Lyapunov函数方法判定T-S模糊系统稳定性存在的保守性和难度,利用T-S系统的模糊前提规则和隶属度函数分别构造分段二次Lyapunov函数和模糊Lyapunov函数,且通过将模糊Lyapunov函数引入到分段二次Lyapunov函数所得到的分段模糊区域中定义了分段模糊Lyapunov函数;研究一类T-S模糊系统的鲁棒控制问题,以线性矩阵不等式的形式给出了单一与非单一鲁棒控制器的参数化设计方法。仿真结果表明,非线性系统在非单一鲁棒控制器作用下能够获得比单一鲁棒控制器更好的控制性能。