范数下无容量限制设施选址逆问题的求解方法

一个优化问题的逆问题是这样一类问题,在给定该优化问题的一个可行解时,通过最小化目标函数中参数的改变量(在某个范数下)使得该可行解成为改变参数后的该优化问题的最优解。对于本是NP-难问题的无容量限制设施选址问题,证明了其逆问题仍是NP-难的。研究了使用经典的行生成算法对无容量限制设施选址的逆问题进行计算,并给出了求得逆问题上下界的启发式方法。两种方法分别基于对子问题的线性松弛求解给出上界和利用邻域搜索以及设置迭代循环次数的方式给出下界。数值结果表明线性松弛法得到的上界与最优值差距较小,但求解效率提升不大;而启发式方法得到的下界与最优值差距极小,极大地提高了求解该逆问题的效率。     

On the adjustment problem for linear programs

We propose a generalization of the inverse problem which we will call the adjustment problem. For an optimization problem with linear objective function and its restriction defined by a given subset of feasible solutions, the adjustment problem consists in finding the least costly perturbations of the original objective function coefficients, which guarantee that an optimal solution of the perturbed problem is also feasible for the considered restriction. We describe a method of solving the adjustment problem for continuous linear programming problems when variables in the restriction are required to be binary.     

Inverse Problems of Submodular Functions on Digraphs

In this paper, we study the inverse problem of submodular functions on digraphs. Given a feasible solution x* for a linear program generated by a submodular function defined on digraphs, we try to modify the coefficient vector c of the objective function, optimally and within bounds, such that x* becomes an optimal solution of the linear program. It is shown that the problem can be formulated as a combinatorial linear program and can be transformed further into a minimum cost circulation problem. Hence, it can be solved in strongly polynomial time. We also give a necessary and sufficient condition for the feasibility of the problem. Finally, we extend the discussion to the version of the inverse problem with multiple feasible solutions.     

A Class of Inverse Dominant Problems under Weighted l ∞ Norm and an Improved Complexity Bound for Radzik’s Algorithm

In this paper, we first discuss a class of inverse dominant problems under weighted l_∞ norm, which is how to change the original weights of elements with bounds in a finite ground set so that a given set becomes a weakly dominant set with respect to a given collection of subsets under the new weights and the largest change of the weights is minimum. This model includes a large class of improvement problems in combinatorial optimization. We propose a Newton-type algorithm for the model. This algorithm can solve the model in strongly polynomial time if the subproblem involved is solvable in strongly polynomial time. In the second part of the paper, we improve the complexity bound for Radzik’s Newton-type method which is designed to solve linear fractional combinatorial optimization problems. As Radzik’s method is closely related to our algorithm, this bound also estimates the complexity of our algorithm. Supported by the Hong Kong Universities Grant Council (CERG CITYU 9040883 and 9041091). Xiaoguang Yang - The author is also grateful for the support by the National Key Research and Development Program of China (Grant No. 2002CB312004) and the National Natural Science Foundation of China (Grant No. 70425004).     

工件带简单线性恶化函数和共同交货期单机排序问题

本文研究单机排序问题,其中工件加工时间具有简单线性恶化函数.同时,所有工件均具有一个给定共同交货期.目标函数为最小化提前有奖延误受罚之和.在逆一致性条件下,给出了求解该排序问题的一个伪多项式时间动态规划算法.同时借助于几何舍入技巧,对求解这类排序问题给出了一个充分多项式时间的近似算法(FPTAS)。     

对称的运输问题及其逆问题

本文对［１,２,６］中提出的运输问题进行了推广,并提出了一个强多项式算法,从而改进了原有的结果．同时对对称的运输问题的逆问题进行了研究,并借助于最小费用循环流技术得到了一个强多项式算法．     

The inverse solution of the atomic mixing equations by an operator-splitting method

The quantification problem of recovering the original material distribution from secondary ion mass spectrometry (SIMS) data is considered in this paper. It is an inverse problem, is ill-posed and hence it requires a special technique for its solution. The quantification problem is essentially an inverse diffusion or (classically) a backward heat conduction problem. In this paper an operator-splitting method (that is proposed in a previous paper by the first author for the solution of inverse diffusion problems) is developed for the solution of the problem of recovering the original structure from the SIMS data. A detailed development of the quantification method is given and it is applied to typical data to demonstrate its effectiveness.     

On degree sequence optimization

We consider the problem of finding a subgraph of a given graph maximizing a given function evaluated at its degree sequence. While it is intractable already for convex functions, we show it is polynomial time solvable for convex multi-criteria objectives. We also consider a colored extension of the problem with separable objectives, which includes the notorious exact matching problem as a special case, and show that it is polynomial time solvable on graphs of bounded tree-depth for any vertex functions.     

反对称正交对称矩阵反问题

本文讨论一类反对称正交对称矩阵反问题及其最佳逼近．研究了这类矩阵的一些性质，利用这些性质给出了反问题解存在的一些条件和解的一般表达式，不仅证明了最佳逼近解的存在唯一性，而且给出了此解的具体表达式．     

强部分逆网络流问题

Abstract. In this paper,a new model for inverse network flow problems,robust partial inverseproblem is presented. For a given partial solution,the robust partial inverse problem is to modify the coefficients optimally such that all full solutions containing the partial solution becomeoptimal under new coefficients. It has been shown that the robust partial inverse spanning treeproblem can be formulated as a combinatorial linear program,while the robust partial inverseminimum cut problem and the robust partial inverse assignment problem can be solved by combinatorial strongly polynomial algorithms.     

On Center Cycles in Grid Graphs

Finding good cycles in graphs is a problem of great interest in graph theory as well as in locational analysis. We show that the center and median problems are NP-hard in general graphs. This result holds both for the variable cardinality case (i.e., all cycles of the graph are considered) and the fixed cardinality case (i.e., only cycles with a given cardinality p are feasible). Hence it is of interest to investigate special cases where the problem is solvable in polynomial time. In grid graphs, the variable cardinality case is, for instance, trivially solvable if the shape of the cycle can be chosen freely. If the shape is fixed to be a rectangle one can analyze rectangles in grid graphs with, in sequence, fixed dimension, fixed cardinality, and variable cardinality. In all cases a complete characterization of the optimal cycles and closed form expressions of the optimal objective values are given, yielding polynomial time algorithms for all cases of center rectangle problems. Finally, it is shown that center cycles can be chosen as rectangles for bounded cardinalities such that the center cycle problem in grid graphs is in these cases completely solved.     

Conic approximation to quadratic optimization with linear complementarity constraints

This paper proposes a conic approximation algorithm for solving quadratic optimization problems with linear complementarity constraints.We provide a conic reformulation and its dual for the original problem such that these three problems share the same optimal objective value. Moreover, we show that the conic reformulation problem is attainable when the original problem has a nonempty and bounded feasible domain. Since the conic reformulation is in general a hard problem, some conic relaxations are further considered. We offer a condition under which both the semidefinite relaxation and its dual problem become strictly feasible for finding a lower bound in polynomial time. For more general cases, by adaptively refining the outer approximation of the feasible set, we propose a conic approximation algorithm to identify an optimal solution or an \(\epsilon \)-optimal solution of the original problem. A convergence proof is given under simple assumptions. Some computational results are included to illustrate the effectiveness of the proposed algorithm.     

Minimum cut bases in undirected networks

Given an undirected, connected network G=(V,E) with weights on the edges, the cut basis problem is asking for a maximal number of linear independent cuts such that the sum of the cut weights is minimized. Surprisingly, this problem has not attained as much attention as another graph theoretic problem closely related to it, namely, the cycle basis problem. We consider two versions of the problem: the unconstrained and the fundamental cut basis problem.For the unconstrained case, where the cuts in the basis can be of an arbitrary kind, the problem can be written as a multiterminal network flow problem, and is thus solvable in strongly polynomial time. In contrast, the fundamental cut basis problem, where all cuts in the basis are obtained by deleting an edge, each from a spanning tree T, is shown to be NP-hard. In this proof, we also show that a tree which induces the minimum fundamental cycle basis is also an optimal solution for the minimum fundamental cut basis problem in unweighted graphs.We present heuristics, integer programming formulations and summarize first experiences with numerical tests.     

WELL-POSEDNESS OF A PARABOLIC INVERSE PROBLEM

1.IntroductionItiswellknownthatinverseproblemsinpartialdifferentialequations,mostofwhichhavenotyetbeensolveduptonow,remainasachallengeinappliedmathematics.Therefore,manymathematiciansstudiedvariousinverseproblemsforparabolicequa-tions.FOrasimplesurveywereferto[1,2,8,9]foridentifyingcoefficients,[7]foridentifyingboundaryvalues,[4,10]foridentifyingsourcetermsofparabolicequations.Wehavenotincludedalotofpapersconcerningthecomputationalmethodsusedforsolvinginverseparabolicproblems.Inthispaperthein…     

Inverse minimum flow problem

In this paper we consider the inverse minimum flow (ImF) problem, where lower and upper bounds for the flow must be changed as little as possible so that a given feasible flow becomes a minimum flow. A linear time and space method to decide if the problem has solution is presented. Strongly and weakly polynomial algorithms for solving the ImF problem are proposed. Some particular cases are studied and a numerical example is given.     

Weighted matching with pair restrictions

The weighted matroid parity problems for the matching matroid and gammoids are among the very few cases for which the weighted matroid parity problem is polynomial time solvable. In this work we extend these problems to a general revenue function for each pair, and show that the resulting problem is still solvable in polynomial time via a standard weighted matching algorithm. We show that in many other directions, extending our results further is impossible (unless P = NP). One consequence of the new polynomial time algorithm is that it demonstrates, for the first time, that a prize-collecting assignment problem with “pair restriction” is solved in polynomial time. The prize collecting assignment problem is a relaxation of the prize-collecting traveling salesman problem which requires, for any prescribed pair of nodes, either both nodes of the pair are matched or none of them are. It is shown that the prize collecting assignment problem is equivalent to the prize collecting cycle cover problem which is hence solvable in polynomial time as well.     

An inverse model for the most uniform problem

In this paper, we consider a kind of inverse model for the most uniform problem. This model has some practical background. It is shown that the model can be solved in polynomial time whenever an associated min-sum problem can be solved in polynomial time.     

A recognition problem for a vector alphabet generating a sequence with a quasiperiodic structure

An off-line recognition problem is analyzed for a vector alphabet generating sequences with quasiperiodic vector fragments that coincide with alphabet vectors. It is shown that the solution of this problem reduces to solving a special optimization problem. It is proved that the problem considered is solvable in polynomial time, and an algorithm for its exact solution is justified. The algorithm ensures the maximum likelihood recognition of a vector alphabet for the case of additive noise which is a Gaussian sequence of independent random values having an identical distribution.     

Two level hierarchical time minimizing transportation problem

A two level hierarchical balanced time minimizing transportation problem is considered in this paper. The whole set of source-destination links consists of two disjoint partitions namely Level-I links and Level-II links. Some quantity of a homogeneous product is first shipped from sources to destinations by Level-I decision maker using only Level-I links, and on its completion the Level-II decision maker transports the remaining quantity of the product in an optimal fashion using only Level-II links. Transportation is assumed to be done in parallel in both the levels. The aim is to find that feasible solution for Level-I decision maker corresponding to which the optimal feasible solution for Level-II decision maker is such that the sum of shipment times in Level-I and Level-II is the least. To obtain the global optimal feasible solution of this non-convex optimization problem, related balanced time minimizing transportation problems are defined. Based upon the optimal feasible solutions of these related problems, standard cost minimizing transportation problems are constructed whose optimal feasible solutions provide various pairs for shipment times for Level-I and Level-II decision makers. The best out of these pairs is finally selected. Being dependent upon solutions of a finite number of balanced time minimizing and cost minimizing transportation problems, the proposed algorithm is a polynomial bound algorithm. The developed algorithm has been implemented and tested on a variety of test problems and performance is found to be quite encouraging.