Showing papers on "Linear programming published in 2006"

A computationally efficient mixed-integer linear formulation for the thermal unit commitment problem

Abstract: This paper presents a new mixed-integer linear formulation for the unit commitment problem of thermal units. The formulation proposed requires fewer binary variables and constraints than previously reported models, yielding a significant computational saving. Furthermore, the modeling framework provided by the new formulation allows including a precise description of time-dependent startup costs and intertemporal constraints such as ramping limits and minimum up and down times. A commercially available mixed-integer linear programming algorithm has been applied to efficiently solve the unit commitment problem for practical large-scale cases. Simulation results back these conclusions

A Robust Optimization Approach to Inventory Theory

Abstract: We propose a general methodology based on robust optimization to address the problem of optimally controlling a supply chain subject to stochastic demand in discrete time. This problem has been studied in the past using dynamic programming, which suffers from dimensionality problems and assumes full knowledge of the demand distribution. The proposed approach takes into account the uncertainty of the demand in the supply chain without assuming a specific distribution, while remaining highly tractable and providing insight into the corresponding optimal policy. It also allows adjustment of the level of robustness of the solution to trade off performance and protection against uncertainty. An attractive feature of the proposed approach is its numerical tractability, especially when compared to multidimensional dynamic programming problems in complex supply chains, as the robust problem is of the same difficulty as the nominal problem, that is, a linear programming problem when there are no fixed costs, and a mixed-integer programming problem when fixed costs are present. Furthermore, we show that the optimal policy obtained in the robust approach is identical to the optimal policy obtained in the nominal case for a modified and explicitly computable demand sequence. In this way, we show that the structure of the optimal robust policy is of the same base-stock character as the optimal stochastic policy for a wide range of inventory problems in single installations, series systems, and general supply chains. Preliminary computational results are very promising.

An introduction to convex optimization for communications and signal processing

Abstract: Convex optimization methods are widely used in the design and analysis of communication systems and signal processing algorithms. This tutorial surveys some of recent progress in this area. The tutorial contains two parts. The first part gives a survey of basic concepts and main techniques in convex optimization. Special emphasis is placed on a class of conic optimization problems, including second-order cone programming and semidefinite programming. The second half of the survey gives several examples of the application of conic programming to communication problems. We give an interpretation of Lagrangian duality in a multiuser multi-antenna communication problem; we illustrate the role of semidefinite relaxation in multiuser detection problems; we review methods to formulate robust optimization problems via second-order cone programming techniques

Robust Branch-and-Cut-and-Price for the Capacitated Vehicle Routing Problem

Abstract: The best exact algorithms for the Capacitated Vehicle Routing Problem (CVRP) have been based on either branch-and-cut or Lagrangean relaxation/column generation. This paper presents an algorithm that combines both approaches: it works over the intersection of two polytopes, one associated with a traditional Lagrangean relaxation over q-routes, the other defined by bound, degree and capacity constraints. This is equivalent to a linear program with exponentially many variables and constraints that can lead to lower bounds that are superior to those given by previous methods. The resulting branch-and-cut-and-price algorithm can solve to optimality all instances from the literature with up to 135 vertices. This more than doubles the size of the instances that can be consistently solved.

Decomposition Techniques in Mathematical Programming: Engineering and Science Applications

Abstract: Motivation and Introduction.- Motivating Examples: Models with Decomposable Structure.- Decomposition Techniques.- Decomposition in Linear Programming: Complicating Constraints.- Decomposition in Linear Programming: Complicating Variables.- Duality.- Decomposition in Nonlinear Programming.- Decomposition in Mixed-Integer Programming.- Other Decomposition Techniques.- Local Sensitivity Analysis.- Local Sensitivity Analysis.- Applications.- Applications.- Computer Codes.- Some GAMS Implementations.- Solution to Selected Exercises.- Exercise Solutions.

On Distributionally Robust Chance-Constrained Linear Programs

Abstract: In this paper, we discuss linear programs in which the data that specify the constraints are subject to random uncertainty. A usual approach in this setting is to enforce the constraints up to a given level of probability. We show that, for a wide class of probability distributions (namely, radial distributions) on the data, the probability constraints can be converted explicitly into convex second-order cone constraints; hence, the probability-constrained linear program can be solved exactly with great efficiency. Next, we analyze the situation where the probability distribution of the data is not completely specified, but is only known to belong to a given class of distributions. In this case, we provide explicit convex conditions that guarantee the satisfaction of the probability constraints for any possible distribution belonging to the given class.

Coverage by directional sensors in randomly deployed wireless sensor networks

Abstract: We study a novel “coverage by directional sensors” problem with tunable orientations on a set of discrete targets. We propose a Maximum Coverage with Minimum Sensors (MCMS) problem in which coverage in terms of the number of targets to be covered is maximized whereas the number of sensors to be activated is minimized. We present its exact Integer Linear Programming (ILP) formulation and an approximate (but computationally efficient) centralized greedy algorithm (CGA) solution. These centralized solutions are used as baselines for comparison. Then we provide a distributed greedy algorithm (DGA) solution. By incorporating a measure of the sensors residual energy into DGA, we further develop a Sensing Neighborhood Cooperative Sleeping (SNCS) protocol which performs adaptive scheduling on a larger time scale. Finally, we evaluate the properties of the proposed solutions and protocols in terms of providing coverage and maximizing network lifetime through extensive simulations. Moreover, for the case of circular coverage, we compare against the best known existing coverage algorithm.

Distributed algorithms for maximum lifetime routing in wireless sensor networks

Abstract: A sensor network of nodes with wireless transceiver capabilities and limited energy is considered. We propose distributed algorithms to compute an optimal routing scheme that maximizes the time at which the first node in the network drains out of energy. The problem is formulated as a linear programming problem and subgradient algorithms are used to solve it in a distributed manner. The resulting algorithms have low computational complexity and are guaranteed to converge to an optimal routing scheme that maximizes the network lifetime. The algorithms are illustrated by an example in which an optimal flow is computed for a network of randomly distributed nodes. We also show how our approach can be used to obtain distributed algorithms for many different extensions to the problem. Finally, we extend our problem formulation to more general definitions of network lifetime to model realistic scenarios in sensor networks

Combinatorial Benders' Cuts for Mixed-Integer Linear Programming

Abstract: Mixed-integer programs (MIPs) involving logical implications modeled through big-M coefficients are notoriously among the hardest to solve. In this paper, we propose and analyze computationally an automatic problem reformulation of quite general applicability, aimed at removing the model dependency on the big-M coefficients. Our solution scheme defines a master integer linear problem (ILP) with no continuous variables, which contains combinatorial information on the feasible integer variable combinations that can be “distilled” from the original MIP model. The master solutions are sent to a slave linear program (LP), which validates them and possibly returns combinatorial inequalities to be added to the current master ILP. The inequalities are associated to minimal (or irreducible) infeasible subsystems of a certain linear system, and can be separated efficiently in case the master solution is integer. The overall solution mechanism closely resembles the Benders' one, but the cuts we produce are purely co...

Understanding and Using Linear Programming

Abstract: What Is It, and What For?- Examples- Integer Programming and LP Relaxation- Theory of Linear Programming: First Steps- The Simplex Method- Duality of Linear Programming- Not Only the Simplex Method- More Applications- Software and Further Reading

Underdetermined blind source separation based on sparse representation

Abstract: This paper discusses underdetermined (i.e., with more sources than sensors) blind source separation (BSS) using a two-stage sparse representation approach. The first challenging task of this approach is to estimate precisely the unknown mixing matrix. In this paper, an algorithm for estimating the mixing matrix that can be viewed as an extension of the DUET and the TIFROM methods is first developed. Standard clustering algorithms (e.g., K-means method) also can be used for estimating the mixing matrix if the sources are sufficiently sparse. Compared with the DUET, the TIFROM methods, and standard clustering algorithms, with the authors' proposed method, a broader class of problems can be solved, because the required key condition on sparsity of the sources can be considerably relaxed. The second task of the two-stage approach is to estimate the source matrix using a standard linear programming algorithm. Another main contribution of the work described in this paper is the development of a recoverability analysis. After extending the results in , a necessary and sufficient condition for recoverability of a source vector is obtained. Based on this condition and various types of source sparsity, several probability inequalities and probability estimates for the recoverability issue are established. Finally, simulation results that illustrate the effectiveness of the theoretical results are presented.

Linear-programming-based techniques for synthesis of network-on-chip architectures

Abstract: Application-specific system-on-chip (SoC) design offers the opportunity for incorporating custom network-on-chip (NoC) architectures that are more suitable for a particular application, and do not necessarily conform to regular topologies. This paper presents novel mixed integer linear programming (MILP) formulations for synthesis of custom NoC architectures. The optimization objective of the techniques is to minimize the power consumption subject to the performance constraints. We present a two-stage approach for solving the custom NoC synthesis problem. The power consumption of the NoC architecture is determined by both the physical links and routers. The power consumption of a physical link is dependent upon the length of the link, which in turn, is governed by the layout of the SoC. Therefore, in the first stage, we address the floorplanning problem that determines the locations of the various cores and the routers. In the second stage, we utilize the floorplan from the first stage to generate topology of the NoC and the routes for the various traffic traces. We also present a clustering-based heuristic technique for the second stage to reduce the run times of the MILP formulation. We analyze the quality of the results and solution times of the proposed techniques by extensive experimentation with realistic benchmarks and comparisons with regular mesh-based NoC architectures.

Approximation algorithms for allocation problems: Improving the factor of 1 - 1/e

Abstract: Combinatorial allocation problems require allocating items to players in a way that maximizes the total utility. Two such problems received attention recently, and were addressed using the same linear programming (LP) relaxation. In the Maximum Submodular Welfare (SMW) problem, utility functions of players are submodular, and for this case Dobzinski and Schapira [SODA 2006] showed an approximation ratio of 1 - 1/e. In the Generalized Assignment Problem (GAP) utility functions are linear but players also have capacity constraints. GAP admits a (1 - 1/e)- approximation as well, as shown by Fleischer, Goemans, Mirrokni and Sviridenko [SODA 2006]. In both cases, the approximation ratio was in fact shown for a more general version of the problem, for which improving 1 - 1/e is NPhard. In this paper, we show how to improve the 1 - 1/e approximation ratio, both for SMW and for GAP. A common theme in both improvements is the use of a new and optimal Fair Contention Resolution technique. However, each of the improvements involves a different rounding procedure for the above mentioned LP. In addition, we prove APX-hardness results for SMW (such results were known for GAP). An important feature of our hardness results is that they apply even in very restricted settings, e.g. when every player has nonzero utility only for a constant number of items.

Mixed integer models for the stationary case of gas network optimization

Abstract: A gas network basically consists of a set of compressors and valves that are connected by pipes. The problem of gas network optimization deals with the question of how to optimize the flow of the gas and to use the compressors cost-efficiently such that all demands of the gas network are satisfied. This problem leads to a complex mixed integer nonlinear optimization problem. We describe techniques for a piece-wise linear approximation of the nonlinearities in this model resulting in a large mixed integer linear program. We study sub-polyhedra linking these piece-wise linear approximations and show that the number of vertices is computationally tractable yielding exact separation algorithms. Suitable branching strategies complementing the separation algorithms are also presented. Our computational results demonstrate the success of this approach.

Linear Programming Relaxations and Belief Propagation -- An Empirical Study

Abstract: The problem of finding the most probable (MAP) configuration in graphical models comes up in a wide range of applications. In a general graphical model this problem is NP hard, but various approximate algorithms have been developed. Linear programming (LP) relaxations are a standard method in computer science for approximating combinatorial problems and have been used for finding the most probable assignment in small graphical models. However, applying this powerful method to real-world problems is extremely challenging due to the large numbers of variables and constraints in the linear program. Tree-Reweighted Belief Propagation is a promising recent algorithm for solving LP relaxations, but little is known about its running time on large problems. In this paper we compare tree-reweighted belief propagation (TRBP) and powerful general-purpose LP solvers (CPLEX) on relaxations of real-world graphical models from the fields of computer vision and computational biology. We find that TRBP almost always finds the solution significantly faster than all the solvers in CPLEX and more importantly, TRBP can be applied to large scale problems for which the solvers in CPLEX cannot be applied. Using TRBP we can find the MAP configurations in a matter of minutes for a large range of real world problems.

Use of linear programming for dynamic subcarrier and bit allocation in multiuser OFDM

Abstract: An adaptive subcarrier allocation and an adaptive modulation for multiuser orthogonal frequency-division multiplexing (OFDM) are considered. The optimal subcarrier and bit allocation problems, which are previously formulated as nonlinear optimizations, are reformulated into and solved by integer programming (IP). A suboptimal approach that performs subcarrier allocation and bit loading separately is proposed. It is shown that the subcarrier allocation in this approach can be optimized by the linear-programming (LP) relaxation of IP, while the bit loading can be performed in a manner similar to a single-user OFDM. In addition, a heuristic method for solving the LP problem is presented. The LP-based suboptimal and heuristic algorithms are considerably simpler to implement than the optimal IP, plus their performances are close to those of the optimal approach

Minimum layout perturbation-based artwork legalization with grid constraints for hierarchical designs

Abstract: A method comprises extracting a hierarchical grid constraint set and modeling one or more critical objects of at least one cell as a variable set. The method further comprises solving a linear programming problem based on the hierarchical grid constraint set with the variable set to provide initial locations of the critical objects of the at least one cell and determining target on-grid locations of the one or more critical objects in the at least one cell using the results of the linear programming solution.

A probabilistic approach to optimal robust path planning with obstacles

Abstract: Autonomous vehicles need to plan trajectories to a specified goal that avoid obstacles. Previous approaches that used a constrained optimization approach to solve for finite sequences of optimal control inputs have been highly effective. For robust execution, it is essential to take into account the inherent uncertainty in the problem, which arises due to uncertain localization, modeling errors, and disturbances. Prior work has handled the case of deterministically bounded uncertainty. We present here an alternative approach that uses a probabilistic representation of uncertainty, and plans the future probabilistic distribution of the vehicle state so that the probability of collision with obstacles is below a specified threshold. This approach has two main advantages; first, uncertainty is often modeled more naturally using a probabilistic representation (for example in the case of uncertain localization); second, by specifying the probability of successful execution, the desired level of conservatism in the plan can be specified in a meaningful manner. The key idea behind the approach is that the probabilistic obstacle avoidance problem can be expressed as a disjunctive linear program using linear chance constraints. The resulting disjunctive linear program has the same complexity as that corresponding to the deterministic path planning problem with no representation of uncertainty. Hence the resulting problem can be solved using existing, efficient techniques, such that planning with uncertainty requires minimal additional computation. Finally, we present an empirical validation of the new method with a number of aircraft obstacle avoidance scenarios.

Decomposition with branch-and-cut approaches for two-stage stochastic mixed-integer programming

Abstract: Decomposition has proved to be one of the more effective tools for the solution of large-scale problems, especially those arising in stochastic programming. A decomposition method with wide applicability is Benders' decomposition, which has been applied to both stochastic programming as well as integer programming problems. However, this method of decomposition relies on convexity of the value function of linear programming subproblems. This paper is devoted to a class of problems in which the second-stage subproblem(s) may impose integer restrictions on some variables. The value function of such integer subproblem(s) is not convex, and new approaches must be designed. In this paper, we discuss alternative decomposition methods in which the second-stage integer subproblems are solved using branch-and-cut methods. One of the main advantages of our decomposition scheme is that Stochastic Mixed-Integer Programming (SMIP) problems can be solved by dividing a large problem into smaller MIP subproblems that can be solved in parallel. This paper lays the foundation for such decomposition methods for two-stage stochastic mixed-integer programs.

Network Coding for Multiple Unicasts: An Approach based on Linear Optimization

Abstract: In this paper we consider the application of network coding to a multiple unicast setup. Wbe present two suboptimal, yet practical code construction techniques. One consists of a linear program and the other of an integer program with fewer variables and constraints. We discuss the performance of the proposed techniques as well as their complexity.

Design and Control of a Large Call Center: Asymptotic Analysis of an LP-Based Method

Abstract: This paper analyzes a call center model with m customer classes and r agent pools. The model is one with doubly stochastic arrivals, which means that the m-vector of instantaneous arrival rates is allowed to vary both temporally and stochastically. Two levels of call center management are considered: staffing the r pools of agents, and dynamically routing calls to agents. The system managers objective is to minimize the sum of personnel costs and abandonment penalties. We consider a limiting parameter regime that is natural for call centers and relatively easy to analyze, but apparently novel in the literature of applied probability. For that parameter regime, we prove an asymptotic lower bound on expected total cost, which uses a strikingly simple distillation of the original system data. We then propose a method for staffing and routing based on linear programming (LP), and show that it achieves the asymptotic lower bound on expected total cost; in that sense the proposed method is asymptotically optimal.