Separating the vertices of N-cubes by hyperplanes and itsapplication to artificial neural networks

A new sufficient condition that a region be classifiable by a two-layer feedforward network using threshold activation functions is found. Briefly, it is either a convex polytope, or that minus the removal of convex polytope from its interior, or. . .recursively. The author refers to these sets as convex recursive deletion regions. The proof of implementability exploits the equivalence of this problem with that of characterizing two set partitions of the vertices of a hypercube which are separable by a hyperplane, for which a new result is obtained.     

Delaunay partitions in Rn applied to non-convex programs and vertex/facet enumeration problems

Using a pair of theorems linking Delaunay partitions and linear programming, we develop a method to generate all simplices in a Delaunay partition of a set of points, and suggest an application to a piecewise linear non-convex optimization problem. The same method is shown to enumerate all facets of a polytope given as the convex hull of a finite set of points. The dual problem of enumerating all vertices of a polytope P defined as the intersection of a finite number of half-spaces is also addressed and solved by sequentially enumerating vertices of expanding polytopes defined within P. None of our algorithms are affected by degeneracy. Examples and computational results are given.     

Efficient collision detection using bounding volume hierarchies ofk-DOPs

Collision detection is of paramount importance for many applications in computer graphics and visualization. Typically, the input to a collision detection algorithm is a large number of geometric objects comprising an environment, together with a set of objects moving within the environment. In addition to determining accurately the contacts that occur between pairs of objects, one needs also to do so at real-time rates. Applications such as haptic force feedback can require over 1000 collision queries per second. We develop and analyze a method, based on bounding-volume hierarchies, for efficient collision detection for objects moving within highly complex environments. Our choice of bounding volume is to use a discrete orientation polytope (k-DOP), a convex polytope whose facets are determined by halfspaces whose outward normals come from a small fixed set of k orientations. We compare a variety of methods for constructing hierarchies (BV-trees) of bounding k-DOPs. Further, we propose algorithms for maintaining an effective BV-tree of k-DOPs for moving objects, as they rotate, and for performing fast collision detection using BV-trees of the moving objects and of the environment. Our algorithms have been implemented and tested. We provide experimental evidence showing that our approach yields substantially faster collision detection than previous methods     

Comprehensive theoretical digging performance analysis for hydraulic excavator using convex polytope method

This paper proposes a method for analyzing comprehensive theoretical digging performance of an excavator based on the convex polytope. The convex polytope is generated by using Newton–Euler’s equations to establish dynamic relationships between the digging capability of the bucket and the driving capability of hydraulic cylinders with the consideration of the excavator tipping and slipping constraints, and is used to identify the excavator’s output capability for digging forces and moments in the bucket force space. A set of indices for theoretically quantifying the digging capability, digging efficiency, as well as the matchability between the manipulator mechanism and the driving capability of hydraulic cylinders are proposed based on the polytope, and they are used to assess the digging trajectory characteristics and the dynamic digging performance of the entire excavator workspace. Two case studies for the excavator’s digging performance assessment and optimal digging trajectory generation are conducted to test and validate this method. The proposed method contributes to comprehensively and deeply understand the design principles of an excavator, and shows the promising application prospect for guiding the design and development of new excavators.

     

A constructive algorithm to solve "convex recursive deletion" (CoRD) classification problems via two-layer perceptron networks.

A sufficient condition that a region be classifiable by a two-layer feedforward neural net (a two-layer perceptron) using threshold activation functions is that either it be a convex polytope or that intersected with the complement of a convex polytope in its interior, or that intersected with the complement of a convex polytope in its interior or... recursively. These have been called convex recursive deletion (CoRD) regions. We give a simple algorithm for finding the weights and thresholds in both layers for a feedforward net that implements such a region. The results of this work help in understanding the relationship between the decision region of a perceptron and its corresponding geometry in input space. Our construction extends in a simple way to the case that the decision region is the disjoint union of CoRD regions (requiring three layers). Therefore this work also helps in understanding how many neurons are needed in the second layer of a general three-layer network. In the event that the decision region of a network is known and is the union of CoRD regions, our results enable the calculation of the weights and thresholds of the implementing network directly and rapidly without the need for thousands of backpropagation iterations.     

On recovering hyperquadrics from range data

This paper discusses the applications of hyperquadric models in computer vision and focuses on their recovery from range data. Hyperquadrics are volumetric shape models that include superquadrics as a special case. A hyperquadric model can be composed of any number of terms and its geometric bound is an arbitrary convex polytope. Thus, hyperquadrics can model more complex shapes than superquadrics. Hyperquadrics also possess many other advantageous properties (compactness, semilocal control, and intuitive meaning). Our proposed algorithm starts with a rough fit using only six terms in 3D (four in 2D) and adds additional terms as necessary to improve fitting. Suitable constraints are used to ensure proper convergence. Experimental results with real 2D and 3D data are presented     

A less conservative LMI condition for the robust stability of discrete-time uncertain systems

In this paper, a less conservative condition for the robust stability of uncertain discrete-time linear systems is proposed. The uncertain parameters, assumed to be time-invariant, are supposed to belong to convex bounded domains (polytope type uncertainty). The stability condition is formulated in terms of a set of linear matrix inequalities involving only the vertices of the polytope domain. A simple and numerically efficient feasibility test provides a set of Lyapunov matrices whose convex combination can be used to assess the stability of any dynamic matrix inside the uncertainty domain. Examples illustrate the results.     

Index Set Splitting

There are many algorithms for the space-time mapping of nested loops. Some of them even make the optimal choices within their framework. We propose a preprocessing phase for algorithms in the polytope model, which extends the model and yields space-time mappings whose schedule is, in some cases, orders of magnitude faster. These are cases in which the dependence graph has small irregularities. The basic idea is to split the index set of the loop nests into parts with a regular dependence structure and apply the existing space-time mapping algorithms to these parts individually. This work is based on a seminal idea in the more limited context of loop parallelization at the code level. We elevate the idea to the model level (our model is the polytope model), which increases its applicability by providing a clearer and wider range of choices at an acceptable analysis cost. Index set splitting is one facet in the effort to extend the power of the polytope model and to enable the generation of competitive target code.     

Decomposition of arbitrarily shaped binary morphologicalstructuring elements using genetic algorithms

A number of different algorithms have been described in the literature for the decomposition of both convex binary morphological structuring elements and a specific subset of nonconvex ones. Nevertheless, up to now no deterministic solutions have been found to the problem of decomposing arbitrarily shaped structuring elements. This work presents a new stochastic approach based on genetic algorithms, in which no constraints are imposed on the shape of the initial structuring element nor assumptions are made on the elementary factors, which are selected within a given set     

A Computational Study of a Cutting Plane Algorithm for University Course Timetabling

In this paper, we describe a case-study where a Branch-and-Cut algorithm yields the “optimal” solution of a real-world timetabling problem of University courses (University Course Timetabling problem). The problem is formulated as a Set Packing problem with side constraints. To tighten the initial formulation, we utilize well-known valid inequalities of the Set Packing polytope, namely Clique and Lifted Odd-Hole inequalities. We also analyze the combinatorial properties of the problem to introduce new families of cutting planes that are not valid for the Set Packing polytope, and their separation algorithms. These cutting planes turned out to be very effective to yield the optimal solution of a set of real-world instances with up to 69 courses, 59 teachers, and 15 rooms.     

Convex-set-based fuzzy clustering

Prototype-based methods are commonly used in cluster analysis and the results may be highly dependent on the prototype used. We propose a two-level fuzzy clustering method that involves adaptively expanding and merging convex polytopes, where the convex polytopes are considered as a “flexible” prototype. Therefore, the dependency on the use of a specified prototype can be eliminated. Also, the proposed method makes it possible to effectively represent an arbitrarily distributed data set without a priori knowledge of the number of clusters in the data set. In the first level of our proposed method, each cluster is represented by a convex polytope which is described by its set of vertices. Specifically, nonlinear membership functions are utilized to determine whether an input pattern creates a new cluster or whether an existing cluster should be modified. In the second level, the expandable clusters that are selected by an intercluster distance measure are merged to improve clustering efficiency and to reduce the order dependency of the incoming input patterns. Several experimental results are given to show the validity of our method     

First-order sequential convex programming using approximate diagonal QP subproblems

Optimization algorithms based on convex separable approximations for optimal structural design often use reciprocal-like approximations in a dual setting; CONLIN and the method of moving asymptotes (MMA) are well-known examples of such sequential convex programming (SCP) algorithms. We have previously demonstrated that replacement of these nonlinear (reciprocal) approximations by their own second order Taylor series expansion provides a powerful new algorithmic option within the SCP class of algorithms. This note shows that the quadratic treatment of the original nonlinear approximations also enables the restatement of the SCP as a series of Lagrange-Newton QP subproblems. This results in a diagonal trust-region SQP type of algorithm, in which the second order diagonal terms are estimated from the nonlinear (reciprocal) intervening variables, rather than from historic information using an exact or a quasi-Newton Hessian approach. The QP formulation seems particularly attractive for problems with far more constraints than variables (when pure dual methods are at a disadvantage), or when both the number of design variables and the number of (active) constraints is very large.     

On bounds of response time performance achievable by multiclass single-server queues

MulticlassM/G/1 systems in steady-state with work-conserving scheduling strategies are studied. Restricting a system's scheduling strategy to making no direct use of the required service times, every time the server becomes idle its memory is cleared, and service may only be interrupted by newly arriving customers, a conservation law is developed by means of inequalities. The conservation law states that if a response time vector composed of the expected response times of the different classes of a system in steady-state is achievable, then it must belong to a well-defined convex polytope (a set bounded by hyperplanes). Furthermore, on each hyperplane bounding the relevant polytope there lies at least one vertex of the convex set of achievable response time vectors. Therefore, this polytope is the least one including the set of all achievable response time vectors.     

Constrained Optimal Hybrid Control of a Flow Shop System

We consider an optimal control problem for the hybrid model of a deterministic flow shop system, in which the jobs are processed in the order they arrive at the system. The problem is decomposed into a higher-level discrete-event system control problem of determining the optimal service times, and a set of lower-level classical control problems of determining the optimal control inputs for given service times. We focus on the higher-level problem which is nonconvex and nondifferentiable. The arrival times are known and the decision variables are the service times that are controllable within constraints. We present an equivalent convex optimization problem with linear constraints. Under some cost assumptions, we show that no waiting is observed on the optimal sample path. This property allows us to simplify the convex optimization problem by eliminating variables and constraints. We also prove, under an additional strict convexity assumption, the uniqueness of the optimal solution and propose two algorithms to decompose the simplified convex optimization problem into a set of smaller convex optimization problems. The effects of the simplification and the decomposition on the solution times are shown on an example problem.     

命令式动态规划类算法程序推导及机械化验证

动态规划是一种递归求解问题最优解的方法,主要通过求解子问题的解并组合这些解来求解原问题.由于其子问题之间存在大量依赖关系和约束条件,所以验证过程繁琐,尤其对命令式动态规划类算法程序正确性验证是一个难点.基于动态规划类算法Isabelle/HOL函数式建模与验证,通过证明命令式动态规划类算法程序与其的等价性,避免证明正确性时处理复杂的依赖关系和约束条件,提出命令式动态规划类算法程序设计框架及其机械化验证.首先,根据动态规划类算法的优化方法(备忘录方法)和性质(最优子结构性质和子问题重叠性质)描述问题规约、归纳递推关系式和形式化构造出循环不变式,并且基于递推关系式生成IMP (Minimalistic Imperative Programming Language)代码;其次,将问题规约、循环不变式和生成的IMP代码输入VCG (Verification Condition Generator),自动生成正确性的验证条件;然后,在Isabelle/HOL定理证明器中对验证条件进行机械化验证.算法首先设计为命令式动态规划类算法的一般形式,并进一步实例化得到具体算法.最后,例证了所提框架的有效性,为动态规划类算法的自动化推导和验证提供参考价值.     

DC optimization approach to robust controls: the optimal scaling value problem

The optimal scaling problem (OSP) for constant scaling in output feedback control is an inherently difficult nonconvex problem for which in general existing local search algorithms can at best locate a local solution. However, it can be restated as a problem of globally minimizing a convex function under DC constraints, i.e., constraints that can be expressed in terms of differences of convex functions. A particular structure of this DC optimization problem is that it becomes convex when a relatively small number of "complicating" variables are held fixed. We propose alternative branch and bound algorithms for OSP, which exploit this structure by branching upon the complicating variables and use adaptive sub-division strategies to speed-up the convergence to the global solution.     

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.     

Refined Inequalities for Stable Marriage

We consider two approaches to the stable marriage problem: proposal algorithms and describing the stable matching polytope using linear inequalities. We illuminate the relationship between the two approaches. Beginning with a set of linear inequalities that describe the stable matching polytope, we describe a process of refining the set of linear inequalities by eliminating redundant constraints and pruning the preference lists to eliminate unattainable assignments. We show that it is trivial to use the pruned preference lists to find the firm-optimal and worker-optimal stable matchings. We then describe a new procedure that combines a proposal algorithm and our refining process to find a stable matching that does not favor one group over the other. Finally, we apply our refining process to problems in which couples submit preferences over pairs of positions.     

Recursive subdivision without the convex hull property

Recursive subdivision is a standard technique in computer aided geometric design for intersecting and rendering curves and surfaces. The convergence of recursive subdivision is critical for its effective use. Bézier and B-spline curves and surfaces have recursive subdivision algorithms that are known to converge. We show more generally that if a recursive subdivision algorithm exists for a given curve or surface type, then convergence is guaranteed if the blending functions are continuous, form a partition of unity, and are linearly independent. Thus, convergence of recursive subdivision does not depend on the convex hull property. We also show that even in the absence of the convex hull property, it is possible to define termination tests based on the flatness of control polygons, and to construct intersection algorithms based on recursive subdivision. Examples are given of polynomial curves to which our theorems apply.     

基于凸优化算法的无人水下航行器协同定位

In this paper, a cooperative localization algorithm for autonomous underwater vehicles (AUVs) is proposed. A ``parallel" model is adopted to describe the cooperative localization problem instead of the traditional ``leader-follower" model, and a linear programming associated with convex optimization method is used to deal with the problem. After an unknown-but-bounded model for sensor noise is assumed, bearing and range measurements can be modeled as linear constraints on the configuration space of the AUVs. Merging these constraints induces a convex polyhedron representing the set of all configurations consistent with the sensor measurements. Estimates for the uncertainty in the position of a single AUV or the relative positions of two or more nodes can then be obtained by projecting this polyhedron onto appropriate subspaces of the configuration space. Two different optimization algorithms are given to recover the uncertainty region according to the number of the AUVs. Simulation results are presented for a typical localization example of the AUV formation. The results show that our positioning method offers a good localization accuracy, although a small number of low-cost sensors are needed for each vehicle, and this validates that it is an economical and practical positioning approach compared with the traditional approach.