Zero-one Linear Programming Problem Research Articles

A ZEROONE PROGRAMMING MODEL FOR RNA STRUCTURES WITH ARCLENGTH≥4

In this paper, we consider RNA structures with arc-length 4  . First, we represent these structures as matrix models and zero-one linearprogramming problems. Then, we obtain an optimal solution for this problemusing an implicit enumeration method. The optimal solution corresponds toan RNA structure with the maximum number of hydrogen bonds.

Conflict graphs in implicit enumeration

A zero-one linear program is a global discrete optimization problem. Namely, it is a linear programming problem with zero-one variables. The real relaxation of a zero-one program is again a continuous linear programming problem. Simply, the zero-one variables are replaced by continuous variables varying between zero and one independently of each other. If the real relaxation, as a continuous problem, solved with the simplex method happens to have a zero-one optimal solution, then this particular solution is also an optimal solution of the original zero-one programming problem. Suppose that x1,…,xn are all the variables of a given zero-one linear programming problem. For convenience we introduce further variables y1,…,yn defined by y1=1−x1,…, yn=1−xn respectively. For a unified notation the variables x1,…,xn and y1,…, yn together will be denoted by z1,…,z2n. These variables will play the roles the nodes of the conflict graph Г associated with the given zero-one linear programming problem. Fixing the variables zi and zj both to be equal to one simultaneously reduces the number of variables in the problem. If the new smaller optimization problem does not have any feasible solution, then the nodes zi and zj will be connected by an edge in the graph Γ. The graph Γ simply records that there is a conflict between the assignments zi =1 and zj =1 which explains the name conflict graph. In other words the non-directed edge between the nodes zi and zj codes the fact that the inequality zi+zj≤1holds. The conflict graph was designed to generate additional constraints the so-called valid inequalities or cuts in order to expedite the solution of the zero-one linear program via its real relaxation. This particular solution strategy is aptly named the branch and cut method. It will be shown that the conflict graph besides its intended use can also be applied in another solution technique the method of implicit enumeration. We will illustrate by examples that the conflict graph provides us with rules to prune the search tree that are not offered by the commonly applied pruning rules. In addition the computations involved can easily be organized into a highly parallel computational scheme.

Robust portfolio modeling with incomplete cost information and project interdependencies

Abstract Robust portfolio modeling (RPM) [Liesio, J., Mild, P., Salo, A., 2007. Preference programming for robust portfolio modeling and project selection. European Journal of Operational Research 181, 1488–1505] supports project portfolio selection in the presence of multiple evaluation criteria and incomplete information. In this paper, we extend RPM to account for project interdependencies, incomplete cost information and variable budget levels. These extensions lead to a multi-objective zero-one linear programming problem with interval-valued objective function coefficients for which all non-dominated solutions are determined by a tailored algorithm. The extended RPM framework permits more comprehensive modeling of portfolio problems and provides support for advanced benefit–cost analyses. It retains the key features of RPM by providing robust project and portfolio recommendations and by identifying projects on which further attention should be focused. The extended framework is illustrated with an example on product release planning.

Least trimmed squares regression, least median squares regression, and mathematical programming

In this paper, we study LTS and LMS regression, two high breakdown regression estimators, from an optimization point of view. We show that LTS regression is a nonlinear optimization problem that can be treated as a concave minimization problem over a polytope. We derive several important properties of the corresponding objective function that can be used to obtain algorithms for the exact solution of LTS regression problems, i.e., to find a global optimum to the problem. Because of today's limited problem-solving capabilities in exact concave minimization, we give an easy-to-implement pivoting algorithm to determine regression parameters corresponding to local optima of the LTS regression problem. For the LMS regression problem, we briefly survey the existing solution methods which are all based on enumeration. We formulate the LMS regression problem as a mixed zero-one linear programming problem which we analyze in depth to obtain theoretical insights required for future algorithmic and computational work.

The optimal location of replicas in a network using a READ-ONE-WRITE-ALL policy

We consider the problem of locating replicas in a network to minimize communications costs. Under the assumption that the READ-ONE-WRITE-ALL policy is used to ensure data consistency, an optimization problem is formulated in which the cost function estimates the total communications costs. The paper concentrates on the study of the optimal communications cost as a function of the ratio between the frequency of the read and write operations. The problem is reformulated as a zero-one linear programming problem, and its connection to the p-median problem is explained. The general problem is proved to be NP-complete. For path graphs a dynamic programming algorithm for the problem is presented.

Obtaining coteries that optimize the availability of replicated databases

Techniques for implementing the coterie scheme and for obtaining optimal coteries for a system are presented. Central to the techniques is the notion of an acceptance set, which is an alternative representation of the information contained in a coterie. Using this concept, the coterie scheme can be implemented efficiently, and an optimal coterie for a system can be obtained more directly. The problem of determining an optimal acceptance set is formulated as a sparse zero-one linear programming problem. Hence, the optimization problem can be handled using the very rich class of existing techniques for solving such problems. Experimental results indicate that the optimization approach is feasible for up to eight nodes at least. The ways in which the scheme and the optimization approach can be used for systems that distinguish between read and write operations are indicated.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Improving LP-Representations of Zero-One Linear Programs for Branch-and-Cut

We present various techniques for automatically improving the LP-representation of general zero-one linear programming problems. These include detection of redundant rows and blatant infeasibilities, coefficient reduction using the Euclidean algorithm, optimality fixing and variable elimination. Extensions to the case where special-ordered-set constraints are present are discussed as well. A summary of the branch-and-cut approach to general zero-one problems (including flowcharts) is given. We report numerical experiments to test the effect of such preprocessing within a branch-and-cut algorithm for eleven large-scale real-world zero-one linear-programming problems. An illustrative example is included in the Appendix. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

The use of a mathematical modelling method to determine the optimal retrofitting option of an oil refinery steam system

In this paper, as a practical example, the use of mathematical modelling procedure to determine the optimal retrofitting option of an oil refinery steam system is presented. The basis for setting up a mathematical model is to use a network diagram to describe all potential practicable alternatives, therefore, in principle this method is appropriate to optimal design or optimal modification of any complex energy flow or material flow system. The nature of this model is a multi-objective zero-one linear mixed integer programming problem.

A comparison of a constraint aggregation and partitioning zero-one goal programming algorithm with the lee and Morris algorithm

This paper presents the computational results of a new zero-one goal programming algorithm. These results are compared, based upon established criteria, to those obtained using the Lee and Morris algorithm. The applications found in the literature in which sufficient data existed were used as test data. The results of this study suggest that this new approach generates more solution alternatives, identifies better solutions, and takes less than 10% of the CPU time required by the Lee and Morris algorithm. Improvements in the algorithm are presented, and the use of this approach to solve zero-one linear multiple objective programming and zero-one linear programming problems is also discussed.

Optimal operation schedule of irrigation distribution systems

The ‘constant frequency-variable depth’ method of water delivery is discussed. The problem of clustering and sequencing outlets on a distributary channel to deliver the required quantity of water is described and formulated as a zero-one linear programming problem. The optimal operation schedule of the outlets was obtained for distributary 3 of the Meena branch canal in the Kukadi Irrigation Project in Maharashtra, India.

Solving Large-Scale Zero-One Linear Programming Problems

In this paper we report on the solution to optimality of 10 large-scale zero-one linear programming problems. All problem data come from real-world industrial applications and are characterized by sparse constraint matrices with rational data. About half of the sample problems have no apparent special structure; the remainder show structural characteristics that our computational procedures do not exploit directly. By today's standards, our methodology produced impressive computational results, particularly on sparse problems having no apparent special structure. The computational results on problems with up to 2,750 variables strongly confirm our hypothesis that a combination of problem preprocessing, cutting planes, and clever branch-and-bound techniques permit the optimization of sparse large-scale zero-one linear programming problems, even those with no apparent special structure, in reasonable computation times. Our results indicate that cutting-planes related to the facets of the underlying polytope are an indispensable tool for the exact solution of this class of problem. To arrive at these conclusions, we designed an experimental computer system PIPX that uses the IBM linear programming system MPSX/370 and the IBM integer programming system MIP/370 as building blocks. The entire system is automatic and requires no manual intervention.

A finitely convergent procedure for facial disjunctive programs

This paper addresses an important class of disjunctive programs called facial disjunctive programs, examples of which include the zero-one linear integer programming problem and the linear complementarity problem. Balas has characterized some fundamental properties of such problems, one of which has been used by Jeroslow to obtain a finitely convergent procedure. This paper exploits another basic property of facial disjunctive programs in order to develop an alternative finitely convergent algorithm.

Solving Large-Scale Symmetric Travelling Salesman Problems to Optimality

We report the solution to optimality of ten large-scale symmetric travelling salesman problems. The travelling salesman problem (TSP) is one of the standard problems of the Operations Research/Management Science literature and is cited in virtually every textbook on this subject. The TSP is a hard combinatorial optimization problem for which to date no technically good algorithm is known. All known algorithmic approaches to the TSP that are amenable to a worst-case analysis have yielded nothing for this problem that is as satisfactory as the existing solution procedures for, as an example, flow problems in networks. Furthermore, recent theoretical work has shown that the TSP is algorithmically equivalent to the general zero-one decision problem with linear constraints and objective function. The phrase “This problem is as hard as a travelling salesman problem” reflects this fact and expresses the slim chances of finding an optimal solution to the problem at hand. It is therefore all the more surprising that in the current study, all of the test problems were solved to optimality—i.e., the proposed methodology did not fail on any one of the ten large-scale test problems of the computational study. Besides being a prototype of a difficult combinatorial optimization problem, the travelling salesman problem has a variety of applications in its own right, mostly in the areas of production management, vehicle routing, and vehicle scheduling. A number of related problems, such as, for example, the multiple travelling salesman problem with m travelling salesmen located at a central depot and with fixed charges for their deployment, have been shown elsewhere to fit the standard form of the TSP treated in this paper. The smallest problem of this computational study has 48 cities and the largest one has 318 cities, i.e., the corresponding zero-one linear programming problems have between 1,128 and 50,403 zero-one variables. The algorithmic procedure is a cutting-plane approach coupled with branch-and-bound. For the latter we use the IBM software system MPSX-MIP/370 in a novel (iterative) way. The cutting-planes that we use are subtour-elimination and comb constraints, i.e., they belong to the class of valid inequalities for the symmetric travelling salesman problem which define facets for the associated polytope. This fact, which has been proven mathematically elsewhere, makes the cutting-planes encountered here different from other arbitrary, but valid, cutting-planes encountered in integer programming. In fact, cutting-planes have been virtually abandoned as an algorithmic tool in integer programming because of the erratic behavior of such algorithms exhibited in earlier computational studies. The present study convincingly establishes the usefulness of mathematically proven good cutting-planes as an invaluable algorithmic tool for difficult combinatorial optimization problems. Here the word optimization is taken in its very meaning—we are interested solely in finding a solution that minimizes a cost function and in proving it to be an optimal solution.

The Journal Selection Problem in a University Library System

The problem of selecting which journals to acquire in order to best satisfy library objectives is examined and modeled as a zero-one linear programming problem This is done using an objective function based on expected usage as a measure of journal worth and on cost constraints which account for the scarcity of capital. A dynamic programming algorithm is used to break down the larger problem into smaller sub-problems and to generate a feasible solution. Special cases where the solution is optimal are presented and discussed in terms of their implications for the library. An example problem is presented to illustrate the algorithm.

Further Reduction of Zero-One Polynomial Programming Problems to Zero-One linear Programming Problems

This paper gives rules that enable the transformation of a 0-1 polynomial programming problem into a 0-1 linear programming problem to be effected with reduced numbers of constraints. Rules are also given that provide reduced numbers of variables when the true variables of interest are not individual cross-product terms, but sums of such terms or polynomials of the form (∑xj)p.

A journal selection model and its implications for a library system

The problem of selecting which journals to acquire in order to best satisfy library objectives is modeled as a zero-one linear programming problem and examined in detail. The model incorporates an objective function based on expected usage as a measure of journal worth and cost constraints which account for the scarcity of capital. The model can be used to aid the librarian in making better selection decisions, since the objective function can be shown to reflect the evaluation of the library as an information retrieval system and as a service organization. Moreover, the model is seen to be related to inventory problems and scheduling models in industrial operations. Journal usage is discussed as a measure of journal worth and is contrasted to journal productivity. Constraints are considered for scarce resources other than capital and for journal interrelationships.

Technical Note—A Note on Zero-One Integer and Concave Programming

It has been shown that the zero-one linear integer programming problem can be formulated as a minimization of a concave quadratic objective function subject to linear constraints. This note extends the concept to a broad class of concave objective functions.

A Balasian-Based Algorithm for Zero-One Polynomial Programming

A new algorithm is developed for solving zero-one polynomial problems. The algorithm is a direct extension of the well-known Balasian algorithm for zero-one linear programs. The basic idea underlying the development is that every zero-one polynomial programming problem can be converted into an equivalent zero-one linear system with nonlinear secondary constraints representing the polynomial terms. Moreover, the optimal solution to the original polynomial problem must be a feasible solution to the converted linear problem. Thus, the optimal solution is determined by searching for the best solution among the feasible solutions to the linear problem such that the secondary constraints are satisfied. The important point is that the secondary constraints are considered implicitly so that the size of the problem is determined by its equivalent linear form. Computational experience indicates that the new algorithm compares favorably with the existing ones due to Lawler and Bell [Lawler, E., M. Bell. 1967. A method for solving discrete optimization problems. Oper. Res. 15 1098–1112.] and to Watters [Watters, L. J. 1967. Reduction of integer polynomial problems to zero-one linear programming problems. Oper. Res. 15 1171–1174.]. The application of the algorithm to a special class of nonlinear binary problems is also investigated.

Method of shifting units for solving the zero-one linear programming problem

Method of shifting units for solving the zero-one linear programming problem

A Branch-and-Bound Algorithm for Zero-One Mixed Integer Programming Problems

This paper presents the results of experimentation on the development of an efficient branch-and-bound algorithm for the solution of zero-one linear mixed integer programming problems. An implicit enumeration is employed using bounds that are obtained from the fractional variables in the associated linear programming problem. The principal mathematical result used in obtaining these bounds is the piecewise linear convexity of the criterion function with respect to changes of a single variable in the interval [0, 1]. A comparison with the computational experience obtained with several other algorithms on a number of problems is included.