Solving nonconvex nonlinear programs with reverse convex constraints by sequential linear programming

The sequential linear programming (SLP) method for solving nonlinear problems was introduced in the 1960s. Many papers that attempted to use SLP reported poor performance and convergence issues. We found that nonlinear programs with reverse convex constraints, which are the most difficult nonlinear programs with many local optima, are solved (heuristically) very well by SLP. We proved that for this type of problems, the solutions to the sequence of the linear programming problems converge to a local optimum. Since the final solution depends on the starting solution, we propose to apply SLP in a multistart approach starting from randomly generated solutions. This multistart SLP is very easy to implement. We recommend that the research community reconsiders the application of SLP for this type of problems.     

Solving geometric constraints by homotopy

Numerous methods have been proposed in order to solve geometric constraints, all of them having their own advantages and drawbacks. In this article, we propose an enhancement to the classical numerical methods, which, up to now, are the only ones that apply to the general case     

Properties of tree convex constraints

It is known that a tree convex network is globally consistent if it is path consistent. However, if a tree convex network is not path consistent, enforcing path consistency on it may not make it globally consistent. In this paper, we investigate the properties of some tree convex constraints under intersection and composition. As a result, we identify a sub-class of tree convex networks that are locally chain convex and strictly union closed. This class of problems can be made globally consistent by arc and path consistency and thus is tractable. Interestingly, we also find that some scene labeling problems can be modeled by tree convex constraints in a natural and meaningful way.     

Solving problems with unilateral constraints by DAE methods

We study trajectories of mechanical systems with unilateral constraints under the additional assumption that always a given number of constraints is active. A reformulation as a problem with bilateral conditions yields a drastic reduction in the number of constraints, but in general, we are faced with regularity problems. We illustrate our approach in the special case of a dynamical rigid body contact problem. In particular, we present a regularization technique which leads to the definition of generalized solutions and a quite effective numerical method on the basis of algorithms for differential–algebraic systems. The results are applied to a wheel–rail contact problem of actual interest to railway engineers.     

Solving the Multidimensional Multiple-choice Knapsack Problem by constructing convex hulls

This paper presents a heuristic to solve the Multidimensional Multiple-choice Knapsack Problem (MMKP), a variant of the classical 0–1 Knapsack Problem. We apply a transformation technique to map the multidimensional resource consumption to single dimension. Convex hulls are constructed to reduce the search space to find the near-optimal solution of the MMKP. We present the computational complexity of solving the MMKP using this approach. A comparative analysis of different heuristics for solving the MMKP has been presented based on the experimental results.     

Solving hierarchies of finite-domain constraints

Abstract. In the past we presented an algorithm to solve ordered constraint hierarchies based on a non-trivial error function using standard constraint satisfaction techniques. We extended this previous work and herewith present a new method to transform hierarchies of inequalities over the integers based on global comparators into equivalent ordered constraint hierarchies. The correctness of this method is proven. Using the results of our previous work, we present another method transforming the resulting ordered constraint hierarchies into ordinary constraint systems. These systems of algebraic equalities and inequalities over the integer domain are soluble with available finite-domain constraint solvers. Finally, we propose some modifications and simplifications of the considered constraint hierarchies, improving the search for solutions and being useful in practical applications like job-shop scheduling. The modifications proposed are the combination of consecutive hierarchy levels in one level where the constraints are considered with different priorities weights and the simplifications proposed are incomplete search strategies resulting in run-time improvements and also in sub-optimal solutions.     

Solving systems of algebraic equations by a general elimination method

A method of factorisation of a U-resultant into linear factors is given. Using this method we can obtain solutions and their multiplicities of a system of algebraic equations, provided the system of algebraic equations has finitely many solutions. We directly calculate a matrix à which gives all solutions of the system by using a Gröbner basis of the ideal generated by the polynomials of the system of algebraic equations.     

Solving general convex nonlinear optimization problems by an efficient neurodynamic model

In this paper, a neural network model is constructed on the basis of the duality theory, optimization theory, convex analysis theory, Lyapunov stability theory and LaSalle invariance principle to solve general convex nonlinear programming (GCNLP) problems. Based on the Saddle point theorem, the equilibrium point of the proposed neural network is proved to be equivalent to the optimal solution of the GCNLP problem. By employing Lyapunov function approach, it is also shown that the proposed neural network model is stable in the sense of Lyapunov and it is globally convergent to an exact optimal solution of the original problem. The simulation results also show that the proposed neural network is feasible and efficient.     

变约束高效预测控制

为改善离线鲁棒预测控制算法的最优性引入了变终端约束集的思想,即在原离线方法基础上通过在线优化方法获得构建当前反馈矩阵的凸组合系数,然后再产生当前控制量.与原方法相比增加了在线求解的自由度,改善了系统的最优性.在此基础上,又提出了变约束预测控制(VC-MPC)算法,根据状态所处的不同稳定椭圆域确定其对应的控制约束,在取得...     

Solving geometric constraints in a parallel network

The paper describes a network implementation of the SUP-INF method for solving sets of inequalities, giving several advantages over previous implementations. The cost of symbolic manipulation is transferred to compile time allowing an increase in speed at run time due to parallel evaluation. Further, allowing iteration in the network improves the competence of the method when working with nonlinear expressions. The network is used to implement a geometric reasoner for a computer vision program and this is shown to meet the general requirements for such a system.     

Solving polynomial systems using no-root elimination blending schemes

Searching for the roots of (piecewise) polynomial systems of equations is a crucial problem in computer-aided design (CAD), and an efficient solution is in strong demand. Subdivision solvers are frequently used to achieve this goal; however, the subdivision process is expensive, and a vast number of subdivisions is to be expected, especially for higher-dimensional systems. Two blending schemes that efficiently reveal domains that cannot contribute by any root, and therefore significantly reduce the number of subdivisions, are proposed. Using a simple linear blend of functions of the given polynomial system, a function is sought after to be no-root contributing, with all control points of its Bernstein–Bézier representation of the same sign. If such a function exists, the domain is purged away from the subdivision process. The applicability is demonstrated on several CAD benchmark problems, namely surface–surface–surface intersection (SSSI) and surface–curve intersection (SCI) problems, computation of the Hausdorff distance of two planar curves, or some kinematic-inspired tasks.     

Solving highly nonlinear convex separable programs using successive approximation

Owing to its rapid convergence, ease of computer implementation[1], and applicability to a wide class of practical problems[2, 3], separable programming is well established among the more useful non-linear programming techniques[2, 4]. Yet at the same time, its impracticality for highly nonlinear problems, pointed out repeatedly[1, 5, 6], constitutes a severe limitation of this important approach. This emerges even more strongly when one observes the essential failure of the method for some of the very small (2 × 2) problems included in this report. In this context of high nonlinearity, we examine the performance of a convergent (to within a given ε> 0 of the optimal) alternative procedure based on Refs.[7, 8] which obviates the major difficulties effectively by solving a series of non-heuristic, rigorously determined small separable programs as opposed to a single large one in the standard separable programming technique given, e.g., in Refs.[1, 2, 5]. Specifically, this paper, first, in absence of any such study in the literature, demonstrates the extreme degree of vulnerability of standard separable programming to high nonlinearity, then states the algorithm and some of its important characteristics, and shows its effectiveness for computational examples. Problems requiring up to about 10,000 nonzero elements in their specifications and about 45,000 nonzero elements in the intermediate separable programs, resulting from up to 70 original nonlinear variables and 70 nonlinear constraints are included in these examples.     

贝叶斯网最优消元顺序的近似构造算法

变量消元(VE)法是贝叶斯网推理的一个基本方法,然而不同的消元顺序会导致相差悬殊的计算复杂度,寻找最优消元顺序问题是一个NP难问题,因此在实际应用中多采用近似算法求解。通过对贝叶斯网对应的端正图的分析,综合考虑了消元过程中消去的边和增加的边对剩余图的影响,进而提出了一些降低图的复杂度从而控制消元成本的方法,在此基础上提出了一个最优消元顺序的近似构造算法,最后通过随机仿真实验分析比较了算法的性能。实验结果表明,新算法较最小缺边搜索算法有明显的优势。     

Solving linear constraints over real and rational fields

Practical methods are proposed to solve linear constraints over real and rational fields with quantifiers. Secondary problems are observed together with the set of possible solutions in the context of automatic software model verification.     

Solving topological constraints for declarative families of objects

Parametric and feature-based CAD models can be considered to represent families of similar objects. In current modelling systems, however, the semantics of such families are unclear and ambiguous.We present the Declarative Family of Objects Model (DFOM), which enables us to adequately specify and maintain family semantics. In this model, not only geometry, but also topology is specified declaratively, by means of constraints. A family of objects is modelled by a DFOM with multiple realizations. A member of the family is modelled by adding constraints, e.g. to set dimension variables, until a single realization remains. The declarative approach guarantees that the realization of a family member is also a realization of the family.The realization of a family member is found by solving first the geometric constraints, and then the topological constraints. From the geometric solution, a cellular model is constructed. Topological constraints indirectly specify which combinations of cellular model entities are allowed in the realization. The system of topological constraints is mapped to a Boolean constraint satisfaction problem. The realization is found by solving this problem using a SAT solver.     

Solving two-machine assembly scheduling problems with inventory constraints

This paper considers a scheduling problem with component availability constraints in a supply chain consisting of two manufacturing facilities and a merge-in-transit facility. Three mixed-integer programming (MIP) models and a constraint programming (CP) model are compared in an extensive numerical study. Results show that when there are no components shared among the two manufacturers, the MIP model based on time-index variables is the best for proving optimality for problems with short processing times whereas the CP model tends to perform better than the others for problems with a large range of processing times. When shared components are added, the performance of all models deteriorates, with the time-indexed MIP providing the best results. By explicitly modelling the dependence of scheduling decisions on the availability of component parts, we contribute to the literature on the integration of inventory and scheduling decisions, which is necessary for solving realistic supply chain problems.     

基于运动平滑约束项的快速误匹配剔除算法

针对图像拼接时用随机抽样一致性（RANSAC）算法迭代计算过程中计算量大、匹配正确率低的问题,提出了一种基于运动平滑约束项的误匹配剔除算法。首先采用快速旋转不变特征（ORB）算法提取特征点,基于汉明距离实现特征点初匹配;其次,基于运动平滑约束项统计邻域支持估计量实现误匹配粗剔除;然后,进一步采用空间几何约束关系实现误匹配精剔除;最后,利用分组排序采样求解模型参数,采用加权平均实现图像融合。实验结果表明,该算法的误匹配剔除率相比缩小抽样点总量算法提升了75.6%,相比自适应阈值算法提升了24%,此方法能有效剔除误匹配,实现图像精确拼接。     

Parameter constraints in generalized linear latent variable models

Parameter constraints in generalized linear latent variable models are discussed. Both linear equality and inequality constraints are considered. Maximum likelihood estimators for the parameters of the constrained model and corrected standard errors are derived. A significant reduction in the dimension of the optimization problem is achieved with the proposed methodology for fitting models subject to linear equality constraints.