We present a semide nite programming (SDP) approach for the problem of ordering vertices of a layered graph such that the edges of the graph are drawn as vertical as possible. This Multi-Level Vertical Ordering (MLVO) problem is a quadratic ordering problem and conceptually related to the well-studied problem of Multi-Level Crossing Minimization (MLCM). In contrast … Read more
We consider the question of finding deep cuts from a model constructed with two rows of a simplex tableau. To do that, we show how to reduce the complexity of setting up the polar of such model from a quadratic number of integer hull computations to a linear number of integer hull computations. Furthermore we … Read more
Recent developments have drawn focus towards the efficient calculation of flows in AC power grids, which are difficult to solve systems of nonlinear equations. The common linearization approach leads to the well known and often used DC formulation, which has some major drawbacks. To overcome these drawbacks we revisit an alternative linearization of the AC … Read more
In this paper branching for attacking MILP is investigated. Under certain circumstances branches can be done concurrently. By introducing a new calculus it is shown there are restrictions for certain dual values and reduced costs. As a second unexpected result of this study a new class of cuts for MILP is found, which are defined … Read more
Pochet and Wolsey [Y. Pochet, L.A. Wolsey, Integer knapsack and flow covers with divisible coefficients: polyhedra, optimization and separation. Discrete Applied Mathematics 59(1995) 57-74] introduced partition inequalities for three substructures arising in various mixed integer programs, namely the integer knapsack set with nonnegative divisible/arbitrary coefficients and two forms of single-node capacitated flow set with divisible … Read more
In this paper, we investigate the recoverable robust knapsack problem, where the uncertainty of the item weights follows the approach of Bertsimas and Sim (2003,2004). In contrast to the robust approach, a limited recovery action is allowed, i.e., upto k items may be removed when the actual weights are known. This problem is motivated by … Read more
We describe a procedure to reduce variable bounds in Mixed Integer Nonlinear Programming (MINLP) as well as Mixed Integer Linear Programming (MILP) problems. The procedure works by combining pairs of inequalities of a linear programming (LP) relaxation of the problem. This bound reduction technique extends the implied bounds procedure used in MINLP and MILP and … Read more
Mathematical modelling languages such as AMPL, GAMS, and Xpress-MP enable mathematical models such as mixed-integer linear programmes (MILPs) to be expressed clearly for solution in solvers such as CPLEX, MINOS and Gurobi. However some models are sufficiently difficult that they cannot be solved using “out-of-the-box” solvers, and customisation of the solver framework to exploit model-specific … Read more
This paper is concerned with the improved constraints handling in mixed-integer optimization problems. The novel element is the reduction of the number of binary variables used for expressing the complement of a convex (polytopic) region. As a generalization, the problem of representing the complement of a possibly non-connected union of such convex sets is detailed. … Read more
Anderson, Cornuejols and Li (2005) show that for a polyhedral mixed integer set defined by a constraint system Ax >= b, where x is n-dimensional, along with integrality restrictions on some of the variables, any split cut is in fact a split cut for a “basic relaxation”, i.e., one defined by a subset of linearly … Read more