On strongly polynomial algorithms for some classes of quadratic programming problems

In this paper we survey some results concerning polynomial and/or strongly polynomial solvability of some classes of quadratic programming problems. The discussion on polynomial solvability of continuous convex quadratic programming is followed by a couple of models for quadratic integer programming which, due to their special structure, allow polynomial (or even strongly polynomial) solvability. The theoretical merit of those results stems from the fact that a running time (i.e. the number of elementary arithmetic operations) of a strongly polynomial algorithm is independent of the input size of the problem

A MATHEMATICAL PROGRAMMING MODEL FOR OPTIMAL MANAGEMENT ZONE DELINEATION IN PRECISION AGRICULTURE

The identification of optimal management zones, including optimal uniform grid size, is a complex issue central to the successful implementation of variable rate input application. A novel economic optimization model is developed and applied that identifies the economically optimal management zone. Variable rate seeding can increase profits and reduce risk.Farm Management,

Locating a bioenergy facility using a hybrid optimization method

In this paper, the optimum location of a bioenergy generation facility for district energy applications is sought. A bioenergy facility usually belongs to a wider system, therefore a holistic approach is adopted to define the location that optimizes the system-wide operational and investment costs. A hybrid optimization method is employed to overcome the limitations posed by the complexity of the optimization problem. The efficiency of the hybrid method is compared to a stochastic (genetic algorithms) and an exact optimization method (Sequential Quadratic Programming). The results confirm that the hybrid optimization method proposed is the most efficient for the specific problem. (C) 2009 Elsevier B.V. All rights reserved

Near-Linear Time Algorithm for n-fold ILPs via Color Coding

We study an important case of ILPs max {c^Tx | Ax = b, l <= x <= u, x in Z^{n t}} with n * t variables and lower and upper bounds l, u in Z^{nt}. In n-fold ILPs non-zero entries only appear in the first r rows of the matrix A and in small blocks of size s x t along the diagonal underneath. Despite this restriction many optimization problems can be expressed in this form. It is known that n-fold ILPs can be solved in FPT time regarding the parameters s, r, and Delta, where Delta is the greatest absolute value of an entry in A. The state-of-the-art technique is a local search algorithm that subsequently moves in an improving direction. Both, the number of iterations and the search for such an improving direction take time Omega(n), leading to a quadratic running time in n. We introduce a technique based on Color Coding, which allows us to compute these improving directions in logarithmic time after a single initialization step. This leads to the first algorithm for n-fold ILPs with a running time that is near-linear in the number nt of variables, namely (rs Delta)^{O(r^2s + s^2)} L^2 * nt log^{O(1)}(nt), where L is the encoding length of the largest integer in the input. In contrast to the algorithms in recent literature, we do not need to solve the LP relaxation in order to handle unbounded variables. Instead, we give a structural lemma to introduce appropriate bounds. If, on the other hand, we are given such an LP solution, the running time can be decreased by a factor of L

Parametrisierte Algorithmen für Ganzzahlige Lineare Programme und deren Anwendungen für Zuweisungsprobleme

This thesis is concerned with solving NP-hard problems. We consider two prominent strategies of coping with such computationally hard questions efficiently. The first approach aims to design approximation algorithms, that is, we are content to find good, but non-optimal solutions in polynomial time. The second strategy is called Fixed-Parameter Tractability (FPT) and considers parameters of the instance to capture the hardness of the problem and by that, obtain efficient algorithms with respect to the remaining input. This thesis employs both strategies jointly to develop efficient approximation and exact algorithms using parameterization and modeling the problem as structured integer linear programs (ILPs), which can be solved in FPT. In the first part of this work, we concentrate on these well-structured ILPs. On the one hand, we develop an efficient algorithm for block-structured integer linear programs called n-fold ILPs. On the other hand, we investigate the similarly block-structured 2-stage stochastic ILPs and prove conditional lower bounds regarding the running time of any algorithm solving them that match the best known upper bounds. We also prove the tightness of certain structural parameters called sensitivity and proximity for ILPs which arise from combinatorial questions such as allocation problems. The second part utilizes n-fold ILPs and structural properties to add to and improve upon known results for Scheduling and Bin Packing problems. We design exact FPT algorithms for the Scheduling With Clique Incompatibilities, Bin Packing, and Multiple Knapsack problems. Further, we provide constant-factor approximation algorithms and polynomial time approximation schemes (PTAS) for the Class Constraint Scheduling problems. Broadening our scope, we also investigate this problem and the closely related Cardinality Constraint Scheduling problem in the online setting and derive lower bounds for the approximation ratios as well as a PTAS for them. Altogether, this thesis contributes to the knowledge about structured ILPs, proves their limits and reaffirms their usefulness for a plethora of allocation problems. In doing so, various new and improved algorithms with respect to the running time or approximation quality emerge