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
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. CitationAn 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 2016ArticleDownload View PDF
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
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
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
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
The essential structure of the mixed–integer programming formulation for chance–constrained program (CCP) is the intersection of multiple mixing sets with a $0-1$ knapsack. To improve our computational capacity on CCP, an underlying substructure, the (single) mixing set with a $0-1$ knapsack, has received substantial attentions recently. In this study, we consider a CCP problem with … Read more
This work is inspired by very challenging issues arising in space logistics. The problem of scheduling a number of activities, in a given time elapse, optimizing the resource exploitation is discussed. The available resources are not constant, as well as the request, relative to each job. The mathematical aspects are illustrated, providing a time-indexed MILP … Read more
This chapter examines the problem of packing tetris-like items, orthogonally, with the possibility of rotations, into a convex domain, in the presence of additional conditions. An MILP (Mixed Integer Linear Programming) and an MINLP (Mixed Integer Nonlinear Programming) models, previously studied by the author, are surveyed. An efficient formulation of the objective function, aimed at … Read more