Reducing Huge Gyroscopic Eigenproblems by Automated Multi-level Substructuring

Simulating numerically the sound radiation of a rolling tire requires the solution of a very large and sparse gyroscopic eigenvalue problem. Taking advantage of the automated multi-level substructuring (AMLS) method it can be projected to a much smaller gyroscopic problem, the solution of which however is still quite costly since the eigenmodes are non-real and complex arithmetic is necessary. This paper discusses the application of AMLS to huge gyroscopic problems and the numerical solution of the AMLS reduction. A numerical example demonstrates the efficiency of AMLS.     

A 2-approximation algorithm for interval data minmax regret sequencing problems with the total flow time criterion

In this paper we discuss a minmax regret version of the single-machine scheduling problem with the total flow time criterion. Uncertain processing times are modeled by closed intervals. We show that if the deterministic problem is polynomially solvable, then its minmax regret version is approximable within 2.     

Minmax scheduling with job-classes and earliness–tardiness costs

We address scheduling problems with job-dependent due-dates and general (possibly nonlinear and asymmetric) earliness and tardiness costs. The number of distinct due-dates is substantially smaller than the number of jobs, thus jobs are partitioned to classes, where all jobs of a given class share a common due-date. We consider the settings of a single machine and parallel identical machines. Our objective is of a minmax type, i.e., we seek a schedule that minimizes the maximum earliness/tardiness cost among all jobs.     

The minmax regret gradual covering location problem on a network with incomplete information of demand weights

The gradual covering location problem seeks to establish facilities on a network so as to maximize the total demand covered, allowing partial coverage. We focus on the gradual covering location problem when the demand weights associated with nodes of the network are random variables whose probability distributions are unknown. Using only information on the range of these random variables, this study is aimed at finding the “minmax regret” location that minimizes the worst-case coverage loss. We show that under some conditions, the problem is equivalent to known location problems (e.g. the minmax regret median problem). Polynomial time algorithms are developed for the problem on a general network with linear coverage decay functions.     

Minmax regret approach and optimality evaluation in combinatorial optimization problems with interval and fuzzy weights

This paper deals with a general combinatorial optimization problem in which closed intervals and fuzzy intervals model uncertain element weights. The notion of a deviation interval is introduced, which allows us to characterize the optimality and the robustness of solutions and elements. The problem of computing deviation intervals is addressed and some new complexity results in this field are provided. Possibility theory is then applied to generalize a deviation interval and a solution concept to fuzzy ones.     

Saddle points and Lagrangian-type duality for discrete minmax fractional subset programming problems with generalized convex functions

In this paper we present four sets of saddle-point-type optimality conditions, construct two Lagrangian-type dual problems, and prove weak and strong duality theorems for a discrete minmax fractional subset programming problem. We establish these optimality and duality results under appropriate (b,?,ρ,θ)-convexity hypotheses.     

On minquantile and maxcovering optimisation

In this paper we introduce the parametric minquantile problem, a weighted generalisation ofkth maximum minimisation. It is shown that, under suitable quasiconvexity assumptions, its resolution can be reduced to solving a polynomial number of minmax problems.It is also shown how this simultaneously solves (parametric) maximal covering problems. It follows that bicriteria problems, where the aim is to both maximize the covering and minimize the cover-level, are reducible to a discrete problem, on which any multiple criteria method may be applied.Corresponding author.Visiting researcher at the Center for Industrial Location of the Vrije Universiteit Brussel during this research.     

On the complexity of the continuous unbounded knapsack problem with uncertain coefficients

In this paper, a linear-time algorithm is developed for the minmax-regret version of the continuous unbounded knapsack problem with n items and uncertain objective function coefficients, where the interval estimates for these coefficients are known. This improves the previously known bound of time for this optimization problem.     

Reference variable methods of solving min–max optimization problems

In this paper, reference variable methods are proposed for solving nonlinear Minmax optimization problems with unconstraint or constraints for the first time, it uses reference decision vectors to improve the methods in Vincent and Goh (J Optim Theory Appl 75:501–519, 1992) such that its algorithm is convergent. In addition, a new method based on KKT conditions of min or max constrained optimization problems is also given for solving the constrained minmax optimization problems, it makes the constrained minmax optimization problems a problem of solving nonlinear equations by a complementarily function. For getting all minmax optimization solutions, the cost function f(x, y) can be constrained as M ₁ < f(x, y) < M ₂ by using different real numbers M ₁ and M ₂. To show effectiveness of the proposed methods, some examples are taken to compare with results in the literature, and it is easy to find that the proposed methods can get all minmax optimization solutions of minmax problems with constraints by using different M ₁ and M ₂, this implies that the proposed methods has superiority over the methods in the literature (that is based on different initial values to get other minmax optimization solutions).     

Comments on an ancient Greek racecourse: finding minimum width annuluses

For a set of measured points, we describe a linear-programming model that enables us to find concentric circumscribed and inscribed circles whose annulus encompasses all the points and whose width tends to be minimum in a Chebychev minmax sense. We illustrate the process using the data of Rorres and Romano (SIAM Rev. 39: 745–754, 1997) that is taken from an ancient Greek stadium in Corinth. The stadium’s racecourse had an unusual circular arc starting line, and measurements along this arc form the basic data sets of Rorres and Romano (SIAM Rev. 39: 745–754, 1997). Here we are interested in finding the center and radius of the circle that defined the starting line arc. We contrast our results with those found in Rorres and Romano (SIAM Rev. 39: 745–754, 1997).