Epsilon-proximal decomposition method

We propose a modification of the proximal decomposition method investigated by Spingarn [30] and Mahey et al. [19] for minimizing a convex function on a subspace. For the method to be favorable from a computational point of view, particular importance is the introduction of approximations in the proximal step. First, we couple decomposition on the graph of the epsilon-subdifferential mapping and cutting plane approximations to get an algorithmic pattern that falls in the general framework of Rockafellar inexact proximal-point algorithms [26]. Recently, Solodov and Svaiter [27] proposed a new proximal point-like algorithm that uses improved error criteria and an enlargement of the maximal monotone operator defining the problem. We combine their idea with bundle mecanism to devise an inexact proximal decomposition method with error condition which is not hard to satisfy in practice. Then, we present some applications favorable to our development. First, we give a new regularized version of Benders decomposition method in convex programming called the proximal convex Benders decomposition algorithm. Second, we derive a new algorithm for nonlinear multicommodity flow problems among which the message routing problem in telecommunications data networks.     

Depth-Optimized Convexity Cuts

This paper presents a general, self-contained treatment of convexity or intersection cuts. It describes two equivalent ways of generating a cut—via a convex set or a concave function—and a partial-order notion of cut strength. We then characterize the structure of the sets and functions that generate cuts that are strongest with respect to the partial order. Next, we specialize this analytical framework to the case of mixed-integer linear programming (MIP). For this case, we formulate two kinds of the deepest cut generation problem, via sets or via functions, and subsequently consider some special cases which are amenable to efficient computation. We conclude with computational tests of one of these procedures on a large set of MIPLIB problems.     

Quantum cutting in Gd<Subscript>2</Subscript>SiO<Subscript>5</Subscript>: Eu<Superscript>3+</Superscript> by VUV excitation

The emission and excitation spectra of Gd₂SiO₅∶Eu³⁺ were investigated using the VUV beam line of the Beijing Synchrotron Radiation Facility (BSRF). The experimental results were discussed in the frame of visible quantum cutting process involved in Gd³⁺−Eu³⁺ system. Upon direct excitation into the⁶G _J states of Gd³⁺, two visible photon emissions from Eu³⁺ were observed. Cursory evaluation proved that Gd₂SiO₅∶Eu³⁺ is an efficient visible quantum cutter.     

Estimation of failure probability using semi-definite logit model

We will propose a new and practical method for estimating the failure probability of a large number of small to medium scale companies using their balance sheet data. We will use the maximum likelihood method to estimate the best parameters of the logit function, where the failure intensity function in its exponent is represented as a convex quadratic function instead of a commonly used linear function. The reasons for using this type of function are : (i) it can better represent the observed nonlinear dependence of failure probability on financial attributes, (ii) the resulting likelihood function can be maximized using a cutting plane algorithm developed for nonlinear semi-definite programming problems.We will show that we can achieve better prediction performance than the standard logit model, using thousands of sample companies.Revised: December 2002,     

Lift-and-project cuts and perfect graphs

We analyze the application of lift-and-project to the clique relaxation of the stable set polytope. We characterize all the inequalities that can be generated through the application of the lift-and-project procedure, introduce the concept of 1-perfection and prove its equivalence to minimal imperfection. This characterization of inequalities and minimal imperfection leads to a generalization of the Perfect Graph Theorem of Lovász, as proved by Aguilera, Escalante and Nasini [1].Mathematics Subject Classification:05C17, 90C57     

Multiple Cuts with a Homogeneous Analytic Center Cutting Plane Method

This paper analyzes the introduction of multiple central cuts in a conic formulation of the analytic center cutting plane method (in short ACCPM). This work extends earlier work on the homogeneous ACCPM, and parallels the analysis of the multiple cut process in the standard ACCPM. The main issue is the calculation of a direction that restores feasibility after introducing p new cutting planes at the query point. We prove that the new analytic center can be recovered in O(p log p) damped Newton iterations, where is a parameter depending of the data. We also present two special cases where the complexity can be decreased to O (p log p). Finally, we show that the number of calls to the oracle is the same as in the single cut case, up to a factor .     

A Combined Algorithm for Fractional Programming

In this paper, we present an outer approximation algorithm for solving the following problem: max _xS {f(x)/g(x)}, where f(x)0 and g(x)>0 are d.c. (difference of convex) functions over a convex compact subset S of R ⁿ . Let ()=max _xS (f(x)–g(x)), then the problem is equivalent to finding out a solution of the equation ()=0. Though the monotonicity of () is well known, it is very time-consuming to solve the previous equation, because that maximizing (f(x)–g(x)) is very hard due to that maximizing a convex function over a convex set is NP-hard. To avoid such tactics, we give a transformation under which both the objective and the feasible region turn to be d.c. After discussing some properties, we propose a global optimization approach to find an optimal solution for the encountered problem.     

激光切割

本文叙述了激光切割的机理,常用激光切割的方法,激光切割的质量及其主要影响因素。     

Solving binary cutting stock problems by column generation and branch-and-bound

We present an algorithm for the binary cutting stock problem that employs both column generation and branch-and-bound to obtain optimal integer solutions. We formulate a branching rule that can be incorporated into the subproblem to allow column generation at any node in the branch-and-bound tree. Implementation details and computational experience are discussed.This research was supported by NSF and AFOSR grant DDM-9115768