Finding an efficient solution to linear bilevel programming problem: An effective approach

Multilevel programming is developed to solve the decentralized problem in which decision makers (DMs) are often arranged within a hierarchical administrative structure. The linear bilevel programming (BLP) problem, i.e., a special case of multilevel programming problems with a two level structure, is a set of nested linear optimization problems over polyhedral set of constraints. Two DMs are located at the different hierarchical levels, both controlling one set of decision variables independently, with different and perhaps conflicting objective functions. One of the interesting features of the linear BLP problem is that its solution may not be Paretooptimal. There may exist a feasible solution where one or both levels may increase their objective values without decreasing the objective value of any level. The result from such a system may be economically inadmissible. If the decision makers of the two levels are willing to find an efficient compromise solution, we propose a solution procedure which can generate effcient solutions, without finding the optimal solution in advance. When the near-optimal solution of the BLP problem is used as the reference point for finding the efficient solution, the result can be easily found during the decision process.     

Finding duplicate rows in a linear programming model

We present an algorithm for identifying and disposing of duplicate rows in an LP matrix — that is, rows which are identical except for a scalar multiplier. This algorithm is a useful tool for debugging models as well as a means of improving efficiency. Implementation in a large scale mathematical programming system and computational experience are discussed.     

Corrected sequential linear programming for sparse minimax optimization

We present a new algorithm for nonlinear minimax optimization which is well suited for large and sparse problems. The method is based on trust regions and sequential linear programming. On each iteration a linear minimax problem is solved for a basic step. If necessary, this is followed by the determination of a minimum norm corrective step based on a first-order Taylor approximation. No Hessian information needs to be stored. Global convergence is proved. This new method has been extensively tested and compared with other methods, including two well known codes for nonlinear programming. The numerical tests indicate that in many cases the new method can find the solution in just as few iterations as methods based on approximate second-order information. The tests also show that for some problems the corrective steps give much faster convergence than for similar methods which do not employ such steps.Research supported partly by The Nordic Council of Ministers, The Icelandic Science Council, The University of Iceland Research Fund and The Danish Science Research Council.     

Finding normal solutions in piecewise linear programming

Letf: ⁿ (–, ] be a convex polyhedral function. We show that if any standard active set method for quadratic programming (QP) findsx(t)= arg min _x ¦x¦²/2+t f(x) for somet> 0, then its final working set defines a simple equality QP subproblem, whose Lagrange multiplier can be used both for testing ift is large enough forx(t) to coincide with the normal minimizer off, and for increasingt otherwise. The QP subproblem may easily be solved via the matrix factorizations used for findingx(t). This opens up the way for efficient implementations. We also give finite methods for computing the whole trajectory {x(t)} _t 0, minimizingf over an ellipsoid, and choosing penalty parameters inL ₁QP methods for strictly convex QP.This research was supported by the State Committee for Scientific Research under Grant 8S50502206.     

Intelligent gradient search in linear programming

The simplex algorithm is still the best known and most frequently used way to solve LP problems. Khachian has suggested a method to solve these problems in polynomial time. The average behaviour of his method, however, is still inferior to the modern simplex based LP codes. A new gradient based approach which also has polynomial worst-case behaviour has been suggested by Karmarkar. This method was modified, programmed and compared with other available LP codes. It is shown that the numerical efficiency of Karmarkar's method compares favourably with other LP codes, particularly for problems with high numbers of variables and few constraints.     

A linear programming approach for linear multi-level programming problems

In an earlier paper, we proposed a modified fuzzy programming method to handle higher level multi-level decentralized programming problems (ML(D)PPs). Here we present a simple and practical method to solve the same. This method overcomes the subjectivity inherent in choosing the tolerance values and the membership functions. We consider a linear ML(D)PP and apply linear programming (LP) for the optimization of the system in a supervised search procedure, supervised by the higher level decision maker (DM). The higher level DM provides the preferred values of the decision variables under his control to enable the lower level DM to search for his optimum in a narrower feasible space. The basic idea is to reduce the feasible space of a decision variable at each level until a satisfactory point is sought at the last level.     

Unconstrained convex programming approach to linear programming

Recently, Fang proposed approximating a linear program in the Karmarkar standard form by adding an entropic barrier function to the objective function and derived an unconstrained dual concave program. We present in this note a necessary and sufficient condition for the existence of a dual optimal solution to the perturbed problem. In addition, a sharp upper bound of error estimation in this approximation scheme is provided.     

An integer linear programming approach and a hybrid variable neighborhood search for the car sequencing problem

In this paper we present two major approaches to solve the car sequencing problem, in which the goal is to find an optimal arrangement of commissioned vehicles along a production line with respect to constraints of the form “no more than l_ccars are allowed to require a component c in any subsequence of m_cconsecutive cars”. The first method is an exact one based on integer linear programming (ILP). The second approach is hybrid: it uses ILP techniques within a general variable neighborhood search (VNS) framework for examining large neighborhoods. We tested the two methods on benchmark instances provided by CSP_LIB and the automobile manufacturer RENAULT for the ROADEF Challenge 2005. These tests reveal that our approaches are competitive to previous reported algorithms. For the CSP_LIB instances we were able to shorten the required computation time for reaching and proving optimality. Furthermore, we were able to obtain tight bounds on some of the ROADEF instances. For two of these instances the proposed ILP-method could provide new optimality proofs for already known solutions. For the VNS, the individual contributions of the used neighborhoods are also experimentally analyzed. Results highlight the significant impact of each structure. In particular the large ones examined using ILP techniques enhance the overall performance significantly, so that the hybrid approach clearly outperforms variants including only commonly defined neighborhoods.     

Finding all maximal efficient faces in multiobjective linear programming

An algorithm for finding the whole efficient set of a multiobjective linear program is proposed. From the set of efficient edges incident to a vertex, a characterization of maximal efficient faces containing the vertex is given. By means of the lexicographic selection rule of Dantzig, Orden and Wolfe, a connectedness property of the set of dual optimal bases associated to a degenerate vertex is proved. An application of this to the problem of enumerating all the efficient edges incident to a degenerate vertex is proposed. Our method is illustrated with numerical examples and comparisons with Armand—Malivert's algorithm show that this new algorithm uses less computer time.     

Duality on a nondifferentiable minimax fractional programming

We establish the necessary and sufficient optimality conditions on a nondifferentiable minimax fractional programming problem. Subsequently, applying the optimality conditions, we constitute two dual models: Mond-Weir type and Wolfe type. On these duality types, we prove three duality theorems??weak duality theorem, strong duality theorem, and strict converse duality theorem.     

Dynamic programming and the backpacker''s linear search problem

A backpacker approaches a road with a marker on it, desirous of finding the marker but having only a rough idea of where it is located. It is well known among backpackers that it is best to aim either right or left of the marker, since otherwise it will not be clear which way to turn upon reaching the road. The problem of deciding exactly where to aim can be formalized as a modification of the Linear Search Problem. This paper does so, and also discusses dynamic programming as a solution method.     

On trend estimation of time-series: a simple linear programming approach

This paper presents a simple linear programme for the solution of the trend-estimation problem in time-series. We studied the trade-off between two important properties of the trend component: smoothness and fidelity (closeness to the data). The linear programming solution is a monotone sequence optimizing some (weighted) combination of both properties. A single coefficient in the objective function is user-dependent and represents the user’s preference with respect to the two properties. The method is illustrated on several empirical series.     

Intuitionistic fuzzy linear programming and duality: a level sets approach

The paper is concerned with linear programming problems whose input data may be intuitionistic fuzzy (IF) while the values of variables are always real numbers. We propose rather general approach to this type of problems based on level sets, and present recent results for problems in which the notions of feasibility and optimality are based on the IF relations. Special attention is devoted to the weak and strong duality.     

Experimental results on Hillier's linear search

This paper reports an empirical discovery in integer programming. A version of the branch-and-bound approach is used as a control and tested against the same algorithm augmented by the use of Hillier's linear search performed at every node of the search tree. It is shown that the imbedded linear search locates solutions within 1%, and solutions within 5% of the theoretical optimum, which in fact can be seen to have this proximity to the theoretical optimum at the time of termination of computation, over ten times faster than the control.     

An integer linear programming approach for bilinear integer programming

We introduce a new Integer Linear Programming (ILP) approach for solving Integer Programming (IP) problems with bilinear objectives and linear constraints. The approach relies on a series of ILP approximations of the bilinear IP. We compare this approach with standard linearization techniques on random instances and a set of real-world product bundling problems.     

Selecting tolerances in chance-constrained programming: A multiple objective linear programming approach

An importance issue concerning the practical application of chance-constrained programming is the lack of a rational method for choosing risk levels or tolerances on the chance constraints. While there has also been much recent debate on the relationship, equivalence, usefulness, and other characteristics of chance-constrained programming relative to stochastic programming with recourse, this paper focuses on the problem of improving the selection of tolerances within the chance-constrained framework. An approach is presented, based on multiple objective linear programming, which allows the decision maker to be more involved in the tolerance selection process, but does not demand a priori decisions on appropriate tolerances. An example is presented which illustrates the approach.     

A linear programming approach to test efficiency in multi-objective linear fractional programming problems

In a multi-objective linear fractional programming problem (MOLFPP), it is often useful to check the efficiency of a given feasible solution, and if the solution is efficient, it is useful to check strong or weak efficiency. In this paper, by applying a geometrical interpretation, a linear programming approach is achieved to test weak efficiency. Also, in order to test strong efficiency for a given weakly efficient point, a linear programming approach is constructed.     

Optimal cargo allocation on board a plane: a sequential linear programming approach

We examine a case study of an airline company whose problem is to plan cargo allocations on board a plane. Given the volume, weight, and structural constraints, the problem of finding the optimal load layout is formulated as a fractional programming problem. An algorithm is suggested to solve the linearized problem as a sequence of linear programming problems whose optimal solutions converge to the optimum (with a predetermined level of tolerance).     

Continuous-time linear programming problems revisited: A perturbation approach

The class of continuous-time linear programming problems under the assumption that the constraints are satisfied almost everywhere in the time interval [0,?T]?is taken into account in this article. Under this assumption, its corresponding discretized problems cannot be formulated by equally dividing the time interval [0,?T]?as subintervals of [0,?T]?. In this article, we also introduce the perturbed continuous-time linear programming problems to prove the strong duality theorem when the constraints are assumed to be satisfied a.e. in [0,?T]?.