We consider the following integer feasibility problem: “Given positive integer numbers $a_0,a_1,\dots,a_n,$ with $\gcd(a_1,\dots,a_n)=1$ and $\va=(a_1,\dots,a_n)$, does there exist a vector $\vx\in\bbbz^n_{\ge \zero}$ satisfying $\va\vx = a_0$?” Some instances of this type have been found to be extremely hard to solve by standard methods such as branch-and-bound, even if the number of variables is as … Read more
We study the issue of updating the analytic center after multiple cutting planes have been added through the analytic center of the current polytope in Euclidean n-space. This is an important issue that arises at every `stage’ in a cutting plane algorithm. If q cuts are to be added, with q no larger than n, … Read more
We investigate the polyhedral structure of an integer program with a single functional constraint: the integer capacity-cover polyhedron. Such constraints arise in telecommunications planning and facility location applications, and feature the use of general integer (rather than just binary) variables. We derive a large class of facet-defining inequalities by using an augmenting technique that builds … Read more
We prove a sufficient global optimality condition for quadratic optimization with quadratic constraints where the variables are allowed to take -1 and 1 values. We extend the condition to quadratic programs with matrix variables and orthogonality conditions, and in particular, to the quadratic assignment problem. CitationBilkent University Technical Report, September 2002.ArticleDownload View PDF
When combinatorial bidding is permitted in Spectrum Auctions, such as the upcoming FCC auction #31, the resulting winner-determination problem can be computationally challenging. We present a branch-and-price algorithm based on a set-packing formulation originally proposed by Dietrich and Forrest (2002). This formulation has a variable for every possible combination of winning bids for each bidder. … Read more
Given vectors $a,c\in Z^n$ and $b\in Z$, we consider the (unbounded) knapsack optimization problem $P:\,\min\{c’x\,\vert\, a’x=b;\,x\in N^n\}$. We compute the minimum value $p^*$ using techniques from complex analysis, namely Cauchy residue technique to integrate a function in $C^2$, the $Z$-transform of an appropriate function related to $P$. The computational complexity depends on $s:=\sum_{a_j} a_j$, not … Read more
Hierarchies of semidefinite relaxations for $0/1$ polytopes have been constructed by Lasserre (2001a) and by Lov\’asz and Schrijver (1991), permitting to find the cut polytope of a graph on $n$ nodes in $n$ steps. We show that $\left\lceil {n\over 2} \right\rceil$ iterations are needed for finding the cut polytope of the complete graph $K_n$. CitationMathematics … Read more
Current mixed-integer linear programming solvers are based on linear programming routines that use floating point arithmetic. Occasionally, this leads to wrong solutions, even for problems where all coefficients and all solution components are small integers. It is shown how, using directed rounding and interval arithmetic, cheap pre- and postprocessing of the linear programs arising in … Read more
The basic principle of the cutting plane techniques is to chop away the portions of the solution space of the linear programming relaxation of an integer program that contain no integer solutions. this is true for both Gomory’s cutting planes, and other more recent cuts based on valid inequalities. Obtaining a partial or full description … Read more
In one of fundamental work in combinatorial optimization Edmonds gave a complete linear description of the matching polytope. Matchings in a graph are equivalent to stable sets its line graph. Also the neighborhood of any vertex in a line graph partitions into two cliques: graphs with this latter property are called quasi-line graphs. Quasi-line graphs … Read more