Showing papers on "Linear-fractional programming published in 2006"
Book•
05 Oct 2006
TL;DR: This chapter discusses the Simplex Method, a very simple and straightforward way of programming that can be applied to Integer Programming and LP Relaxation.
Abstract: What Is It, and What For?- Examples- Integer Programming and LP Relaxation- Theory of Linear Programming: First Steps- The Simplex Method- Duality of Linear Programming- Not Only the Simplex Method- More Applications- Software and Further Reading
358 citations
Book•
20 Apr 2006
TL;DR: Theorem A: solvability of systems of interval linear equations and inequalities and optimization problems over max-algebras.
Abstract: Matrices.- Solvability of systems of interval linear equations and inequalities.- Interval linear programming.- Linear programming with set coefficients.- Fuzzy linear optimization.- Interval linear systems and optimization problems over max-algebras.
306 citations
TL;DR: By use of a linear ranking function, the dual of fuzzy number linear programming primal problems is introduced and several duality results are presented.
147 citations
TL;DR: The proposed approach to find the optimal solutions of a class of fuzzy linear programming problems called fully fuzzified linear programming (FFLP), where all decision parameters and variables are fuzzy numbers, is constructed on the basis of comparison of mean and standard deviation of fuzzy numbers.
Abstract: In this paper, we propose a two-phase approach to find the optimal solutions of a class of fuzzy linear programming problems called fully fuzzified linear programming (FFLP), where all decision parameters and variables are fuzzy numbers. Our approach is constructed on the basis of comparison of mean and standard deviation of fuzzy numbers. In this approach, the first phase maximizes the possibilistic mean value of fuzzy objective function and obtains a set of feasible solutions. The second phase minimizes the standard deviation of the original fuzzy objective function, by considering all basic feasible solutions obtained at the end of the first phase. The advantage of the proposed approach is its simplicity in programming and computation. Moreover, we also generalize the concept of linear programming duality and extend the duality as well as the weak duality theory to FFLP.
102 citations
TL;DR: This paper shows that the point location problem can be written as an additively weighted nearest neighbour search that can be solved in time linear in the dimension of the state space and logarithmic in the number of regions.
73 citations
21 May 2006
TL;DR: This work presents the first randomized polynomial-time simplex algorithm for linear programming, reducing the input linear program to a special form in which it merely needs to certify boundedness, and runs the shadow-vertex simplex method with a random right-hand-side vector.
Abstract: We present the first randomized polynomial-time simplex algorithm for linear programming. Like the other known polynomial-time algorithms for linear programming, its running time depends polynomially on the number of bits used to represent its input.We begin by reducing the input linear program to a special form in which we merely need to certify boundedness. As boundedness does not depend upon the right-hand-side vector, we run the shadow-vertex simplex method with a random right-hand-side vector. Thus, we do not need to bound the diameter of the original polytope.Our analysis rests on a geometric statement of independent interest: given a polytope A x ≤ b in isotropic position, if one makes a polynomially small perturbation to b then the number of edges of the projection of the perturbed polytope onto a random 2-dimensional subspace is expected to be polynomial.
66 citations
TL;DR: An algorithm and few concrete examples are presented in order to optimize linear objective function and the feasible solution set with two schemes and study relationship between maximum and minimum points.
59 citations
TL;DR: A bound is established on the performance of the resulting policy that scales gracefully with the number of states without imposing the strong Lyapunov condition required by its counterpart in de Farias and Van Roy.
Abstract: We introduce a new algorithm based on linear programming for optimization of average-cost Markov decision processes (MDPs). The algorithm approximates the differential cost function of a perturbed MDP via a linear combination of basis functions. We establish a bound on the performance of the resulting policy that scales gracefully with the number of states without imposing the strong Lyapunov condition required by its counterpart in de Farias and Van Roy [de Farias, D. P., B. Van Roy. 2003. The linear programming approach to approximate dynamic programming. Oper. Res.51(6) 850--865]. We investigate implications of this result in the context of a queueing control problem.
59 citations
TL;DR: In this article, an interval fuzzy two-stage stochastic mixed-integer linear programming (IFTSIP) method is developed for planning waste-management systems under uncertainty, which has advantages in uncertainty reflection, policy analysis and capacity expansion.
Abstract: In this study, an interval fuzzy two-stage stochastic mixed-integer linear programming (IFTSIP) method is developed for planning waste-management systems under uncertainty. As a new extension of mathematical programming methods, the developed IFTSIP approach has advantages in uncertainty reflection, policy analysis and capacity expansion. Methods of two-stage stochastic programming and interval fuzzy linear programming are introduced to a mixed-integer linear programming framework to effectively tackle uncertainties described in terms of probability density functions, fuzzy membership functions and discrete intervals. The IFTSIP method can incorporate pre-defined waste management policies directly into its optimization process and can be used for analyzing various policy scenarios associated with different levels of economic penalties when the promised policy targets are violated. Moreover, it can obtain optimal decisions of capacity-expansion schemes for waste management within a multi-stage context. The...
49 citations
TL;DR: This paper focuses on these kind problems in which the solutions region is the fuzzy relation equation with max-prod composition and the objective function is linear and has given an algorithm to optimize the linear objective function on such these regions.
42 citations
DOI•
01 Jan 2006TL;DR: A mixed- integer linear programming solution to coordinate multiple heterogenenous robots for detecting and controlling multiple regions of interest in an unknown environment and various extensions to objective function and constraints to show the flexibility of mixed-integer linear programming formulation is introduced.
Abstract: Multi-robot systems require efficient and accurate planning in order to perform mission-critical tasks. This paper introduces a mixed-integer linear programming solution to coordinate multiple heterogenenous robots for detecting and controlling multiple regions of interest in an unknown environment. The objective function contains four basic requirements of a multi-robot system serving this purpose: control regions of interest, provide communication between robots, control maximum area and detect regions of interest. Our solution defines optimum locations of robots in order to maximize the objective function while efficiently satisfying some constraints such as avoiding obstacles and staying within the speed capabilities of the robots. We implemented and tested our approach under realistic scenarios. We showed various extensions to objective function and constraints to show the flexibility of mixed-integer linear programming formulation. Type of Report: Other Department of Computer Science & Engineering Washington University in St. Louis Campus Box 1045 St. Louis, MO 63130 ph: (314) 935-6160 Mixed-Integer Linear Programming Solution to Multi-Robot Task Allocation Problem Nuzhet Atay Department of Computer Science and Engineering Washington University in St. Louis Email: atay@cse.wustl.edu Burchan Bayazit Department of Computer Science and Engineering Washington University in St. Louis Email: bayazit@cse.wustl.edu Abstract— Multi-robot systems require efficient and accurate planning in order to perform mission-critical tasks. This paper introduces a mixed-integer linear programming solution to coordinate multiple heterogenenous robots for detecting and controlling multiple regions of interest in an unknown environment. The objective function contains four basic requirements of a multi-robot system serving this purpose: control regions of interest, provide communication between robots, control maximum area and detect regions of interest. Our solution defines optimum locations of robots in order to maximize the objective function while efficiently satisfying some constraints such as avoiding obstacles and staying within the speed capabilities of the robots. We implemented and tested our approach under realistic scenarios. We showed various extensions to objective function and constraints to show the flexibility of mixed-integer linear programming formulation. Multi-robot systems require efficient and accurate planning in order to perform mission-critical tasks. This paper introduces a mixed-integer linear programming solution to coordinate multiple heterogenenous robots for detecting and controlling multiple regions of interest in an unknown environment. The objective function contains four basic requirements of a multi-robot system serving this purpose: control regions of interest, provide communication between robots, control maximum area and detect regions of interest. Our solution defines optimum locations of robots in order to maximize the objective function while efficiently satisfying some constraints such as avoiding obstacles and staying within the speed capabilities of the robots. We implemented and tested our approach under realistic scenarios. We showed various extensions to objective function and constraints to show the flexibility of mixed-integer linear programming formulation.
01 Jan 2006
TL;DR: In this article, the authors define the value of a feasible solution of a linear program (3.3) and introduce the value value for a given linear program, where the value is defined as the ratio of the probability that the problem is feasible and infeasible.
Abstract: (notice the use of the upper case in "Min" to denote a problem in contrast to "min" which denotes minimum when applicable). A vector x satisfying (3.2) is called a feasible solution of (3.3). A problem (3.3) having a feasible solution is said to be feasible, and infeasible in the opposite case. Hence, the problem (3.3) is feasible if and only if the system Ax = b is feasible in the terminology of Section 2.4. For a given linear program (3.3) we introduce the value
TL;DR: Convergent property of the presented algorithm is proved and numerical results are given to show the feasibility of the proposed algorithm.
TL;DR: A branch and bound algorithm is proposed for globally solving the sum of linear ratios problem with coefficients by utilizing an equivalent problem and linearization technique, and the initial nonconvex programming problem is reduced to a sequence of linear programming problems.
TL;DR: An outcome-space finite algorithm for solving linear multiplicative programming, in each iteration of which a convex quadratic programming is only solved, which can be shown by the numerical results that the proposed algorithm is effective and computational results can be gained in short time.
TL;DR: Two approximate dynamic programming methods to optimize the distribution operations of a company manufacturing a certain product at multiple production plants and shipping it to different customer locations for sale are proposed.
Abstract: We propose two approximate dynamic programming methods to optimize the distribution operations of a company manufacturing a certain product at multiple production plants and shipping it to different customer locations for sale. We begin by formulating the problem as a dynamic program. Our first approximate dynamic programming method uses a linear approximation of the value function and computes the parameters of this approximation by using the linear programming representation of the dynamic program. Our second method relaxes the constraints that link the decisions for different production plants. Consequently, the dynamic program decomposes by the production plants. Computational experiments show that the proposed methods are computationally attractive, and in particular, the second method performs significantly better than standard benchmarks.
TL;DR: The proposed algorithm is convergent to the globally optimal solution of MPE by means of the subsequent solutions of a series of linear programming problems.
TL;DR: A cutting plane technique may be used to compute all the efficient solutions of the last model leaving the decision maker to choose a solution according to his preferences.
TL;DR: An extension of the simplex algorithm, where each successive relaxed problem is obtained by adding a single constraint chosen from the constraints violated by the solution to the previous relaxed problem, maximizes the cosine of the angle that the gradient of any violated constraint forms with thegradient of the objective function.
Abstract: An extension of the simplex algorithm is presented. For a given linear programming problem, a sequence of relaxed linear programming problems is solved until a solution to the original problem is reached. Each successive relaxed problem is obtained from the previous one by adding a single constraint chosen from the constraints violated by the solution to the previous relaxed problem. This added constraint maximizes the cosine of the angle that the gradient of any violated constraint forms with the gradient of the objective function. In other words, each successive relaxed problem is obtained by adding the violated constraint most parallel to the objective function. The proposed algorithm terminates when no constraints are violated. Preliminary results indicate that this cosine simplex algorithm reduces both the number of simplex iterations and the number of computations at each iteration.
01 Jan 2006
TL;DR: In this paper, the optimality conditions for a linear parametric optimization problem with parameters b in the right hand side and c in the objective function are checked in polynomial time.
Abstract: Let Ψ(b, c) be the solution set mapping of a linear parametric optimization problem with parameters b in the right hand side and c in the objective function. Then, given a point x0 we search for parameter values b and c as well as for an optimal solution x ∈ Ψ (b, c) such that ‖x − x0‖ is minimal. This problem is formulated as a bilevel programming problem. Focus in the paper is on optimality conditions for this problem. We show that, under mild assumptions, these conditions can be checked in polynomial time.
Proceedings Article•
13 Jul 2006TL;DR: This paper addresses several questions to enhance the applicability of recent work on approximate linear programming techniques for first-order Markov Decision Processes, and proposes answers to how to decompose intractable problems with universally quantified rewards into tractable subproblems.
Abstract: Recent work on approximate linear programming (ALP) techniques for first-order Markov Decision Processes (FOMDPs) represents the value function linearly w.r.t. a set of first-order basis functions and uses linear programming techniques to determine suitable weights. This approach offers the advantage that it does not require simplification of the first-order value function, and allows one to solve FOMDPs independent of a specific domain instantiation. In this paper, we address several questions to enhance the applicability of this work: (1) Can we extend the first-order ALP framework to approximate policy iteration and if so, how do these two algorithms compare? (2) Can we automatically generate basis functions and evaluate their impact on value function quality? (3) How can we decompose intractable problems with universally quantified rewards into tractable subproblems? We propose answers to these questions along with a number of novel optimizations and provide a comparative empirical evaluation on problems from the ICAPS 2004 Probabilistic Planning Competition.
TL;DR: A simple and reliable test is introduced to establish whether a linear fractional goal programming problem has solutions that verify all goals and, if so, how to find them by solving a linear programming problem.
Posted Content•
TL;DR: A polynomial-sized linear programming formulation of the Traveling Salesman Problem (TSP) is presented and the proposed linear program is a network flow-based model.
Abstract: In this paper, we present a polynomial-sized linear programming formulation of the Traveling Salesman Problem (TSP). The proposed linear program is a network flow-based model. Numerical implementation issues and results are discussed. (The exposition and proofs are much more detailed in an edition which I wrote in collaboration with Dr. M.H. Karwan in 2012-2014 . That edition is available at this http URL)
TL;DR: The feasible solution set of the fuzzy relation equations is characterized, and an efficient solution procedure for solving such problems appears to be necessary.
TL;DR: The aim of this article is to consider a new linear programming and two goal programming models for two-group classification problems that perform well both in separating the groups and the group-membership predictions of new objects.
TL;DR: The paper describes how CP can be used to exploit linear programming within different kinds of hybrid algorithm, and how it can enhance techniques such as Lagrangian relaxation, Benders decomposition and column generation.
Abstract: This paper presents constraint programming (CP) as a natural formalism for modelling problems, and as a flexible platform for solving them. CP has a range of techniques for handling constraints including several forms of propagation and tailored algorithms for global constraints. It also allows linear programming to be combined with propagation and novel and varied search techniques which can be easily expressed in CP. The paper describes how CP can be used to exploit linear programming within different kinds of hybrid algorithm. In particular it can enhance techniques such as Lagrangian relaxation, Benders decomposition and column generation.
TL;DR: In this paper, the problem of minimizing a separable nonlinear objective function under linear constraints is considered and a systematic approach is proposed to obtain an approximately globally optimal solution via piecewise linear approximation.
01 Jan 2006
TL;DR: The basic framework of a primal-dual interior point method is given, and the numerical issues involved in calculating the search direction in each iteration are considered, including the use of factorization methods and/or preconditioned conjugate gradient methods.
Abstract: We discuss interior point methods for large-scale linear programming, with an emphasis on methods that are useful for problems arising in telecommunications. We give the basic framework of a primal-dual interior point method, and consider the numerical issues involved in calculating the search direction in each iteration, including the use of factorization methods and/or preconditioned conjugate gradient methods. We also look at interior point column generation methods which can be used for very large scale linear programs or for problems where the data is generated only as needed.
11 Sep 2006
TL;DR: It is shown that this optimization problem can be divided into two subproblems by separating the decision variables associated with negative and nonnegative coefficients in the objective function.
Abstract: In this paper minimizing a linear objective function subject to a continuous max-i-norm fuzzy relational equation is considered. Our contributions are two folds. First, We show that this optimization problem can be divided into two subproblems by separating the decision variables associated with negative and nonnegative coefficients in the objective function. A 0-1 integer programming problem as an equivalent model can be derived for our current study. Our second contribution is to present an efficient procedure for solving a subclass of the max-t-norm-type optimization problems in which the max-product-type one is a special case, yet, the max-min-type one is not included. Numerical examples are provided to illustrate the procedure.
01 Jan 2006
TL;DR: In this paper, an iterative approach for topology optimization of load-carrying structures is presented, which is guaranteed to find a local optimum of the original problem, but not necessarily a global optimum.
Abstract: We present a new iterative approach for topology optimization of load-carrying structures. In each iteration an integer linear programming problem is generated and solved. The method is guaranteed to find a local optimum of the original problem, but not necessarily a global optimum. Numerical results for some stress constrained problems are presented.