Showing papers on "Integer programming published in 2007"

Operations Research an Introduction

Abstract: 1. Overview of Operations Research. I. DETERMINISTIC MODELS. 2. Introduction to Linear Programming. 3. The Simplex Method. 4. Duality and Sensitivity Analysis. 5. Transportation Model and Its Variants. 6. Network Models. 7. Advanced Linear Programming. 8. Goal Programming. 9. Integer Linear Programming. 10. Deterministic Dynamic Programming. 11. Deterministic Inventory Models. II. PROBABILISTIC MODELS. 12. Review of Basic Probability. 13. Forecasting Models. 14. Decision Analysis and Games. 15. Probabilistic Dynamic Programming. 16. Probabilistic Inventory Models. 17. Queueing Systems. 18. Simulation Modeling. 19. Markovian Decision Process. III. NONLINEAR MODELS. 20. Classical Optimization Theory. 21. Nonlinear Programming Algorithms. Appendix A: Review of Matrix Algebra. Appendix B: Introduction to Simnet II. Appendix C: Tora and Simnet II Installation and Execution. Appendix D: Statistical Tables. Appendix E: Answers to Odd-Numbered Problems. Index.

Constraint Integer Programming

Abstract: This thesis introduces the novel paradigm of constraint integer programming (CIP), which integrates constraint programming (CP) and mixed integer programming (MIP) modeling and solving techniques. It is supplemented by the software SCIP, which is a solver and framework for constraint integer programming that also features SAT solving techniques. SCIP is freely available in source code for academic and non-commercial purposes. Our constraint integer programming approach is a generalization of MIP that allows for the inclusion of arbitrary constraints, as long as they turn into linear constraints on the continuous variables after all integer variables have been fixed. The constraints, may they be linear or more complex, are treated by any combination of CP and MIP techniques: the propagation of the domains by constraint specific algorithms, the generation of a linear relaxation and its solving by LP methods, and the strengthening of the LP by cutting plane separation. The current version of SCIP comes with all of the necessary components to solve mixed integer programs. In the thesis, we cover most of these ingredients and present extensive computational results to compare different variants for the individual building blocks of a MIP solver. We focus on the algorithms and their impact on the overall performance of the solver. In addition to mixed integer programming, the thesis deals with chip design verification, which is an important topic of electronic design automation. Chip manufacturers have to make sure that the logic design of a circuit conforms to the specification of the chip. Otherwise, the chip would show an erroneous behavior that may cause failures in the device where it is employed. An important subproblem of chip design verification is the property checking problem, which is to verify whether a circuit satisfies a specified property. We show how this problem can be modeled as constraint integer program and provide a number of problem-specific algorithms that exploit the structure of the individual constraints and the circuit as a whole. Another set of extensive computational benchmarks compares our CIP approach to the current state-of-the-art SAT methodology and documents the success of our method.

Energy minimizing vehicle routing problem

Abstract: This paper proposes a new cost function based on distance and load of the vehicle for the Capacitated Vehicle Routing Problem. The vehicle-routing problem with this new load-based cost objective is called the Energy Minimizing Vehicle Routing Problem (EMVRP). Integer linear programming formulations with O(n2) binary variables and O(n2) constraints are developed for the collection and delivery cases, separately. The proposed models are tested and illustrated by classical Capacitated Vehicle Routing Problem (CVRP) instances from the literature using CPLEX 8.0.

Optimization Methods in Finance

Abstract: 1. Introduction 2. Linear programming: theory and algorithms 3. LP models: asset/liability cash flow matching 4. LP models: asset pricing and arbitrage 5. Nonlinear programming: theory and algorithms 6. NLP volatility estimation 7. Quadratic programming: theory and algorithms 8. QP models: portfolio optimization 9. Conic optimization tools 10. Conic optimization models in finance 11. Integer programming: theory and algorithms 12. IP models: constructing an index fund 13. Dynamic programming methods 14. DP models: option pricing 15. DP models: structuring asset backed securities 16. Stochastic programming: theory and algorithms 17. SP models: value-at-risk 18. SP models: asset/liability management 19. Robust optimization: theory and tools 20. Robust optimization models in finance Appendix A. Convexity Appendix B. Cones Appendix C. A probability primer Appendix D. The revised simplex method Bibliography Index.

What You Should Know about the Vehicle Routing Problem

Abstract: In the Vehicle Routing Problem (VRP), the aim is to design a set of m minimum cost vehicle routes through n customer locations, so that each route starts and ends at a common location and some side constraints are satisfied. Common applications arise in newspaper and food delivery, and in milk collection. This article summarizes the main known results for the classical VRP in which only vehicle capacity constraints are present. The article is structured around three main headings: exact algorithms, classical heuristics, and metaheuristics. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007

Power system transmission network expansion planning using AC model

Abstract: An optimisation technique to solve transmission network expansion planning problem, using the AC model, is presented. This is a very complex mixed integer nonlinear programming problem. A constructive heuristic algorithm aimed at obtaining an excellent quality solution for this problem is presented. An interior point method is employed to solve nonlinear programming problems during the solution steps of the algorithm. Results of the tests, carried out with three electrical energy systems, show the capabilities of the method and also the viability of using the AC model to solve the problem.

Valid inequalities for mixed integer linear programs

Abstract: This tutorial presents a theory of valid inequalities for mixed integer linear sets. It introduces the necessary tools from polyhedral theory and gives a geometric understanding of several classical families of valid inequalities such as lift-and-project cuts, Gomory mixed integer cuts, mixed integer rounding cuts, split cuts and intersection cuts, and it reveals the relationships between these families. The tutorial also discusses computational aspects of generating the cuts and their strength.

Hybrid Particle Swarm Optimization Approach for Solving the Discrete OPF Problem Considering the Valve Loading Effects

Abstract: This paper presents a hybrid particle swarm optimization algorithm (HPSO) as a modern optimization tool to solve the discrete optimal power flow (OPF) problem that has both discrete and continuous optimization variables. The problem is classified as constrained mixed integer nonlinear programming with multimodal characteristics. The objective functions considered are the system real power losses, fuel cost, and the gaseous emissions of the generating units. Two different types of fuel cost functions are considered in this study, namely the conventional quadratic function and the augmented quadratic function to introduce more accurate modeling that incorporates the valve loading effects. The latter model presents nondifferentiable and nonconvex regions that challenge most gradient-based optimization algorithms. The proposed algorithm makes use of the PSO, known for its global searching capabilities, to allocate the optimal control settings while Newton-Raphson algorithm minimizes the mismatch of the power flow equations. A hybrid inequality constraint handling mechanism that preserves only feasible solutions without the need to augment the original objective function is incorporated in the proposed approach. To demonstrate its robustness, the proposed algorithm was tested on the IEEE 30-bus system with six generating units. Several cases were investigated to test and validate the consistency of detecting optimal or near optimal solution for each objective. Results are compared to solutions obtained of MATPOWER software outcomes that employs sequential quadratic programming algorithm to solve the OPF. The impact of the proposed inequality constraint handling method in improving the HPSO performance is illustrated. Furthermore, a study of HPSO parameters tuning with regard to the OPF problem is presented and analyzed.

Exact and Heuristic Algorithms for the Weapon-Target Assignment Problem

Abstract: The weapon-target assignment (WTA) problem is a fundamental problem arising in defense-related applications of operations research. This problem consists of optimally assigning n weapons to m targets so that the total expected survival value of the targets after all the engagements is minimal. The WTA problem can be formulated as a nonlinear integer programming problem and is known to be NP-complete. No exact methods exist for the WTA problem that can solve even small-size problems (for example, with 20 weapons and 20 targets). Although several heuristic methods have been proposed to solve the WTA problem, due to the absence of exact methods, no estimates are available on the quality of solutions produced by such heuristics. In this paper, we suggest integer programming and network flow-based lower-bounding methods that we obtain using a branch-and-bound algorithm for the WTA problem. We also propose a network flow-based construction heuristic and a very large-scale neighborhood (VLSN) search algorithm. We present computational results of our algorithms, which indicate that we can solve moderately large instances (up to 80 weapons and 80 targets) of the WTA problem optimally and obtain almost optimal solutions of fairly large instances (up to 200 weapons and 200 targets) within a few seconds.

An Optimal PMU Placement Method Against Measurement Loss and Branch Outage

Abstract: This paper presents a new method for an optimal measurement placement of phasor measurement units (PMUs) for power system state estimation. The proposed method considers two types of contingency conditions (i.e., single measurement loss and single-branch outage) in order to obtain a reliable measurement system. First, the minimum condition number of the normalized measurement matrix is used as the criteria in conjunction with the sequential elimination approach to obtain a completely determined condition. Next, a sequential addition approach is used to search for necessary candidates for single measurement loss and single-branch outage conditions. These redundant measurements are optimized by binary integer programming. Finally, in order to minimize the number of PMU placement sites, a heuristic technique to rearrange measurement positions is also proposed. Numerical results on the IEEE test systems are demonstrated

Optimal Spectrum Sharing for Multi-Hop Software Defined Radio Networks

Abstract: Software defined radio (SDR) capitalizes advances in signal processing and radio technology and is capable of reconfiguring RF and switching to desired frequency bands. It is a frequency-agile data communication device that is vastly more powerful than recently proposed multi-channel multi-radio (MC-MR) technology. In this paper, we investigate the important problem of multi-hop networking with SDR nodes. For such network, each node has a pool of frequency bands (not necessarily of equal size) that can be used for communication. The uneven size of bands in the radio spectrum prompts the need of further division into sub-bands for optimal spectrum sharing. We characterize behaviors and constraints for such multi-hop SDR network from multiple layers, including modeling of spectrum sharing and sub-band division, scheduling and interference constraints, and flow routing. We give a formal mathematical formulation with the objective of minimizing the required network-wide radio spectrum resource for a set of user sessions. Since such problem formulation falls into mixed integer non-linear programming (MINLP), which is NP-hard in general, we develop a lower bound for the objective by relaxing the integer variables and linearization. Subsequently, we develop a near-optimal algorithm to this MINLP problem. This algorithm is based on a novel sequential fixing procedure, where the integer variables are determined iteratively via a sequence of linear programming. Simulation results show that solutions obtained by this algorithm are very close to lower bounds obtained via relaxation, thus suggesting that the solution produced by the algorithm is near-optimal.

Using Decomposition Techniques and Constraint Programming for Solving the Two-Dimensional Bin-Packing Problem

Abstract: The two-dimensional bin-packing problem is the problem of orthogonally packing a given set of rectangles into a minimum number of two-dimensional rectangular bins. The problem is NP-hard and very difficult to solve in practice as no good mixed integer programming (MIP) formulation has been found for the packing problem. We propose an algorithm based on the well-known Dantzig-Wolfe decomposition where the master problem deals with the production constraints on the rectangles while the subproblem deals with the packing of rectangles into a single bin. The latter problem is solved as a constraint-satisfaction problem (CSP), which makes it possible to formulate a number of additional constraints that may be difficult to formulate as MIP models. This includes guillotine-cutting requirements, relative positions, fixed positions and irregular bins. The CSP approach uses forward propagation to prune inferior arrangements of rectangles. Unsuccessful attempts to pack rectangles into a bin are brought back to the master model as valid inequalities. Hence, CSP is used not only to solve the pricing problem but also to generate valid inequalities in a branch-and-cut system. Using delayed column-generation, we obtain lower bounds of very good quality in reasonable time. In all instances considered, we obtain similar or better bounds than previously published. Several instances with up to n = 100 rectangles are solved to optimality through the developed branch-and-price-and-cut algorithm.

Using a Mixed Integer Quadratic Programming Solver for the Unconstrained Quadratic 0-1 Problem

Abstract: In this paper, we consider problem (P) of minimizing a quadratic function q(x)=x tQx+ctx of binary variables. Our main idea is to use the recent Mixed Integer Quadratic Programming (MIQP) solvers. But, for this, we have to first convexify the objective function q(x). A classical trick is to raise up the diagonal entries of Q by a vector u until (Q+diag(u)) is positive semidefinite. Then, using the fact that xi2=xi, we can obtain an equivalent convex objective function, which can then be handled by an MIQP solver. Hence, computing a suitable vector u constitutes a preprocessing phase in this exact solution method. We devise two different preprocessing methods. The first one is straightforward and consists in computing the smallest eigenvalue of Q. In the second method, vector u is obtained once a classical SDP relaxation of (P) is solved.We carry out computational tests using the generator of (Pardalos and Rodgers, 1990) and we compare our two solution methods to several other exact solution methods. Furthermore, we report computational results for the max-cut problem.

Memetic particle swarm optimization

Abstract: We propose a new Memetic Particle Swarm Optimization scheme that incorporates local search techniques in the standard Particle Swarm Optimization algorithm, resulting in an efficient and effective optimization method, which is analyzed theoretically. The proposed algorithm is applied to different unconstrained, constrained, minimax and integer programming problems and the obtained results are compared to that of the global and local variants of Particle Swarm Optimization, justifying the superiority of the memetic approach.

An Exact Solution Approach for Portfolio Optimization Problems under Stochastic and Integer Constraints

Abstract: In this paper, we study extensions of the classical Markowitz' mean-variance portfolio optimization model. First, we consider that the expected asset returns are stochastic by introducing a probabilistic constraint imposing that the expected return of the constructed portfolio must exceed a prescribed return level with a high confidence level. We study the deterministic equivalents of these models. In particular, we define under which types of probability distributions the deterministic equivalents are second-order cone programs, and give exact or approximate closed-form formulations. Second, we account for real-world trading constraints, such as the need to diversify the investments in a number of industrial sectors, the non-profitability of holding small positions and the constraint of buying stocks by lots, modeled with integer variables. To solve the resulting problems, we propose an exact solution approach in which the uncertainty in the estimate of the expected returns and the integer trading restrictions are simultaneously considered. The proposed algorithmic approach rests on a non-linear branch-and-bound algorithm which features two new branching rules. The first one is a static rule, called idiosyncratic risk branching, while the second one is dynamic and called portfolio risk branching. The proposed branching rules are implemented and tested using the open-source framework of the solver Bonmin. The comparison of the computational results obtained with standard MINLP solvers and with the proposed approach shows the effectiveness of this latter which permits to solve to optimality problems with up to 200 assets in a reasonable amount of time.

Optimal versus Heuristic Global Code Scheduling

Abstract: We present a global instruction scheduler based on inte- ger linear programming (ILP) that was implemented exper- imentally in the Intel Itanium® product compiler. It features virtually the full scale of known EPIC scheduling optimiza- tions, more than its heuristic counterpart in the compiler, GCS, and in contrast to the latter it computes optimal so- lutions in the form of schedules with minimal length. Due to our highly efficient ILP model it can solve problem in- stances with 500-750 instructions, and in combination with region scheduling we are able to schedule routines of arbi- trary size. In experiments on five SPEC® CPU2006 integer bench- marks, ILP-scheduled code exhibits a 32% schedule length advantage and a 10% runtime speedup over GCS-scheduled code, at the highest compiler optimization levels typically used for SPEC submissions. We further study the impact of different code motion classes, region sizes, and target microarchitectures, gaining insights into the nature of the global instruction scheduling problem.