Jørgen Tind

L -shaped decomposition of two-stage stochastic programs with integer recourse

We consider two-stage stochastic programming problems with integer recourse. The L-shaped method of stochastic linear programming is generalized to these problems by using generalized Benders decomposition. Nonlinear feasibility and optimality cuts are determined via general duality theory and can be generated when the second stage problem is solved by standard techniques. Finite convergence of the method is established when Gomory’s fractional cutting plane algorithm or a branch-and-bound algorithm is applied.

An Elementary Survey of General Duality-theory in Mathematical-programming

We survey some recent developments in duality theory with the idea of explaining and unifying certain basic duality results in both nonlinear and integer programming. The idea of replacing dual variables (prices) by price functions, suggested by Everett and developed by Gould, is coupled with an appropriate dual problem with the consequence that many of the results resemble those used in linear programming.The dual problem adopted has a (traditional) economic interpretation and dual feasibility then provides a simple alternative to concepts such as conjugate functions or subdifferentials used in the study of optimality. In addition we attempt to make precise the relationship between primal, dual and saddlepoint results in both the traditional Lagrangean and the more general duality theories and to see the implications of passing from prices to price functions. Finally, and perhaps surprisingly, it appears that all the standard algorithms terminate by constructing primal and dual feasible solutions of equal value, i.e., by satisfying generalised optimality conditions.

Incentive plans for productive efficiency, innovation and learning

In many industries where production or sales is delegated to a number of subunits, the central management faces the classical problem how to induce continuous efficiency improvements, organizational learning and transfer of knowledge with a minimum of control exercised. This paper draws on recent results regarding regulatory frameworks to construct simple, yet powerful incentive schemes for decentralized production under asymmetric information. The theoretical foundation is based on principal–agent theory (cf. Laffont and Tirole, Econometrica 56 (1986) 614–641) and extensions to production theory by Bogetoft (Management Science 40 (1994) 959–968). The proposed incentive system is operational and makes use of available information to provide positive incentives for participation in the dynamic development of the entire organization.

A Dual Approach to Nonconvex Frontier Models

This paper extends the links between the non-parametric data envelopment analysis (DEA) models for efficiency analysis, duality theory and multi-criteria decision making models for the linear and non-linear case. By drawing on the properties of a partial Lagrangean relaxation, a correspondence is shown between the CCR, BCC and free disposable hull (FDH) models in DEA and the MCDM model. One of the implications is a characterization that verifies the sufficiency of the weighted scalarizing function, even for the non-convex case FDH. A linearization of FDH is presented along with dual interpretations. Thus, an input/output-oriented model is shown to be equivalent to a maximization of the weighted input/output, subject to production space feasibility. The discussion extends to the recent developments: the free replicability hull (FRH), the new elementary replicability hull (ERH) and the non-convex models by Petersen (1990). FRH is shown to be a true mixed integer program, whereas the latter can be characterized as the CCR and BCC models.

Unbiased approximation in multicriteria optimization

Abstract. Algorithms generating piecewise linear approximations of the nondominated set for general, convex and nonconvex, multicriteria programs are developed. Polyhedral distance functions are used to construct the approximation and evaluate its quality. The functions automatically adapt to the problem structure and scaling which makes the approximation process unbiased and self-driven. Decision makers preferences, if available, can be easily incorporated but are not required by the procedure.

A cutting-plane approach to mixed 0–1 stochastic integer programs

Abstract We consider a mixed 0–1 integer programming problem with dual block-angular structure arising in two-stage stochastic programming. A relaxation is proposed such that the problem is decomposed into subproblems each corresponding to the outcomes of the random variable. The convex hull of feasible solutions for the relaxation is characterized using results from disjunctive programming and it is shown how Lift-and-Project cuts can be generated for one subproblem and made valid for different outcomes.

Augmented Lagrangian and Tchebycheff Approaches in Multiple Objective Programming

Relationships between the Tchebycheff scalarization and the augmented Lagrange multiplier technique are examined in the framework of general multiple objective programs (MOPs). It is shown that under certain conditions the Tchebycheff method can be represented as a quadratic weighted-sums scalarization of the MOP, that is, given weight values in the former, the coefficients of the latter can be found so that the same efficient point is selected. Analysis for concave and linear MOPs is included. Resulting applications in multiple criteria decision making are also discussed.

An interior point method in Dantzig-Wolfe decomposition

This paper studies the application of interior point methods in Dantzig–Wolfe decomposition. The main idea is to develop strategies for finding useful interior points in the dual of the restricted master problem as an alternative to finding an optimal solution or the analytic center. The method considers points on the central path between the optimal solution and the analytic center, and thus it includes the previous instances as extreme cases. For a given duality gap there exists a unique primal–dual solution on the central path. We use this solution for some choice of the duality gap. The desired duality gap is either kept fixed in all master iterations or it is updated according to some strategy. We test the method on a number of randomly generated problems of different sizes and with different numbers of subproblems. For most problems our method requires fewer master iterations than the classical Dantzig–Wolfe and the analytic center method. This result is especially true for problems requiring many master iterations. In addition to experiments using an interior point method on the master problems, we have also performed some experiments with an interior point method on the subproblems. Instead of finding an optimal solution for the problems we have developed a strategy that selects a feasible solution having a reduced cost below some prescribed level. Our study focuses on comparative experiments. Scope and purpose Dantzig–Wolfe decomposition is a well established discipline in linear programming, and most textbooks in the area include a chapter on this topic. It is a procedure of fundamental importance for the development of the so-called column generation technique for the solution of large scale problems in optimization as well as for the development of decentralized planning systems in economics. Most presentations and applications of the procedure are done within the context of the simplex method. However with the active development of interior point methods there is an increasing interest also to use those methods in decomposition including the Dantzig–Wolfe decomposition. The present manuscript is a part of this development. Some methods are here proposed linking Dantzig–Wolfe decomposition to some of the decomposition procedures using interior point methods. Comparative experiments are performed pointing out places, where it is advantageous to use an interior point method.

Sequential convexification in reverse convex and disjunctive programming

This paper is about a property of certain combinatorial structures, called sequential convexifiability, shown by Balas (1974, 1979) to hold for facial disjunctive programs. Sequential convexifiability means that the convex hull of a nonconvex set defined by a collection of constraints can be generated by imposing the constraints one by one, sequentially, and generating each time the convex hull of the resulting set. Here we extend the class of problems considered to disjunctive programs with infinitely many terms, also known as reverse convex programs, and give necessary and sufficient conditions for the solution sets of such problems to be sequentially convexifiable. We point out important classes of problems in addition to facial disjunctive programs (for instance, reverse convex programs with equations only) for which the conditions are always satisfied. Finally, we give examples of disjunctive programs for which the conditions are violated, and so the procedure breaks down.

Blocking and antiblocking sets

The purpose of this paper is to study a generalization of the concept of blocking and antiblocking polyhedra that has been introduced by D.R. Fulkerson. The polyhedra studied by Fulkerson are restricted to the nonnegative orthant in Rm. The present generalization considers sets restricted to a cone. This leads to two polarity correspondences, which are related to the Minkowski polarity for convex sets.