Romesh Saigal

Linear Programming: A Modern Integrated Analysis

1 Introduction.- 1.1 The Problem.- 1.2 Prototype Problems.- 1.3 About this Book.- 1.4 Notes.- 2 Background.- 2.1 Real Analysis.- 2.2 Linear Algebra and Matrix Analysis.- 2.3 Numerical Linear Algebra.- 2.4 Convexity and Separation Theorems.- 2.5 Linear Equations and Inequalities.- 2.6 Convex Polyhedral Sets.- 2.7 Nonlinear System of Equations.- 2.8 Notes.- 3 Duality Theory and Optimality Conditions.- 3.1 The Dual Problem.- 3.2 Duality Theorems.- 3.3 Optimality and Complementary Slackness.- 3.4 Complementary Pair of Variables.- 3.5 Degeneracy and Uniqueness.- 3.6 Notes.- 4 Boundary Methods.- 4.1 Introduction.- 4.2 Primal Simplex Method.- 4.3 Bounded Variable Simplex Method.- 4.4 Dual Simplex Method.- 4.5 Primal - Dual Method.- 4.6 Notes.- 5 Interior Point Methods.- 5.1 Primal Affine Scaling Method.- 5.2 Degeneracy Resolution by Step-Size Control.- 5.3 Accelerated Affine Scaling Method.- 5.4 Primal Power Affine Scaling Method.- 5.5 Obtaining an Initial Interior Point.- 5.6 Bounded Variable Affine Scaling Method.- 5.7 Affine Scaling and Unrestricted Variables.- 5.8 Dual Affine Scaling Method.- 5.9 Primal-Dual Affine Scaling Method.- 5.10 Path Following or Homotopy Methods.- 5.11 Projective Transformation Method.- 5.12 Method and Unrestricted Variables.- 5.13 Notes.- 6 Implementation.- 6.1 Implementation of Boundary Methods.- 6.2 Implementation of Interior Point Methods.- 6.3 Notes.- A Tables.

Risk, Financing and the Optimal Number of Suppliers

Should firms in developed economies work with more or fewer suppliers than firms in developing economies? More generally, how does the number of suppliers for a firm depend on the firm’s economic environment? To answer these questions we identify several economic and business factors that might affect the number of suppliers (and that separate developed and developing economies): supply risk, fixed costs of working with suppliers, and access to financing (particularly trade-credit financing).

A simple proof of a primal affine scaling method

In this paper, we present a simpler proof of the result of Tsuchiya and Muramatsu on the convergence of the primal affine scaling method. We show that the primal sequence generated by the method converges to the interior of the optimum face and the dual sequence to the analytic center of the optimal dual face, when the step size implemented in the procedure is bounded by 2/3. We also prove the optimality of the limit of the primal sequence for a slightly larger step size of 2q/(3q−1), whereq is the number of zero variables in the limit. We show this by proving the dual feasibility of a cluster point of the dual sequence.

An Incomplete Cholesky Factorization for Dense Symmetric Positive Definite Matrices

In this paper, we study the use of an incomplete Cholesky factorization (ICF) as a preconditioner for solving dense symmetric positive definite linear systems. This method is suitable for situations where matrices cannot be explicitly stored but each column can be easily computed. Analysis and implementation of this preconditioner are discussed. We test the proposed ICF on randomly generated systems and large matrices from two practical applications: semidefinite programming and support vector machines. Numerical comparison with the diagonal preconditioner is also presented.

Vehicle Platoon Control in High-Latency Wireless Communications Environment: Model Predictive Control Method

Recent developments in vehicle onboard computers and vehicle-to-vehicle communications technology allow automatic control of vehicles and the organization of vehicles into platoons with short intraplatoon distances. One of the major issues with platoon control is latency in wireless communications. Latency has a negative impact on safety and disrupts the stability of platoons. A decentralized longitudinal platoon-controlling mechanism that uses a model predictive control approach to control vehicles safely, even in harsh communications environments, is proposed. The sensitivity of this method was analyzed to derive the conditions for this method to work safely. A simulation test bed for this control method was implemented to test effectiveness and safety under two communications latency settings. The results showed that the model predictive control method could safely control the platoon even in harsh communications environments.

Validation of stochastic traffic flow model with microscopic traffic simulation

Traffic congestion in urban areas is posing many challenges, and a traffic flow model that accurately predicts traffic conditions can be useful in responding to them. With the limitation of infrastructure and the difficulty for real world experiment, traffic simulation is a good tool for the validation and future potential applications of the traffic flow model. This research presents a stochastic traffic flow model which uses a stochastic partial differential equation to describe the evolution of traffic flow on the highway. The stochastic model is calibrated and validated by real traffic data on a highway and is proved to have better predictive power than the deterministic model. Also, a microscopic traffic simulation is built and calibrated by the same highway data, and the validation result shows that the simulation is capable of representing the real traffic. By using traffic simulation to further explore the capability of the stochastic model, the validation demonstrated that the prediction works well at most locations, and the interpolation error may be improved by considering the influence of ramps and the change in the number of lanes.

Distance-Based Dynamic Pricing Strategy for Managed Toll Lanes

Congestion in highway networks inflicts economic and emotional stress on drivers and a considerable cost to society. To mitigate some of these effects, managed toll lanes are being designed and built in the United States. These managed toll lanes guarantee to drivers that they will be able to travel at free-flow speed during peak hours when the general lanes are congested. The strategy used to price tolls on managed toll lanes has become an important issue; however, few studies have focused on dynamic optimal pricing strategies. This paper formulates the pricing problem on the basis of a stochastic macroscopic traffic flow model and investigates the methodology used to find a pricing strategy to maximize the total expected revenue. A simulation-based numerical algorithm that obtains the optimal prices efficiently in real time is proposed. The methodology is also applicable and readily adjusted for other objective purposes of administrators, such as maximization of total throughput. The general pricing model developed in this paper is not limited to one specific underlying traffic flow model and is readily adapted to other macroscopic traffic models, such as the classical Lighthill–Whitham–Richards model.

The multi-item newsvendor model with cross-selling and the solution when demand is jointly normally distributed

Products are often demanded in tandem because of the cross-selling effect. The demand for an item can increase if sales of its cross-selling-associated items are achieved or decrease when the associated items are out of stock, resulting in lost sales. Therefore, a joint inventory policy should be pursued in a cross-selling system. This paper introduces customer-driven cross-selling into centralized and competitive newsvendor (NV) models by representing an item’s effective demand as a function of other items’ order quantities. We derive first-order optimality conditions for the centralized model in addition to pure-strategy Nash equilibrium conditions and uniqueness conditions of the equilibria for the competitive model. We further develop gradient-based (GB) and iteration-based (IB) algorithms to solve the centralized and competitive models, respectively. A computational study verifies the effectiveness of the proposed algorithms. The computational results show that a larger cross-selling effect leads to a larger order quantity in a centralized NV model but a smaller order quantity in a competitive NV model, and a larger positive correlation between items’ demands leads to higher profits with smaller order quantities in both models. Moreover, NVs will order more items if the demand variance is greater, however resulting in lower profits. In a competitive situation, one will prefer smaller order quantities than in a centralized decision situation.

Combinatorial Auction to Allocate Traffic

This paper proposes an auction system implemented with vehicle-to-infrastructure devices to toll and to allocate traffic. Vehicles bid for paths before entering the network. By solving an optimization problem, the system assigns vehicles to paths and computes the corresponding toll. A mathematical model of this auction is presented and analyzed. The paper proves that this auction mechanism guarantees truthful reporting and maximizes social utility. The auction is then tested on a network with 5,100 vehicles. The paper also discusses the use of the auction as a toll-setting mechanism for high-occupancy vehicle or high-occupancy toll lanes.

Matrix partitioning methods for interior point algorithms

We consider here a linear programming problem whose rows of the constraint matrix can be partitioned into two parts. Such natural partitions exist in several special linear programs, including the assignment problem, the transportation problem, the generalized upper-bounded variable problem, the block diagonal linear program; and can also be used to induce sparsity patterns in Cholesky factorizations. In this paper, we propose a matrix partitioning method for interior point algorithms. The proposed method independently generates Cholesky factorizations of each part, and reduces the complexity to that of solving generally, a dense linear system involving several rank one updates of the identity matrix. Here, we propose solving this linear system by an inductive use of the Sherman-Morrison-Woodbury formula. The proposed method is easily implemented on a vector, parallel machine as well as on a distributed system. Such partitioning methods have been popular in the context of the simplex method, where the special structure of the basis matrix is exploited.