On the stability of some integer programming algorithms

In the present paper we develop our approach for studying the stability of integer programming problems. We prove that the L-class enumeration method is stable on integer linear programming problems in the case of bounded relaxation sets [9]. The stability of some cutting plane algorithms is discussed.     

Using objective values to start multiple objective linear programming algorithms

We introduce in this paper a new starting mechanism for multiple-objective linear programming (MOLP) algorithms. This makes it possible to start an algorithm from any solution in objective space. The original problem is first augmented in such a way that a given starting solution is feasible. The augmentation is explicitly or implicitly controlled by one parameter during the search process, which verifies the feasibility (efficiency) of the final solution. This starting mechanism can be applied either to traditional algorithms, which search the exterior of the constraint polytope, or to algorithms moving through the interior of the constraints. We provide recommendations on the suitability of an algorithm for the various locations of a starting point in objective space. Numerical considerations illustrate these ideas.     

Dynamic programming algorithms for the bi-objective integer knapsack problem

This paper presents two new dynamic programming (DP) algorithms to find the exact Pareto frontier for the bi-objective integer knapsack problem. First, a property of the traditional DP algorithm for the multi-objective integer knapsack problem is identified. The first algorithm is developed by directly using the property. The second algorithm is a hybrid DP approach using the concept of the bound sets. The property is used in conjunction with the bound sets. Next, the numerical experiments showed that a promising partial solution can be sometimes discarded if the solutions of the linear relaxation for the subproblem associated with the partial solution are directly used to estimate an upper bound set. It means that the upper bound set is underestimated. Then, an extended upper bound set is proposed on the basis of the set of linear relaxation solutions. The efficiency of the hybrid algorithm is improved by tightening the proposed upper bound set. The numerical results obtained from different types of bi-objective instances show the effectiveness of the proposed approach.     

An algorithm for the multiple objective integer linear programming problem

A technique is presented for solving the multiple objective integer linear programming problem. The technique can be used to identify some or all efficient solutions. While the technique is applicable with any integer programming algorithm, it is well suited for implementation using integer postoptimality techniques. Such an implementation, based on Balas' Additive algorithm, is described for problems with zero-one variables.     

Multiparametric sensitivity analysis in programming problem with linear-plus-linear fractional objective function

In this paper, we study multiparametric sensitivity analysis for programming problems with linear-plus-linear fractional objective function using the concept of maximum volume in the tolerance region. We construct critical regions for simultaneous and independent perturbations in the objective function coefficients and in the right-hand-side vector of the given problem. Necessary and sufficient conditions are derived to classify perturbation parameters as `focal' and `non-focal'. Non-focal parameters can have unlimited variations, because of their low sensitivity in practice, these parameters can be deleted from the analysis. For focal parameters, a maximum volume tolerance region is characterized. Theoretical results are illustrated with the help of a numerical example.     

Linear programming with a possibilistic objective function

This paper proposes some ways for dealing with a linear program when the coefficients of the objective function are subject to possibilistic imprecision, i.e. they are characterized by fuzzy restrictions. Emphasis is placed upon a passive approach that yields a satisfying solution via an appropriate semi-infinite program, and an active one that allows to reach a solution with a high possibility level of optimality. Extensions to the possibilistic constraints case and to the case of multiple-objective programming problems with possibilistic coefficients are also hinted. We end up with some concluding remarks and indicate axes for further developments.     

Heuristic algorithms for the inverse mixed integer linear programming problem

The recent development in inverse optimization, in particular the extension from the inverse linear programming problem to the inverse mixed integer linear programming problem (InvMILP), provides more powerful modeling tools but also presents more challenges to the design of efficient solution techniques. The only reported InvMILP algorithm, referred to as Alg^InvMILP, although finitely converging to global optimality, suffers two limitations that greatly restrict its applicability: it is time consuming and does not generate a feasible solution except for the optimal one. This paper presents heuristic algorithms that are designed to be implemented and executed in parallel with Alg^InvMILP in order to alleviate and overcome its two limitations. Computational experiments show that implementing the heuristic algorithm on one auxiliary processor in parallel with Alg^InvMILP on the main processor significantly improves its computational efficiency, in addition to providing a series of improving feasible upper bound solutions. The additional speedup of parallel implementation on two or more auxiliary processors appears to be incremental, but the upper bound can be improved much faster.     

A discussion of scalarization techniques for multiple objective integer programming

In this paper we consider solution methods for multiobjective integer programming (MOIP) problems based on scalarization. We define the MOIP, discuss some common scalarizations, and provide a general formulation that encompasses most scalarizations that have been applied in the MOIP context as special cases. We show that these methods suffer some drawbacks by either only being able to find supported efficient solutions or introducing constraints that can make the computational effort to solve the scalarization prohibitive. We show that Lagrangian duality applied to the general scalarization does not remedy the situation. We also introduce a new scalarization technique, the method of elastic constraints, which is shown to be able to find all efficient solutions and overcome the computational burden of the scalarizations that use constraints on objective values. Finally, we present some results from an application in airline crew scheduling as evidence. This research is partially supported by University of Auckland grant 3602178/9275 and by the Deutsche Forschungsgemeinschaft grant Ka 477/27-1.     

Smoothed analysis of integer programming

We present a probabilistic analysis of integer linear programs (ILPs). More specifically, we study ILPs in a so-called smoothed analysis in which it is assumed that first an adversary specifies the coefficients of an integer program and then (some of) these coefficients are randomly perturbed, e.g., using a Gaussian or a uniform distribution with small standard deviation. In this probabilistic model, we investigate structural properties of ILPs and apply them to the analysis of algorithms. For example, we prove a lower bound on the slack of the optimal solution. As a result of our analysis, we are able to specify the smoothed complexity of classes of ILPs in terms of their worst case complexity. This way, we obtain polynomial smoothed complexity for packing and covering problems with any fixed number of constraints. Previous results of this kind were restricted to the case of binary programs.      

On the use of multiple objective linear programming algorithms to solve problems with fractional objectives

Multiple objective linear fractional programming can be used in a wide range of problem formulations in private and public enterprises. Under certain restrictions, a simpler multiple objective linear programming approach can be used for problem solution. In this note we analyse the use of this simplified approach to MOLFP, and the practical effect of the restrictions implied.     

Branch-and-bound algorithms for the partial inverse mixed integer linear programming problem

This paper presents branch-and-bound algorithms for the partial inverse mixed integer linear programming (PInvMILP) problem, which is to find a minimal perturbation to the objective function of a mixed integer linear program (MILP), measured by some norm, such that there exists an optimal solution to the perturbed MILP that also satisfies an additional set of linear constraints. This is a new extension to the existing inverse optimization models. Under the weighted $L_1$ and $L_\infty $ norms, the presented algorithms are proved to finitely converge to global optimality. In the presented algorithms, linear programs with complementarity constraints (LPCCs) need to be solved repeatedly as a subroutine, which is analogous to repeatedly solving linear programs for MILPs. Therefore, the computational complexity of the PInvMILP algorithms can be expected to be much worse than that of MILP or LPCC. Computational experiments show that small-sized test instances can be solved within a reasonable time period.     

Bilevel linear programming with ambiguous objective function of the follower

Bilevel linear optimization problems are the linear optimization problems with two sequential decision steps of the leader and the follower. In this paper, we focus on the ambiguity of coefficients of the follower in his objective function that hinder the leader from exactly calculating the rational response of the follower. Under the assumption that the follower’s possible range of the ambiguous coefficient vector is known as a certain convex polytope, the leader can deduce the possible set of rational responses of the follower. The leader further assumes that the follower’s response is the worst-case scenario to his objective function, and then makes a decision according to the maximin criteria. We thus formulate the bilevel linear optimization problem with ambiguous objective function of the follower as a special kind of three-level programming problem. In our formulation, we show that the optimal solution locates on the extreme point and propose a solution method based on the enumeration of possible rational responses of the follower. A numerical example is used to illustrate our proposed computational method.     

Equivalence of some quadratic programming algorithms

We formulate a general algorithm for the solution of a convex (but not strictly convex) quadratic programming problem. Conditions are given under which the iterates of the algorithm are uniquely determined. The quadratic programming algorithms of Fletcher, Gill and Murray, Best and Ritter, and van de Panne and Whinston/Dantzig are shown to be special cases and consequently are equivalent in the sense that they construct identical sequences of points. The various methods are shown to differ only in the manner in which they solve the linear equations expressing the Kuhn-Tucker system for the associated equality constrained subproblems. Equivalence results have been established by Goldfarb and Djang for the positive definite Hessian case. Our analysis extends these results to the positive semi-definite case. This research was supported by the Natural Sciences and Engineering Research Council of Canada under Grant No. A8189.     

Cutting plane algorithms for the inverse mixed integer linear programming problem

We present cutting plane algorithms for the inverse mixed integer linear programming problem (InvMILP), which is to minimally perturb the objective function of a mixed integer linear program in order to make a given feasible solution optimal.     

Complexity of some parametric integer and network programming problems

Two examples of parametric cost programming problems—one in network programming and one in NP-hard 0-1 programming—are given; in each case, the number of breakpoints in the optimal cost curve is exponential in the square root of the number of variables in the problem. This research is partially supported by the Air Force Office of Scientic Research. Air Force Number AFOSR-78-3646     

The problem of the stability of quasilinear flows with respect to perturbations of the hardening function

A formulation of the linearized boundary-value problem of the stability of a deformation process with respect to small perturbations of the hardening function (of the scalar constitutive relation of the material) is presented. The characteristic vector relations of the medium are assumed to be linear. The occurrence of rigid zones in the domain of the solid and the change in their boundaries in the perturbed motion are taken into account. A perfect rigid plastic deformation and the flow of a Newtonian fluid are considered explicitly as the basic flow. In the latter case, the equation of the asymptotic boundary of the rigid zone, which appears when there is a small variation in the yield stress and a transition to a viscoplastic material, is derived.     

Indefinite cubic programming with standard errors in objective function

Summary This paper concerns with the problem of indefinite cubic programming in which the objective function is the product of two factors, one of which is quadratic and contains the terms with standard errors, the other being a linear factor and the constraints being linear. It has been shown that the solution of such a programming problem can be obtained by solving a convex programming problem. The problem has been extended to the case where both the factors in the objective function are quadratic factors containing terms with standard errors. Some already known results have also been deduced as particular cases of the problem discussed.

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Zusammenfassung Die Arbeit behandelt ein indefinites kubisches Programmierungsproblem unter linearen Nebenbedingungen. Hierbei ist die Zielfunktion das Produkt eines quadratischen und eines linearen Faktors. Es wird gezeigt, daß das Problem auf ein konvexes Programm zurückgeführt werden kann. Ferner wird der Fall betrachtet, daß beide Faktoren der Zielfunktion quadratische Ausdrücke mit Fehlertermen sind. Als Sonderfälle der betrachteten Probleme werden einige bekannte Resultate abgeleitet.

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Vorgel. v.:H. P. Künzi.     

On the convex hull of the simple integer recourse objective function

We consider the objective function of a simple integer recourse problem with fixed technology matrix.Using properties of the expected value function, we prove a relation between the convex hull of this function and the expected value function of a continuous simple recourse program.We present an algorithm to compute the convex hull of the expected value function in case of discrete right-hand side random variables. Allowing for restrictions on the first stage decision variables, this result is then extended to the convex hull of the objective function.Supported by the National Operations Research Network in the Netherlands (LNMB).