We consider the problem of generating inequalities that are valid for one-row relaxations of a simplex tableau, with the integrality constraints preserved for one or more non-basic variables. These relaxations are interesting because they can be used to generate cutting planes for general mixed-integer problems. We first consider the case of a single non-basic integer … Read more
In this paper, I view and present the multiobjective discrete optimisation problem as a particular case of disjunctive programming where one seeks to identify efficient solutions from within a disjunction formed by a set of systems. The proposed approach lends itself to a simple yet effective iterative algorithm that is able to yield the set … Read more
We construct a two-sided discontinuous piecewise linear minimal valid function for the 1-row Gomory–Johnson model which is not extreme, but which is not a convex combination of other piecewise linear minimal valid functions. This anomalous behavior results from combining features of Hildebrand’s two-sided discontinuous extreme functions and Basu–Hildebrand–Koeppe’s piecewise linear extreme function with irrational breakpoints. … Read more
We present software for investigations with cut generating functions in the Gomory-Johnson model and extensions, implemented in the computer algebra system SageMath. Citation An extended abstract of 8 pages appeared under the title “Software for cut-generating functions in the Gomory–Johnson model and beyond” in Proc. International Congress on Mathematical Software 2016 Article Download View Equivariant … Read more
The resolution of integer programming problems is typically performed via branch-and-bound. Nodes of the branch-and-bound tree are pruned whenever the corresponding subproblem is proven not to contain a solution better than the best solution found so far. This is a key ingredient for achieving reasonable solution times. However, since subproblems are solved in floating-point arithmetic, … Read more
We study the chance-constrained vehicle routing problem (CCVRP), a version of the vehicle routing problem (VRP) with stochastic demands, where a limit is imposed on the probability that each vehicle’s capacity is exceeded. A distinguishing feature of our proposed methodologies is that they allow correlation between random demands, whereas nearly all existing exact methods for … Read more
Multistage stochastic integer programming (MSIP) combines the difficulty of uncertainty, dynamics, and non-convexity, and constitutes a class of extremely challenging problems. A common formulation for these problems is a dynamic programming formulation involving nested cost-to-go functions. In the linear setting, the cost-to-go functions are convex polyhedral, and decomposition algorithms, such as nested Benders’ decomposition and … Read more
The optimal design of wireless networks has been widely studied in the literature and many optimization models have been proposed over the years. However, most models directly include the signal-to-interference ratios representing service coverage conditions. This leads to mixed-integer linear programs with constraint matrices containing tiny coefficients that vary widely in their order of magnitude. … Read more
This paper describes the extensions that were added to the constraint integer programming framework SCIP in order to enable it to solve convex and nonconvex mixed-integer nonlinear programs (MINLPs) to global optimality. SCIP implements a spatial branch-and-bound algorithm based on a linear outer-approximation, which is computed by convex over- and underestimation of nonconvex functions. An … Read more
We consider a challenging resource allocation problem arising in mobile wireless communications. The goal is to allocate the available channels and power in a so-called OFDMA system, in order to maximise the transmission rate, subject to quality of service (QoS) constraints. Standard MINLP software struggled to solve even small instances of this problem. Using outer … Read more