Robust variable selection for nonlinear models with diverging number of parameters

We focus on the problem of simultaneous variable selection and estimation for nonlinear models based on modal regression (MR), when the number of coefficients diverges with sample size. With appropriate selection of the tuning parameters, the resulting estimator is shown to be consistent and to enjoy the oracle properties.     

Minimizing the number of machines for minimum length schedules

In this paper, we present a new objective function for scheduling on parallel machines: minimizing the number of machines for schedules of minimum length. We study its complexity and we prove the NP-completeness of this problem, even if there is no precedences or for unitary execution times. We propose several polynomial algorithms for various particular cases.     

The geometry and number of the root invariant regions for linear systems

The stability domain is a feasible set for numerous optimization problems. D-decomposition technique is targeted to describe the stability domain in the parameter space for linear parameter-dependent systems. This technique is very simple and efficient for robust stability analysis and design of low-order controllers. However, the geometry of the arising parameter space decomposition into root invariant regions has not been studied in detail; it is an objective of the present paper. We estimate the number of root invariant regions and provide examples, where this number is attained.     

Minimizing the number of machines for scheduling jobs with equal processing times

In this paper, we consider a parallel machine environment when all jobs have the same processing time and arbitrary release dates and deadlines of the jobs are given. We suppose that the available number of machines, which can be used simultaneously, may vary over time. The aim is to construct a feasible schedule in such a way that the maximal number of simultaneously used machines is minimal. We give a polynomial algorithm for this problem.     

Minimizing the jump number for partially ordered sets: A graph-theoretic approach

The purpose of this paper is to present a graph-theoretic approach to the jump number problem for N-free posets which is based on the observation that the Hasse diagram of an N-free poset is a line digraph. Therefore, to every N-free poset P we can assign another digraph which is the root digraph of the Hasse diagram of P. Using this representation we show that the jump number of an N-free poset is equal to the cyclomatic number of its root digraph and can be found (without producing any linear extension) by an algorithm which tests if a given poset is N-free. Moreover, we demonstrate that there exists a correspondence between optimal linear extensions of an N-free poset and spanning branchings of its root digraph. We provide also another proof of the fact that optimal linear extensions of N-free posets are exactly greedy linear extensions. In conclusion, we discuss some possible generalizations of these results to arbitrary posets.     

Minimizing the number of independent sets in triangle-free regular graphs

Recently, Davies, Jenssen, Perkins, and Roberts gave a very nice proof of the result (due, in various parts, to Kahn, Galvin–Tetali, and Zhao) that the independence polynomial of a $d$-regular graph is maximized by disjoint copies of $K_{d,d}$. Their proof uses linear programming bounds on the distribution of a cleverly chosen random variable. In this paper, we use this method to give lower bounds on the independence polynomial of regular graphs. We also give a new bound on the number of independent sets in triangle-free cubic graphs.     

Minimizing the number of transitions with respect to observation equivalence

Labeled transition systems (lts) provide an operational semantics for many specification languages. In order to abstract unrelevant details of lts's, manybehavioural equivalences have been defined; here observation equivalence is considered. We are interested in the following problem:Given a finite lts, which is the minimal observation equivalent lts corresponding to it? It is well known that the number of states of an lts can be minimized by applying arelational coarsest partition algorithm. However, the obtained lts is not unique (up to the renaming of the states): for an lts there may exist several observation equivalent lts's which have the minimal number of states but varying number of transitions. In this paper we show how the number of transitions can be minimized, obtaining a unique lts.     

Minimizing the condition number of a positive definite matrix by completion

Summary. We consider the problem of minimizing the spectral condition number of a positive definite matrix by completion: \noindent where is an Hermitian positive definite matrix, a matrix and is a free Hermitian matrix. We reduce this problem to an optimization problem for a convex function in one variable. Using the minimal solution of this problem we characterize the complete set of matrices that give the minimum condition number. Received October 15, 1993     

Minimizing the number of workers in a paced mixed-model assembly line

We study a problem of minimizing the maximum number of identical workers over all cycles of a paced assembly line comprised of m stations and executing n parts of k types. There are lower and upper bounds on the workforce requirements and the cycle time constraints. We show that this problem is equivalent to the same problem without the cycle time constraints and with fixed workforce requirements. We prove that the problem is NP-hard in the strong sense if $m=4$ and the workforce requirements are station independent, and present an Integer Linear Programming model, an enumeration algorithm and a dynamic programming algorithm. Polynomial in k and polynomial in n algorithms for special cases with two part types or two stations are also given. Relations to the Bottleneck Traveling Salesman Problem and its generalizations are discussed.     

Minimizing the weighted number of late jobs in UET open shops

We consider a two-machine open shop problem where the jobs have release dates and due dates, and where all single operations have unit processing times. The goal is to minimize the weighted number of late jobs. We derive a polynomial time algorithm for this problem, thereby answering an open question posed in a recent paper by Brucker et al.This research was supported by the Christian Doppler Laboratorium für Diskrete Optimierung.     

Bin packing: On optimizing the number of pieces packed

We consider bin-packing variations related to the well-studied problem of maximizing the total number of pieces packed into a fixed set of bins. We show that, when the objective is to minimize the total number of pieces packed subject to the constraint that no unpacked piece will fit, no polynomial-time relative approximation algorithm exists (unless, of course,P=NP). That is, we prove that it isNP-hard to guarantee packing no more than a constant multiple of the optimal number of pieces, for any constant. This appears to be the first bin-packing problem for which this property has been demonstrated. Furthermore, this result also holds for the allied packing variant which seeks to minimize the maximum number of pieces packed in any single bin. We find the situation to be markedly better for the problem of maximizing the minimum number of pieces in any bin. If all bins possess the same capacity, we prove that the familiar SPF rule is an absolute approximation algorithm with additive constant 1, and can therefore be regarded as a best possible heuristic. For the more general and difficult case in which bin capacities may differ, it turns out that SPF fails to qualify as even a relative approximation algorithm. However, we devise an implementation of the well-known FFD heuristic, which we show to be a relative approximation algorithm, yielding a worst-case performance ratio of 1/2 with additive constant 0. Moreover, we prove that (unlessP=NP) no polynomial-time algorithm can guarantee a higher ratio without sacrificing the additive constant.This author's research is supported in part by the National Science Foundation under grants ECS-8403859 and MIP-8603879.     

A comparison of the performance of artificial intelligence techniques for optimizing the number of kanbans

This paper discusses the use of modern heuristic techniques coupled with a simulation model of a Just in Time system to find the optimum number of kanbans while minimizing cost. Three simulation search heuristic procedures based on Genetic Algorithms, Simulated Annealing, and Tabu Search are developed and compared both with respect to the best results achieved by each algorithm in a limited time span and their speed of convergence to the results. In addition, a Neural Network metamodel is developed and compared with the heuristic procedures according to the best results. The results indicate that Tabu Search performs better than the other heuristics and Neural Network metamodel in terms of computational effort.     

Minimizing the number of machines with limited workload capacity for scheduling jobs with interval constraints

In this paper, we consider a parallel machine scheduling problem in which machines have a limited workload capacity and jobs have deadlines and release dates. The problem is motivated by the operation of energy storage management systems for microgrids under emergency conditions and generalizes some problems that have already been studied in the literature for their theoretical value. In this work, we propose heuristic and exact algorithms to solve the problem. The heuristics are adaptations of classical bin packing heuristics in which additional conditions on the feasibility of a solution are imposed, whereas the exact method is a branch-and-price approach. The results show that the branch-and-price approach is able to optimally solve random instances with up to 250 jobs within a time limit of one hour, while the heuristic procedures provide near optimal solution within reduced running times. Finally, we also provide additional complexity results for a special case of the problem.     

On the number of summands in the asymptotic formula for the number of solutions to Waring's equation

In the paper, an estimate of the number of summands in the asymptotic formula for the number of solutions to Waring's equation is obtained. This is achieved by means of a recurrent process leading to a greater reduction than that in Vinogradov's mean value theorem.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 285–296, August, 1998.The author wishes to thank N. M. Korobov for setting the problem and supervising his work.     

On the number of Fourier coefficients that determine a Hilbert modular form

We estimate the number of Fourier coefficients that determine a Hilbert modular cusp form of arbitrary weight and level. The method is spectral (Rayleigh quotient) and avoids the use of the maximum principle.

     

On the spectral radii of unicyclic graphs with fixed matching number

In this paper, the first five sharp upper bounds on the spectral radii of unicyclic graphs with fixed matching number are presented. The first ten spectral radii over the class of unicyclic graphs on a given number of vertices and the first four spectral radii of unicyclic graphs with perfect matchings are also given, respectively.     

An exact solution for variable coefficients fourth-order wave equation using the Adomian method

This paper presents a novel approach for the analysis of a fourth-order parabolic equation dealing with vibration of beams by using the decomposition method. In this regard, a general approach based on the generalized Fourier series expansion is applied. The obtained analytic solution is simplified in terms of a given set of orthogonal basis functions. The result is compared with the classical modal analysis technique which is widely used in the field of structural dynamics. It is shown that the result of the decomposition method leads to an exact closed-form solution which is equivalent to the result obtained by the modal analysis method. Some examples are given to demonstrate the validity of the present study.