Churlzu Lim

Algorithms for discrete and continuous multicommodity flow network interdiction problems

We consider a network interdiction problem on a multicommodity flow network, in which an attacker disables a set of network arcs in order to minimize the maximum profit that can be obtained from shipping commodities across the network. The attacker is assumed to have some budget for destroying (or “interdicting”) arcs, and each arc is associated with a positive interdiction expense. In this paper, we examine problems in which interdiction must be discrete (i.e., each arc must either be left alone or completely destroyed), and in which interdiction can be continuous (the capacities of arcs may be partially reduced). For the discrete problem, we describe a linearized model for optimizing network interdiction that is similar to previous studies in the field, and compare it to a penalty model that does not require linearization constraints. For the continuous case, we prescribe an optimal partitioning algorithm along with a heuristic procedure for estimating the optimal objective function value. We demonstrate on a set of randomly generated test data that our penalty model for the discrete interdiction problem significantly reduces computational time when compared to that consumed by the linearization model.

Survivable network design under optimal and heuristic interdiction scenarios

We examine the problem of building or fortifying a network to defend against enemy attacks in various scenarios. In particular, we examine the case in which an enemy can destroy any portion of any arc that a designer constructs on the network, subject to some interdiction budget. This problem takes the form of a three-level, two-player game, in which the designer acts first to construct a network and transmit an initial set of flows through the network. The enemy acts next to destroy a set of constructed arcs in the designer’s network, and the designer acts last to transmit a final set of flows in the network. Most studies of this nature assume that the enemy will act optimally; however, in real-world scenarios one cannot necessarily assume rationality on the part of the enemy. Hence, we prescribe optimal network design algorithms for three different profiles of enemy action: an enemy destroying arcs based on capacities, based on initial flows, or acting optimally to minimize our maximum profits obtained from transmitting flows.

Portfolio optimization by minimizing conditional value-at-risk via nondifferentiable optimization

Conditional Value-at-Risk (CVaR) is a portfolio evaluation function having appealing features such as sub-additivity and convexity. Although the CVaR function is nondifferentiable, scenario-based CVaR minimization problems can be reformulated as linear programs (LPs) that afford solutions via widely-used commercial softwares. However, finding solutions through LP formulations for problems having many financial instruments and a large number of price scenarios can be time-consuming as the dimension of the problem greatly increases. In this paper, we propose a two-phase approach that is suitable for solving CVaR minimization problems having a large number of price scenarios. In the first phase, conventional differentiable optimization techniques are used while circumventing nondifferentiable points, and in the second phase, we employ a theoretically convergent, variable target value nondifferentiable optimization technique. The resultant two-phase procedure guarantees infinite convergence to optimality. As an optional third phase, we additionally perform a switchover to a simplex solver starting with a crash basis obtained from the second phase when finite convergence to an exact optimum is desired. This three phase procedure substantially reduces the effort required in comparison with the direct use of a commercial stand-alone simplex solver (CPLEX 9.0). Moreover, the two-phase method provides highly-accurate near-optimal solutions with a significantly improved performance over the interior point barrier implementation of CPLEX 9.0 as well, especially when the number of scenarios is large. We also provide some benchmarking results on using an alternative popular proximal bundle nondifferentiable optimization technique.

Algorithms for Network Interdiction and Fortification Games

This chapter explores models and algorithms applied to a class of Stackelberg games on networks. In these network interdiction games, a network exists over which an operator wishes to execute some function, such as finding a shortest path, shipping a maximum flow, or transmitting a minimum cost multicommodity flow. The role of the interdictor is to compromise certain network elements before the operator acts, by (for instance) increasing the cost of flow or reducing capacity on an arc. We begin by reviewing the field of network interdiction and its related theoretical and mathematical foundations. We then discuss recent applications of stochastic models, valid inequalities, continuous bilinear programming techniques, and asymmetric analysis to network interdiction problems. Next, note that interdiction problems can be extended to a three-stage problem in which the operator fortifies the network (by increasing capacities, reducing flow costs, or defending network elements from the interdictor) before the interdictor takes action. We devote one section to ongoing research in this area and conclude by discussing areas for future research.

Sequential Search with Multiattribute Options

We describe a search problem in which a decision maker (DM) must select among sequentially encountered options. Each option is described by multiple attributes, and the value of an option is given by a separable function of its attribute values. However, the attribute values are not known with certainty, and can only be ascertained in a predefined order, at some fixed cost. During the search the DM can choose to select an option, purchase information about an attribute value, reject (permanently) the current option and continue the search, or terminate the search and accept a status quo outcome. We introduce a threshold policy for this search process, and prove the optimality of this policy for separable value functions. We then furnish a dynamic programming procedure for prescribing an optimal policy for this problem. Finally, we derive analytic solutions to some special cases of the problem, and present a case study that demonstrates a possible use of the proposed approach.

Convergence and Computational Analyses for Some Variable Target Value and Subgradient Deflection Methods

We consider two variable target value frameworks for solving large-scale nondifferentiable optimization problems. We provide convergence analyses for various combinations of these variable target value frameworks with several direction-finding and step-length strategies including the pure subgradient method, the volume algorithm, the average direction strategy, and a generalized Polyak-Kelley cutting plane method. In addition, we suggest a further enhancement via a projected quadratic-fit line-search whenever any of these algorithmic procedures experiences an improvement in the objective value. Extensive computational results on different classes of problems reveal that these modifications and enhancements significantly improve the effectiveness of the algorithms to solve Lagrangian duals of linear programs, even yielding a favorable comparison against the commercial software CPLEX 8.1.

On embedding the volume algorithm in a variable target value method

We employ the volume algorithm as a subgradient deflection strategy in a variable target value method for solving nondifferentiable optimization problems. Focusing on Lagrangian duals for LPs, we exhibit primal nonconvergence of the original method, establish convergence of the proposed algorithm in the dual space, and present related computational results.

Enhancing Lagrangian Dual Optimization for Linear Programs by Obviating Nondifferentiability

We consider nondifferentiable optimization problems that arise when solving Lagrangian duals of large-scale linear programs. Different from traditional subgradient-based approaches, we design two new methods that attempt to circumvent or obviate the nondifferentiability of the objective function, so that standard differentiable optimization techniques could be used. These methods, called the perturbation technique and the barrier-Lagrangian reformulation, are implemented as initialization procedures to provide a warm start to a theoretically convergent nondifferentiable optimization algorithm. Our computational study reveals that this two-phase strategy produces much better solutions with less computation in comparison with both the stand-alone nondifferentiable optimization procedure employed, and the popular Held-Wolfe-Crowder subgradient heuristic. Furthermore, the best version of this composite algorithm is shown to consume only about 3.19% of the CPU time required by the commercial linear programming solver CPLEX 8.1 (using the dual simplex option) to produce the same quality solutions. We also demonstrate that this initialization technique greatly facilitates quick convergence in the primal space when used as a warm start for ergodic-type primal recovery schemes.

Conditions of reverse bullwhip effect in pricing for price-sensitive demand functions

Supply chain mechanisms that exacerbate price variation needs special attention, since price variation is one of the root causes of the bullwhip effect. In this study, we investigate conditions that create an amplification of price variation moving from the upstream suppliers to the downstream customers in a supply chain, which is referred as the “reverse bullwhip effect in pricing” (RBP). Considering initially a single-stage supply chain in which a retailer faces a random and price-sensitive demand, we derive conditions on a general demand function for which the retail price variation is higher than that of the wholesale price. The investigation is extended to a multi-stage supply chain in which the price at each stage is determined by a game theoretical framework. We illustrate the use of the conditions in identifying commonly used demand functions that induce RBP analytically and by means of several numerical examples.

Cost-of-Quality Optimization via Zero-One Polynomial Programming

In this paper, we consider a Cost-of-Quality (CoQ) optimization problem that finds an optimal allocation of prevention and inspection resources to minimize the expected total quality costs under a prevention-appraisal-failure framework, where the quality costs in the proposed model are involved with prevention, inspection, and correction of internal and external failures. Commencing with a simple structure of the problem, we progressively increase the complexity of the problem by accommodating realistic scenarios regarding preventive, appraisal, and corrective actions. The resulting problem is formulated as a zero-one polynomial program, which can be solved either directly using a mixed-integer nonlinear programming solver such as BARON, or using a more conventional mixed-integer linear programming (MILP) solver such as CPLEX after performing an appropriate linearization step. We examine two case studies from the literature (related to a lamp manufacturing context and an order entry process) to illustrate how the proposed model can be utilized to find optimal inspection and prevention strategies, as well as to analyze sensitivity with respect to different cost parameters. We also provide a comparative numerical study of using the aforementioned solvers to optimize the respective model formulations. The results provide insights into the use of such quantitative methods for optimizing the CoQ, and indicate the efficacy of using the linearized MILP model for this purpose.