Linear Cuts Research Articles

Paramètres de découpe et de mensurations des carcasses d’agneaux Martinik élevés en conditions intensives

L’ovin Martinik est sélectionné pour ses capacités d’adaptation et sa productivité. Les filières lui reprochent une faible conformation de carcasse, alors même que ces carcasses ont été peu décrites. Au cours de deux essais d’engraissement intensif, des abattages ont été réalisés et des découpes et des mensurations linéaires ont été relevées. Ces variables ont été analysées et la détermination des équations d’allométrie a été réalisée. La base de données (n = 43 agneaux) a été codifiée pour les niveaux d’alimentation et la classe de poids d’abattage (PA : 28, 32 et 36 kg). Les poids moyens de carcasse froide (CF) ont varié (P < 0,05) de 11,7 à 15,6 kg. Les rendements de carcasse ont été en moyenne de 59 p. 100 (P > 0,05). Les poids des différents morceaux ont augmenté significativement (P < 0,01) avec la gamme de CF. Les proportions d’épaule et de gigot ont représenté respectivement 18 et 34 p. 100 de la CF. Les différentes longueurs mesurées sur la carcasse et la largeur de bassin ont augmenté (P < 0,05) avec la CF (près de 9 p. 100 de différence entre les poids extrêmes). Les index de compacité de carcasse (0,32) et de gigot (0,45) n’ont pas varié selon le PA, alors que les index pondéraux ont été différents. Les coefficients d’allométrie de l’épaule (0,915) et du gigot (0,891) ont été du même ordre de grandeur que ceux de la littérature. Ces premières descriptions des carcasses de l’ovin Martinik pourraient être proposées à la filière afin de poursuivre le travail et mieux baser les choix de race et de systèmes.

A second-order cone cutting surface method: complexity and application

We present an analytic center cutting surface algorithm that uses mixed linear and multiple second-order cone cuts. Theoretical issues and applications of this technique are discussed. From the theoretical viewpoint, we derive two complexity results. We show that an approximate analytic center can be recovered after simultaneously adding p second-order cone cuts in O(plog?(p+1)) Newton steps, and that the overall algorithm is polynomial. From the application viewpoint, we implement our algorithm on mixed linear-quadratic-semidefinite programming problems with bounded feasible region and report some computational results on randomly generated fully dense problems. We compare our CPU time with that of SDPLR, SDPT3, and SeDuMi and show that our algorithm outperforms these software packages on problems with fully dense coefficient matrices. We also show the performance of our algorithm on semidefinite relaxations of the maxcut and Lovasz theta problems.

Convex underestimation for posynomial functions of positive variables

The approximation of the convex envelope of nonconvex functions is an essential part in deterministic global optimization techniques (Floudas in Deterministic Global Optimization: Theory, Methods and Application, 2000). Current convex underestimation algorithms for multilinear terms, based on arithmetic intervals or recursive arithmetic intervals (Hamed in Calculation of bounds on variables and underestimating convex functions for nonconvex functions, 1991; Maranas and Floudas in J Global Optim 7:143–182, (1995); Ryoo and Sahinidis in J Global Optim 19:403–424, (2001)), introduce a large number of linear cuts. Meyer and Floudas (Trilinear monomials with positive or negative domains: Facets of convex and concave envelopes, pp. 327–352, (2003); J Global Optim 29:125–155, (2004)), introduced the complete set of explicit facets for the convex and concave envelopes of trilinear monomials with general bounds. This study proposes a novel method to underestimate posynomial functions of strictly positive variables.

The Analytic-Center Cutting-Plane Method for Variational Inequalities: A Quadratic-Cut Approach

We introduce a cutting-plane, analytic-center algorithm for strongly monotone variational inequalities (VIs). The approach extends that of Goffin et al. (1997) and Denault and Goffin (1999). The VI is still treated as a convex feasibility problem, with linear cuts progressively shrinking a localization set that contains the solution of the VI. However, a quadratic cut is used to improve the positioning of the point at which the next cut will be generated. Our approach uses quadratic, ellipsoidal cuts, based on the symmetrized Jacobian of the VI. Since it cannot be guaranteed that such quadratic cuts do not cut off the solution of the VI, they are used only for direction, and are not integrated as such in the localization set; only the linear part of the quadratic cuts can safely be added to the localization set. The introduction of the quadratic cut together with the drop of the quadratic part of the previous cut is studied carefully. Numerical results are given that illustrate the substantial improvement that quadratic cuts can yield over linear cuts.

Cuts for mixed 0-1 conic programming

In this we paper we study techniques for generating valid convex constraints for mixed 0-1 conic programs. We show that many of the techniques developed for generating linear cuts for mixed 0-1 linear programs, such as the Gomory cuts, the lift-and-project cuts, and cuts from other hierarchies of tighter relaxations, extend in a straightforward manner to mixed 0-1 conic programs. We also show that simple extensions of these techniques lead to methods for generating convex quadratic cuts. Gomory cuts for mixed 0-1 conic programs have interesting implications for comparing the semidefinite programming and the linear programming relaxations of combinatorial optimization problems, e.g. we show that all the subtour elimination inequalities for the traveling salesman problem are rank-1 Gomory cuts with respect to a single semidefinite constraint. We also include results from our preliminary computational experiments with these cuts.

The elastic generalized assignment problem

The generalized assignment problem (GAP) has been studied by numerous researchers over the past 30 years or so. Simply stated, one must find a minimum-cost assignment of tasks to agents such that each task is assigned to exactly one agent and such that each agent's resource capacity is honoured. The problem is known to be NP-hard. In this paper, we study the elastic generalized assignment problem (EGAP). The elastic version of GAP allows agent resource capacity to be violated at additional cost. Another version allows undertime costs to be assessed as well if an agent's resource capacity is not used to its full extent. The EGAP is also NP-hard. We describe a special-purpose branch-and-bound algorithm that utilizes linear programming cuts, feasible solution generators, Lagrangean relaxation and subgradient optimization. We present computational results on a large collection of randomly generated ‘hard’ problems with up to 4000 binary variables.

CVD diamond insert in stone cutting

This work aims to verify the use of chemical vapour deposition (CVD) diamond inserts in a chain saw, instead of traditional tungsten carbide inserts, in order to combine diamond hardness and abrasiveness together with the tungsten carbide tenacity to quarry stone more efficiently. It aims to study if a CVD diamond insert allows to give a faster cutting speed and a longer life of the tool, when compared with a traditional tungsten carbide insert. The present work focuses on a single CVD diamond insert that has been mounted on a single-tip tool. The single-tip tool has been fixed to a NC machine tool in order to carry out a set of linear cuts on a marble slab. Different values of the process parameters have been used for the different cuts. The measured values of the cutting force have been put into relationship with the process parameters. The obtained results have been compared with those of the traditional tungsten carbide inserts. The results show how CVD insert allows to significantly reduce the cutting force compared to tungsten carbide insert does and, therefore, to increase the cutting speed without involving hard stresses on tool. The increase of the cutting speed allows to sensibly reduce the machining time.

Solving the Generalized Assignment Problem: An Optimizing and Heuristic Approach

The classical generalized assignment problem (GAP) may be stated as finding a minimum-cost assignment of tasks to agents such that each task is assigned to exactly one agent and such that each agent's resource capacity is honored. This NP-hard problem has applications that include job scheduling, routing, loading for flexible manufacturing systems, and facility location. Due to the difficulty in solving “hard” GAPs to optimality, most recent papers either describe heuristic methods for generating “good” solutions or, in the case of optimizing methods, computational results are limited to 500 to 1,000 binary variables. In this paper we describe a special purpose branch-and-bound algorithm that utilizes linear programming cuts, feasible-solution generators, Lagrangean relaxation, and subgradient optimization. We present computational results for solving “hard” problems with up to 3,000 binary variables. An unanticipated benefit of the algorithm is its ability to generate good feasible solutions early in the process whose solution quality generally dominates the solutions generated by two recently published heuristics. Furthermore, the computation time required is often less than the time taken by the heuristics. Thus, we have an optimizing algorithm that can be used quite effectively as a heuristic when proof of optimality is not an absolute requirement.

Space curve interpolation for CNC machines

Despite the tremendous development in CNC programming facilities, linear and circular cuts parallel to the coordinate planes continue to be the standard motions of modern CNC machines. However, the increasing industrial demand for parts with intricate shapes cannot be satisfied with only these standard motions. This paper proposes an efficient and accurate method for developing a class of precise interpolation algorithms which can drive the cutter of a CNC machine along three-dimensional (3D) trajectories. Engaging a pair of primitive shapes, the method uses their implicit or parametric definition to interpolate their intersection curve on the basis of Danielson’s step selection rules. The presence of Danielson’s criteria guarantees a position error of at most one step. A general-purpose computer for path generation and real-time control of the machine is employed. The user first chooses a suitable combination of shapes whose intersection gives the desired form of the 3D trajectory and then initiates the algorithm with the desired information of the two shapes.

Visualization of High-Dimensional Systems via Geometric Modeling with Homogeneous Coordinates

A framework is presented for the visualization of high-dimensional systems, particularly multicomponent phase diagrams. It is based on geometric modeling with homogeneous coordinates of the transformations among geometric varieties in high-dimensional space. By reduction of the dimensionality of the system through such transforms, the resulting linear cuts and projections, in their aggregate, provide a mental picture of the system in its entirety. A general way to perform cuts and projections is presented, including how to calculate the transforms and how to define the projective space. Specifically, a procedure is presented for the development of a set of canonical coordinates for the projective space with reduced dimensions. Also, it is shown that the seemingly different transformed coordinates for reactive systems in the literature can be unified under our framework. Examples are provided to highlight the procedure and to demonstrate the visualization of phase diagrams for nonreactive and reactive systems.

Set packing relaxations of some integer programs

This paper is about set packing relaxations of combinatorial optimization problems associated with acyclic digraphs and linear orderings, cuts and multicuts, and set packings themselves. Families of inequalities that are valid for such a relaxation as well as the associated separation routines carry over to the problems under investigation.

A Polynomial Cutting Surfaces Algorithm for the ConvexFeasibility Problem Defined by Self-Concordant Inequalities

Consider a nonempty convex set in R^m which is defined by a finite number of smooth convex inequalities and which admits a self-concordant logarithmic barrier. We study the analytic center based column generation algorithm for the problem of finding a feasible point in this set. At each iteration the algorithm computes an approximate analytic center of the set defined by the inequalities generated in the previous iterations. If this approximate analytic center is a solution, then the algorithm terminatess otherwise either an existing inequality is shifted or a new inequality is added into the system. As the number of iterations increases, the set defined by the generated inequalities shrinks and the algorithm eventually finds a solution of the problem. The algorithm can be thought of as an extension of the classical cutting plane method. The difference is that we use analytic centers and “convex cuts” instead of arbitrary infeasible points and linear cuts. In contrast to the cutting plane method, the algorithm has a polynomial worst case complexity ofO(N\log\frac{1}{\varepsilon}) on the total number of cuts to be used, where N is the number of convex inequalities in the original problem and e is the maximum common slack of the original inequality system.

MINIMUM NUMBER OF PIECES IN A CONVEX PARTITION OF A POLYGONAL DOMAIN

Let [Formula: see text] be a given nonempty family of directions in the plane. For a multiply connected polygonal domain P with polygonal holes, possibly degenerate, we determine the minimum number of convex polygons into which P is partitioned by linear cuts in the directions from [Formula: see text].

Probabilistic μ upper bound using linear cuts

Abstract As a direct extension of the structured singular value μ framework from the worst-case to the probabilistic robustness analysis, probabilistic μ defines a probability measure of performance assuming some distribution on the uncertainty. Exact computation of probabilistic μ is intractable. Standard branch and bound methods with axially aligned cuts had been applied to compute a hard upper bound for probabilistic μ which failed to give satisfactory solutions. This paper explores the use of linear cuts in the upper bound computation, which involves the construction of an implicit system. The bounds achieved are exact for rank-one problems. The algorithm has also been tested on random matrices and the results will be compared to those obtained by the standard B&B algorithm.

An Analytic Center Based Column Generation Algorithm for Convex Quadratic Feasibility Problems

We consider the analytic center based column generation algorithm for the problem of finding a feasible point in a set defined by a finite number of convex quadratic inequalities. At each iteration the algorithm computes an approximate analytic center of the set defined by the intersection of quadratic inequalities generated in the previous iterations. If this approximate analytic center is a solution, then the algorithm terminates; otherwise a quadratic inequality violated at the current center is selected and a new quadratic cut (defined by a convex quadratic inequality) is placednear the approximate center. As the number of cuts increases, the set defined by their intersection shrinks and the algorithm eventually finds a solution to the problem. Previously, similar analytic center based column generation algorithms were studied first for the linear feasibility problem and later for the general convex feasibility problem. Our method differs from these early methods in that we use "quadratic cuts" in the computation instead of linear cuts. Moreover, our method has a polynomial worst case complexity of $O(n\ln \frac{1}{\varepsilon})$ on the total number of cuts to be used, where n is the number of convex quadratic polynomial inequalities in the problem and $\varepsilon$ is the radius of the largest ball contained in the feasible set. In contrast, the early column generation methods using linear cuts can only solve the convex quadratic feasibility problem in pseudopolynomial time.

Logarithmic barrier decomposition methods for semi-infinite programming

A computational study of some logarithmic barrier decomposition algorithms for semi-infinite programming is presented in this paper. The conceptual algorithm is a straightforward adaptation of the logarithmic barrier cutting plane algorithm which was presented recently by den Hartog et al. (Annals of Operations Research, 58, 69–98, 1995), to solve semi-infinite programming problems. Usually decomposition (cutting plane methods) use cutting planes to improve the localization of the given problem. In this paper we propose an extension which uses linear cuts to solve large scale, difficult real world problems. This algorithm uses both static and (doubly) dynamic enumeration of the parameter space and allows for multiple cuts to be simultaneously added for larger/difficult problems. The algorithm is implemented both on sequential and parallel computers. Implementation issues and parallelization strategies are discussed and encouraging computational results are presented.

Logarithmic Barrier Decomposition Methods for Semi‐infinite Programming

A computational study of some logarithmic barrier decomposition algorithms for semi‐infinite programming is presented in this paper. The conceptual algorithm is a straightforward adaptation of the logarithmic barrier cutting plane algorithm which was presented recently by den Hartog et al. (Annals of Operations Research, 58, 69–98, 1995), to solve semi‐infinite programming problems. Usually decomposition (cutting plane methods) use cutting planes to improve the localization of the given problem. In this paper we propose an extension which uses linear cuts to solve large scale, difficult real world problems. This algorithm uses both static and (doubly) dynamic enumeration of the parameter space and allows for multiple cuts to be simultaneously added for larger/difficult problems. The algorithm is implemented both on sequential and parallel computers. Implementation issues and parallelization strategies are discussed and encouraging computational results are presented.

The statistical basis of neural network algorithms: Theory and applications

In a very general framework, neural network and related algorithms are discussed with emphasis on how the use of correlations between variables can improve on linear cuts. Mathematical results for a large class of neural algorithms are shown indicating the generic importance of high-order correlations between input variables. The dangers of the blind application of these newer techniques are stressed and suggestions are made for how to use them wisely.

Use of neural networks in the analysis of τ → K s0ντX inclusive decays at LEP

Neutral Networks (NNW) were used to tag the inclusive τ → K s 0 ν τ X decay at the Z 0 energy. Two feed-forward NNWs were trained with the backpropagation algorithm, to select, respectively, Z 0 → τ + τ − events and K s 0 → π + π − decays. They were used to process the data collected with the DELPHI detector at LEP in 1993. The selection efficiency and purity achieved with the NNW at each step of the analysis were compared with those obtained using linear cuts.

Gap Independence and Lacunarity in Percolation Clusters.

The gaps between occupied sites on linear cuts of two and three dimensional critical percolation clusters are found to be closely described as statistically independent, with a universal scaling distribution close to that of positive L\'evy flights. The moments of the mass distribution of L\'evy flights obey $〈{m}^{k}〉/〈m{〉}^{k}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}k![\ensuremath{\gamma}(\ensuremath{\alpha}+1){]}^{k}/\ensuremath{\gamma}(k\ensuremath{\alpha}+1)$, where $\ensuremath{\alpha}$ is their fractal dimension. Our data on linear cuts of critical percolation clusters are consistent (within the numerical error bars) with these predictions. The property of statistical independence of the gaps characterizes the lacunarity of the percolation clusters as being neutral.