Showing papers on "Linear-fractional programming published in 2009"

Linear and Nonlinear Optimization

Abstract: Preface Part I. Basics: 1. Optimization models 2. Fundamentals of optimization 3. Representation of linear constraints Part II. Linear Programming: 4. Geometry of linear programming 5. The simplex method 6. Duality and sensitivity 7. Enhancements of the simplex method 8. Network problems 9. Computational complexity of linear programming 10. Interior-point methods of linear programming Part III. Unconstrained Optimization: 11. Basics of unconstrained optimization 12. Methods for unconstrained optimization 13. Low-storage methods for unconstrained problems Part IV. Nonlinear Optimization: 14. Optimality conditions for constrained problems 15. Feasible-point methods 16. Penalty and barrier methods Part V. Appendices: Appendix A. Topics from linear algebra Appendix B. Other fundamentals Appendix C. Software Bibliography Index.

A generic dynamic programming Matlab function

Abstract: This paper introduces a generic dynamic programming function for Matlab. This function solves discretetime optimal-control problems using Bellman's dynamic programming algorithm. The function is implemented such that the user only needs to provide the objective function and the model equations. The function includes several options for solving optimal-control problems. The model equations can include several state variables and input variables. Furthermore, the model equations can be time-variant and include time-variant state and input constraints. The syntax of the function is explained using two examples. The first is the well-known Lotka-Volterra fishery problem and the second is a parallel hybrid-electric vehicle optimization problem.

Reformulations in Mathematical Programming: A Computational Approach

Abstract: Mathematical programming is a language for describing optimization problems; it is based on parameters, decision variables, objective function(s) subject to various types of constraints. The present treatment is concerned with the case when objective(s) and constraints are algebraic mathematical expressions of the parameters and decision variables, and therefore excludes optimization of black-box functions. A reformulation of a mathematical program P is a mathematical program Q obtained from P via symbolic transformations applied to the sets of variables, objectives and constraints. We present a survey of existing reformulations interpreted along these lines, some example applications, and describe the implementation of a software framework for reformulation and optimization.

Optimal value range in interval linear programming

Abstract: We deal with the linear programming problem in which input data can vary in some given real compact intervals. The aim is to compute the exact range of the optimal value function. We present a general approach to the situation the feasible set is described by an arbitrary linear interval system. Moreover, certain dependencies between the constraint matrix coefficients can be involved. As long as we are able to characterize the primal and dual solution set (the set of all possible primal and dual feasible solutions, respectively), the bounds of the objective function result from two nonlinear programming problems. We demonstrate our approach on various cases of the interval linear programming problem (with and without dependencies).

Interior-Point Methods for Linear Programming

Abstract: We studied two pivoting algorithms for linear programming in Chapter 4. These algorithms are finite and the simplex method in particular is known to be very efficient practically. Yet, there is no known pivoting algorithm that is polynomial. There are pathological examples of linear programs for which the simplex method (or the criss-cross method) behaves badly, i.e., generates an exponential number of pivots to find an optimal solution. There is a famous example due to Klee and Minty, [28, 35]. An explicit description is given by max x n subject to 0 ≤ x 1 ≤ 1, (10.1) where 0 < < 1/2.

A branch and bound algorithm for globally solving a class of nonconvex programming problems

Abstract: A branch and bound algorithm is proposed for globally solving a class of nonconvex programming problems (NP). For minimizing the problem, linear lower bounding functions (LLBFs) of objective function and constraint functions are constructed, then a relaxation linear programming is obtained which is solved by the simplex method and which provides the lower bound of the optimal value. The proposed algorithm is convergent to the global minimum through the successive refinement of linear relaxation of the feasible region and the solutions of a series of linear programming problems. And finally the numerical experiment is reported to show the feasibility and effectiveness of the proposed algorithm.

Temporal planning in domains with linear processes

Abstract: We consider the problem of planning in domains with continuous linear numeric change. Such change cannot always be adequately modelled by discretisation and is a key facet of many interesting problems. We show how a forward-chaining temporal planner can be extended to reason with actions with continuous linear effects. We extend a temporal planner to handle numeric values using linear programming. We show how linear continuous change can be integrated into the same linear program and we discuss how a temporal-numeric heuristic can be used to provide the search guidance necessary to underpin continuous planning. We present results to show that the approach can effectively handle duration-dependent change and numeric variables subject to continuous linear change.

An interactive method of tackling uncertainty in interval multiple objective linear programming

Abstract: Mathematical programming models for decision support must explicitly take account of the treatment of the uncertainty associated with the model coefficients along with multiple and conflicting objective functions Interval programming just assumes that information about the variation range of some (or all) of the coefficients is available In this paper, we propose an interactive approach for multiple objective linear programming problems with interval coefficients that deals with the uncertainty in all the coefficients of the model The presented procedures provide a global view of the solutions in the best and worst case coefficient scenarios and allow performing the search for new solutions according to the achievement rates of the objective functions regarding both the upper and lower bounds The main goal is to find solutions associated with the interval objective function values that are closer to their corresponding interval ideal solutions It is also possible to find solutions with non-dominance relations regarding the achievement rates of the upper and lower bounds of the objective functions considering interval coefficients in the whole model

Solving Bilevel Linear Programs Using Multiple Objective Linear Programming

Abstract: We present an algorithm for solving bilevel linear programs that uses simplex pivots on an expanded tableau The algorithm uses the relationship between multiple objective linear programs and bilevel linear programs along with results for minimizing a linear objective over the efficient set for a multiple objective problem Results in multiple objective programming needed are presented We report computational experience demonstrating that this approach is more effective than a standard branch-and-bound algorithm when the number of leader variables is small

Branching approaches for integrated vehicle and crew scheduling

Abstract: The integrated multi-depot vehicle and crew scheduling problem simultaneously builds vehicle blocks and crew duties. We present an integer mathematical formulation that combines a multi-commodity flow model with a mixed set partitioning/covering model. We propose solution approaches that start by solving the linear programming relaxation of the model. Whenever the resulting linear programming solution is not integer, three branching alternative strategies can be applied: a branch-and-bound algorithm and two branch-and-price schemes. The branch-and-bound algorithm performs branching over the set of feasible crew duties generated while solving the linear relaxation. In the first branch-and-price scheme the linear programming relaxation is solved approximately, while in the second one it is solved exactly. Computational experience is reported over two different types of problems: randomly generated data publicly available for benchmarking in the Internet and data from a bus company operating in Lisbon.

Set-valued duality theory for multiple objective linear programs and application to mathematical finance

Abstract: We develop a duality theory for weakly minimal points of multiple objective linear programs which has several advantages in contrast to other theories. For instance, the dual variables are vectors rather than matrices and the dual feasible set is a polyhedron. We use a set-valued dual objective map the values of which have a very simple structure, in fact they are hyperplanes. As in other set-valued (but not in vector-valued) approaches, there is no duality gap in the case that the right-hand side of the linear constraints is zero. Moreover, we show that the whole theory can be developed by working in a complete lattice. Thus the duality theory has a high degree of analogy to its classical counterpart. Another important feature of our theory is that the infimum of the set-valued dual problem is attained in a finite set of vertices of the dual feasible domain. These advantages open the possibility of various applications such as a dual simplex algorithm. Exemplarily, we discuss an application to a Markowitz-type bicriterial portfolio optimization problem where the risk is measured by the Conditional Value at Risk.

A Secure Revised Simplex Algorithm for Privacy-Preserving Linear Programming

Abstract: Linear programming is one of the most widely applied solutions to optimization problems. This paper presents a privacy-preserving solution to linear programming for two parties when the cost, or objective, function is known only to one party, and the constraint equations are known only to the other party. An algorithm based on Revised Simplex is given that ensures that neither party gains access to the other’s private information. While this has been proposed before, our solution is significantly more efficient for the given data distribution. This enables collaboration among companies in many domains, enhancing efficiency while maintaining competitiveness.

Interval-parameter Fuzzy-stochastic Semi-infinite Mixed-integer Linear Programming for Waste Management under Uncertainty

Abstract: An interval-parameter fuzzy-stochastic semi-infinite mixed-integer linear programming (IFSSIP) method is developed for waste management under uncertainties. The IFSSIP method integrates the fuzzy programming, chance-constrained programming, integer programming and interval semi-infinite programming within a general optimization framework. The model is applied to a waste management system with three disposal facilities, three municipalities, and three periods. Compared with the previous methods, IFSSIP have two major advantages. One is that it can help generate solutions for the stable ranges of the decision variables and objective function value under fuzzy satisfaction degree and different levels of probability of violating constraints, which are informative and flexible for solution users to interpret/justify. The other is that IFSSIP can not only handle uncertainties through constructing fuzzy and random parameter, but also reflect dynamic features of the system conditions through interval function of time over the planning horizon. By comparing IFSSIP with interval-parameter mixed-integer linear semi-infinite programming and parametric programming, the IFSSIP method is more reasonable than others.

A primal-dual interior-point linear programming algorithm for MPC

Abstract: Constrained optimal control problems for linear systems with linear constraints and an objective function consisting of linear and l1-norm terms can be expressed as linear programs. We develop an efficient primal-dual interior point algorithm for solution of such linear programs. The algorithm is implemented in Matlab and its performance is compared to an active set based LP solver and linprog in Matlab's optimization toolbox. Simulations demonstrate that the new algorithm is more than one magnitude faster than the other LP algorithms applied to this problem.

A penalty function algorithm with objective parameters for nonlinear mathematical programming

Abstract: In this paper, we present a penalty function with objective parameters for inequality constrained optimization problems. We prove that this type of penalty functions has good properties for helping to solve inequality constrained optimization problems. Moreover, based on the penalty function, we develop an algorithm to solve the inequality constrained optimization problems and prove its convergence under some conditions. Numerical experiments show that we can obtain a satisfactorily approximate solution for some constrained optimization problems as the same as the exact penalty function.

Nonlinear Pseudo-Boolean Optimization: Relaxation or Propagation?

Abstract: Pseudo-Boolean problems lie on the border between satisfiability problems, constraint programming, and integer programming In particular, nonlinear constraints in pseudo-Boolean optimization can be handled by methods arising in these different fields: One can either linearize them and work on a linear programming relaxation or one can treat them directly by propagation In this paper, we investigate the individual strengths of these approaches and compare their computational performance Furthermore, we integrate these techniques into a branch-and-cut-and-propagate framework, resulting in an efficient nonlinear pseudo-Boolean solver

A method of network programming in problems of nonlinear optimization

Abstract: A method of network programming for solving problems of nonlinear optimization is used. A notion of dual problem is introduced. It is proved that a dual problem is a problem of convex programming. Necessary and sufficient conditions for optimality of dual problem of integer linear programming are obtained.

Global optimization of multi-parametric MILP problems

Abstract: In this paper, we present a novel global optimisation approach for the general solution of multi-parametric mixed integer linear programs (mp-MILPs). We describe an optimisation procedure which iterates between a (master) mixed integer nonlinear program and a (slave) multi-parametric program. Moreover, we explain how to overcome the presence of bilinearities, responsible for the non-convexity of the multi-parametric program, in two classes of mp-MILPs, with (i) varying parameters in the objective function and (ii) simultaneous presence of varying parameters in the objective function and the right-hand side of the constraints. Examples are provided to illustrate the solution steps.

Solving Fuzzy Linear Programming Problem as Multi Objective Linear Programming Problem

Abstract: This paper proposes the method to the solution of fuzzy linear programming problem with the help of multi objective constrained linear programming problem when constraint matrix and the cost coefficients of an objective function are fuzzy in nature. Also proved is that the solutions are independent of weights. Key words: Fuzzy linear programming problems, multi objective, fuzzy sets.

A mixed 0-1 linear programming formulation for the exact solution of the minimum linear arrangement problem

Abstract: We are concerned with the exact solution of a graph optimization problem known as minimum linear arrangement (MinLA). Define the length of each edge of a graph with respect to a linear ordering of the graph vertices. Then, the MinLA problem asks for a vertex ordering that minimizes the sum of edge lengths. MinLA has several practical applications and is NP-Hard. We present a mixed 0-1 linear programming formulation of the problem, which led to fast optimal solutions for dense graphs of sizes up to n = 23.