We give an elementary proof of optimality conditions for linear programming. The proof is direct, built on a straightforward classical perturbation of the constraints, and does not require either the use of Farkas’ lemma or the use of the simplex method. Citation Technical Report TRITA-MAT-2008-OS6, Department of Mathematics, Royal Institute of Technology (KTH), SE-100 44 … Read more
In many applications in statistics, finance, and insurance/reinsurance, one seeks a solution of finding a covariance matrix satisfying a large number of given linear equality and inequality constraints in a way that it deviates the least from a given symmetric matrix. The difficulty in finding an efficient method for solving this problem is due to … Read more
This paper presents algorithms for single and parallel identical machine scheduling problems. While the overall algorithm can be viewed as a branch-cut-and-price over a very large extended formulation, a number of auxiliary techniques are necessary to make the column generation effective. Those techniques include a powerful fixing by reduced costs and a new proposed dual … Read more
We investigate the problem of simultaneously determining the location of facilities and the design of vehicle routes to serve customer demands under vehicle and facility capacity restrictions. We present a set-partitioning-based formulation of the problem and study the relationship between his formulation and the graph-based formulations that have been used in previous studies of this … Read more
Branching is an important component of the branch-and-cut algorithm for solving mixed integer linear programs. Most solvers branch by imposing a disjunction of the form“$x_i \leq k \vee x_i \geq k+1$” for some integer $k$ and some integer-constrained variable $x_i$. A generalization of this branching scheme is to branch by imposing a more general disjunction … Read more
This paper addresses the one machine scheduling problem in which $n$ jobs have distinct due dates with earliness and tardiness costs. Fast neighborhoods are proposed for the problem. They are based on a block representation of the schedule and can be computed in $O(n^2)$. A timing operator is presented as well as swap and extract-and-reinsert … Read more
Euclidean Jordan-algebra is a commonly used tool in designing interior point algorithms for symmetric cone programs. T-algebra, on the other hand, has rarely been used in symmetric cone programming. In this paper, we use both algebraic characterizations of symmetric cones to extend the target-following framework of linear programming to symmetric cone programming. Within this framework, … Read more
The space of linear programs (LP) can be partitioned into a finite number of sets, each corresponding to a basis. This partition is thus called the basis partition. The closed-form solution on the space of LP can be determined with the basis partition if we can characterize the basis partition. A differential equation on the … Read more
In this paper I investigate several offline and online data transfer scheduling problems and propose efficient algorithms and techniques for addressing them. In the offline case, I present a novel, heuristic, algorithm for scheduling files with divisible sizes on multiple disjoint paths, in order to maximize the total profit (the problem is equivalent to the … Read more
Recently, Andersen et al.(2007), Borozan and Cornuejols (2007) and Cornuejols and Margot(2007) characterized extreme inequalities of a system of two rows with two free integer variables and nonnegative continuous variables. These inequalities are either split cuts or intersection cuts (Balas (1971)) derived using maximal lattice-free convex sets. In order to use these inequalities to obtain … Read more