Using duality to solve generalized fractional programming problems

In this paper we explore the relations between the standard dual problem of a convex generalized fractional programming problem and the partial dual problem proposed by Barros et al. (1994). Taking into account the similarities between these dual problems and using basic duality results we propose a new algorithm to directly solve the standard dual of a convex generalized fractional programming problem, and hence the original primal problem, if strong duality holds. This new algorithm works in a similar way as the algorithm proposed in Barros et al. (1994) to solve the partial dual problem. Although the convergence rates of both algorithms are similar, the new algorithm requires slightly more restrictive assumptions to ensure a superlinear convergence rate. An important characteristic of the new algorithm is that it extends to the nonlinear case the Dinkelbach-type algorithm of Crouzeix et al. (1985) applied to the standard dual problem of a generalized linear fractional program. Moreover, the general duality results derived for the nonlinear case, yield an alternative way to derive the standard dual of a generalized linear fractional program. The numerical results, in case of quadratic-linear ratios and linear constraints, show that solving the standard dual via the new algorithm is in most cases more efficient than applying directly the Dinkelbach-type algorithm to the original generalized fractional programming problem. However, the numerical results also indicate that solving the alternative dual (Barros et al., 1994) is in general more efficient than solving the standard dual.This research was carried out at the Econometric Institute, Erasmus University Rotterdam, the Netherlands and was supported by the Tinbergen Institute Rotterdam     

Using mixed-integer programming to solve power grid blackout problems

We consider optimization problems related to the prevention of large-scale cascading blackouts in power transmission networks subject to multiple scenarios of externally caused damage. We present computation with networks with up to 600 nodes and 827 edges, and many thousands of damage scenarios.     

Using mathematical programming to solve Factored Markov Decision Processes with Imprecise Probabilities

This paper investigates Factored Markov Decision Processes with Imprecise Probabilities (MDPIPs); that is, Factored Markov Decision Processes (MDPs) where transition probabilities are imprecisely specified. We derive efficient approximate solutions for Factored MDPIPs based on mathematical programming. To do this, we extend previous linear programming approaches for linear approximations in Factored MDPs, resulting in a multilinear formulation for robust “maximin” linear approximations in Factored MDPIPs. By exploiting the factored structure in MDPIPs we are able to demonstrate orders of magnitude reduction in solution time over standard exact non-factored approaches, in exchange for relatively low approximation errors, on a difficult class of benchmark problems with millions of states.     

On the application of the flexible polyhedron method to solve mathematical programming problems

Translated fromIssledovaniya po Prikladnoi Matematike, No. 18, 1992, pp. 26–30.     

Using the Dinkelbach-type algorithm to solve the continuous-time linear fractional programming problems

A Dinkelbach-type algorithm is proposed in this paper to solve a class of continuous-time linear fractional programming problems. We shall transform this original problem into a continuous-time non-fractional programming problem, which unfortunately happens to be a continuous-time nonlinear programming problem. In order to tackle this nonlinear problem, we propose the auxiliary problem that will be formulated as parametric continuous-time linear programming problem. We also introduce a dual problem of this parametric continuous-time linear programming problem in which the weak duality theorem also holds true. We introduce the discrete approximation method to solve the primal and dual pair of parametric continuous-time linear programming problems by using the recurrence method. Finally, we provide two numerical examples to demonstrate the usefulness of this practical algorithm.     

Using a facility location algorithm to solve large set covering problems

Erlenkotter has developed an efficient exact (guarantees optimality) algorithm to solve the uncapacitated facility location problem (UFLP). In this paper, we use his algorithm to solve large instances of an important subset of the UFLP; the set covering problem (SCP). In addition, we present further empirical evidence that a heuristic algorithm developed by Vasko and Wilson for the SCP is capable of quickly generating good solutions to large SCP's.     

Some applications of mathematical programming to problems in mathematical physics

Siberian Mathematical Journal -     

On the solution of large engineering design problems via mathematical programming

The method of moving asymptotes (MMA) is known to work well for the solution of mathematical programs arising in engineering design problems. Recently, the method has been extended by using two interior point approaches for the solution of the arising convex subproblems. Besides function and gradient evaluations, i.e. finite element analyses, the main computational work has to be done in the solution of a large sequence of linear systems. The interior point approach provides more flexibility in the formulation of these systems which is advantageous for large scale applications.     

Using Lin's method to solve Bykov's problems

We consider nonwandering dynamics near heteroclinic cycles between two hyperbolic equilibria. The constituting heteroclinic connections are assumed to be such that one of them is transverse and isolated. Such heteroclinic cycles are associated with the termination of a branch of homoclinic solutions, and called T-points   in this context. We study codimension-two T-points and their unfoldings in Rⁿ${\mathbb{R}}^n$. In our consideration we distinguish between cases with real and complex leading eigenvalues of the equilibria. In doing so we establish Lin's method as a unified approach to (re)gain and extend results of Bykov's seminal studies and related works. To a large extent our approach reduces the study to the discussion of intersections of lines and spirals in the plane.     

An iterative approach to solve multiobjective linear fractional programming problems

This paper suggests an iterative parametric approach for solving multiobjective linear fractional programming (MOLFP) problems which only uses linear programming to obtain efficient solutions and always converges to an efficient solution. A numerical example shows that this approach performs better than some existing algorithms. Randomly generated MOLFP problems are also solved to demonstrate the performance of new introduced algorithm.     

Perturbation theory for mathematical programming problems

Mathematical programming (MP) problems depending on a small parameter are investigated. Attention is paid to the cases where the solutions to the reduced program and/or the solutions to the dual reduced program are not unique. Conditions are given for the convergence of perturbed solutions to a point of the reduced problem solution set, if the small parameter tends to zero. It is shown how to find this point and how to construct an approximate solution to the perturbed program. A singular situation may appear if the dual solution set is unbounded. In this case, a gap between perturbed and reduced solutions may arise. However, it is shown that the perturbed solutions are close to the solutions of some modified reduced problem. The practical usefulness of perturbation theory is demonstrated by considering the two LP problems. Decomposition and aggregation procedures are constructed on the base of general results to find suboptimal solutions of these problems.     

Combining linear programming and automated planning to solve intermodal transportation problems

When dealing with transportation problems Operational Research (OR), and related areas as Artificial Intelligence (AI), have focused mostly on uni-modal transport problems. Due to the current existence of bigger international logistics companies, transportation problems are becoming increasingly more complex. One of the complexities arises from the use of intermodal transportation. Intermodal transportation reflects the combination of at least two modes of transport in a single transport chain, without a change of container for the goods. In this paper, a new hybrid approach is described which addresses complex intermodal transport problems. It combines OR techniques with AI search methods in order to obtain good quality solutions, by exploiting the benefits of both kinds of techniques. The solution has been applied to a real world problem from one of the largest spanish companies using intermodal transportation, Acciona Transmediterránea Cargo.     

A neural network to solve quadratic programming problems with fuzzy parameters

In this paper, a representation of a recurrent neural network to solve quadratic programming problems with fuzzy parameters (FQP) is given. The motivation of the paper is to design a new effective one-layer structure neural network model for solving the FQP. As far as we know, there is not a study for the neural network on the FQP. Here, we change the FQP to a bi-objective problem. Furthermore, the bi-objective problem is reduced to a weighting problem and then the Lagrangian dual is constructed. In addition, we consider a neural network model to solve the FQP. Finally, some illustrative examples are given to show the effectiveness of our proposed approach.     

Using convex envelopes to solve the interactive fixed-charge linear programming problem

In the well-known fixed-charge linear programming problem, it is assumed, for each activity, that the value of the fixed charge incurred when the level of the activity is positive does not depend upon which other activities, if any, are also undertaken at a positive level. However, we have encountered several practical problems where this assumption does not hold. In an earlier paper, we developed a new problem, called the interactive fixed-charge linear programming problem (IFCLP), to model these problems. In this paper, we show how to construct the convex envelopes and other convex underestimating functions for the objective function for problem (IFCLP) over various rectangular subsets of its domain. Using these results, we develop a specialized branch-and-bound algorithm for problem (IFCLP) which finds an exact optimal solution for the problem in a finite number of steps. We also discuss the main properties of this algorithm.The authors would like to thank an anonymous referee for his helpful suggestions.     

Using complementary slackness property to solve linear programming with fuzzy parameters

The most popular approach to handle the challenge of solving fuzzy linear programming problems is to convert the fuzzy linear programming into the corresponding deterministic linear programming. Mahdavi-Amiri and Nasseri [15,16] developed the fuzzy dual simplex algorithm to fuzzy linear programming with fuzzy parameters. In this paper, we use the complementary slackness to solve it without the need of a simplex tableau.     

Correction to: A sequential homotopy method for mathematical programming problems

Mathematical Programming - A correction to this paper has been published: https://doi.org/10.1007/s10107-021-01668-5     

Analysis of mathematical programming problems prior to applying the simplex algorithm

Large practical linear and integer programming problems are not always presented in a form which is the most compact representation of the problem. Such problems are likely to posses generalized upper bound(GUB) and related structures which may be exploited by algorithms designed to solve them efficiently. The steps of an algorithm which by repeated application reduces the rows, columns, and bounds in a problem matrix and leads to the freeing of some variables are first presented. The ‘unbounded solution’ and ‘no feasible solution’ conditions may also be detected by this. Computational results of applying this algorithm are presented and discussed. An algorithm to detect structure is then described. This algorithm identifies sets of variables and the corresponding constraint relationships so that the total number of GUB-type constraints is maximized. Comparisons of computational results of applying different heuristics in this algorithm are presented and discussed.