We present semidefinite relaxations for unconstrained non-convex quadratic mixed-integer optimization problems. These relaxations yield tight bounds and are computationally easy to solve for medium-sized instances, even if some of the variables are integer and unbounded. In this case, the problem contains an infinite number of linear constraints; these constraints are separated dynamically. We use this … Read more
We propose a new look at the problem of truss topology optimization with integer or binary variables. We show that the problem can be equivalently formulated as an integer \emph{linear} semidefinite optimization problem. This makes its numerical solution much easier, compared to existing approaches. We demonstrate that one can use an off-the-shelf solver with default … Read more
Resource leveling is a variant of resource-constrained project scheduling in which a non-regular objective function, the resource availability cost, is to be minimized. We present a branch-and-price approach together with a new heuristic to solve the more general turnaround scheduling problem. Besides precedence and resource constraints, also availability periods and multiple modes per job have … Read more
This paper introduces a new cutting plane method for two-stage stochastic mixed-integer programming (SMIP) called Fenchel decomposition (FD). FD uses a class of valid inequalities termed, FD cuts, which are derived based on Fenchel cutting planes from integer programming. First, we derive FD cuts based on both the first and second-stage variables, and devise an … Read more
We consider a knapsack problem with precedence constraints imposed on pairs of items, known as the precedence constrained knapsack problem (PCKP). This problem has applications in manufacturing and mining, and also appears as a subproblem in decomposition techniques for network design and related problems. We present a new approach for determining facets of the PCKP … Read more
Computing the 1-width of the incidence matrix of a Steiner Triple System gives rise to small set covering instances that provide a computational challenge for integer programming techniques. One major source of difficulty for instances of this family is their highly symmetric structure, which impairs the performance of most branch-and-bound algorithms. The largest instance in … Read more
We present a number of new algorithmic ideas for solving LP relaxations of extremely large precedence constrained production scheduling problems. These ideas are used to develop an implementation that is tested on a variety of real-life, large scale instances; yielding optimal solutions in very practicable CPU time. Citation Unpublished. Columbia University, BHP Billiton, August 2009. … Read more
The classical branch-and-bound algorithm for the integer feasibility problem has exponential worst case complexity. We prove that it is surprisingly efficient on reformulations, in which the columns of the constraint matrix are short, and near orthogonal, i.e. a reduced basis of the generated lattice; when the entries of A (i.e. the dense part of the … Read more
We describe techniques combining the S-lemma and computation of projected quadratics which experimentally yield strong bounds on the value of convex quadratic programs with nonconvex constraints Citation unpublished report, Columbia University, March 2009 Article Download View Eigenvalue techniques for proving bounds for convex objective, nonconvex programs
Cybersecurity is a growing concern, especially in open grids, where attack propagation is easy because of prevalent collaborations among thousands of users and hundreds of institutions. The collaboration rules that typically govern large science experiments as well as social networks of scientists span across the institutional security boundaries. A common concern is that the increased … Read more