Siqian Shen

Exact interdiction models and algorithms for disconnecting networks via node deletions

Abstract This paper analyzes the problem of maximizing the disconnectivity of undirected graphs by deleting a subset of their nodes. We consider three metrics that measure the connectivity of a graph: the number of connected components (which we attempt to maximize), the largest component size (which we attempt to minimize), and the minimum cost required to reconnect the graph after the nodes are deleted (which we attempt to maximize). We formulate each problem as a mixed-integer program, and then study valid inequalities for the first two connectivity objectives by examining intermediate dynamic programming solutions to k -hole subgraphs. We randomly generate a set of test instances, on which we demonstrate the computational efficacy of our approaches.

Polynomial-time algorithms for solving a class of critical node problems on trees and series-parallel graphs

We examine variants of the critical node problem on specially structured graphs, which aim to identify a subset of nodes whose removal will maximally disconnect the graph. These problems lie at the intersection of network interdiction and graph theory research and are relevant to several practical optimization problems. The two different connectivity metrics that we consider regard the number of maximal connected components (which we attempt to maximize) and the largest component size (which we attempt to minimize). We develop optimal polynomial-time dynamic programming algorithms for solving these problems on tree structures and on series-parallel graphs, corresponding to each graph-connectivity metric. We also extend our discussion by considering node deletion costs, node weights, and solving the problems on generalizations of tree structures. Finally, we demonstrate the computational efficacy of our approach on randomly generated graph instances.

Expectation and Chance-Constrained Models and Algorithms for Insuring Critical Paths

In this paper, we consider a class of two-stage stochastic optimization problems arising in the protection of vital arcs in a critical path network. A project is completed after a series of dependent tasks are all finished. We analyze a problem in which task finishing times are uncertain but can be insured a priori to mitigate potential delays. A decision maker must trade off costs incurred in insuring arcs with expected penalties associated with late project completion times, where lateness penalties are assumed to be lower semicontinuous nondecreasing functions of completion time. We provide decomposition strategies to solve this problem with respect to either convex or nonconvex penalty functions. In particular, for the nonconvex penalty case, we employ the reformulation-linearization technique to make the problem amenable to solution via Benders decomposition. We also consider a chance-constrained version of this problem, in which the probability of completing a project on time is sufficiently large. We demonstrate the computational efficacy of our approach by testing a set of size-and-complexity diversified problems, using the sample average approximation method to guide our scenario generation.

Distributionally Robust Chance-Constrained Optimal Power Flow With Uncertain Renewables and Uncertain Reserves Provided by Loads

Aggregations of electric loads can provide reserves to power systems, but their available reserve capacities are time-varying and not perfectly known when the system operator computes the optimal generation and reserve schedule. In this paper, we formulate a chance constrained optimal power flow problem to procure minimum cost energy, generator reserves, and load reserves given uncertainty in renewable energy production, load consumption, and load reserve capacities. Assuming that uncertainty distributions are not perfectly known, we solve the problem with distributionally robust optimization, which ensures that chance constraints are satisfied for any distribution in an ambiguity set built upon the first two moments. We use two ambiguity sets to reformulate the model as a semidefinite program and a second-order cone program and run computational experiments on the IEEE 9-bus, 39-bus, and 118-bus systems. We compare the solutions to those given by two benchmark reformulations; the first assumes normally distributed uncertainty and the second uses large numbers of uncertainty samples. We find that the use of load reserves, even when load reserve capacities are uncertain, reduces operational costs. Also, the approach is able to meet reliability requirements, unlike the first benchmark approach and with lower computation times than the second benchmark approach.

Optimizing designs and operations of a single network or multiple interdependent infrastructures under stochastic arc disruption

In this paper, we consider an infrastructure as a network with supply, transshipment, and demand nodes. A subset of potential arcs can be constructed between node pairs for conveying service flows. The paper studies two optimization models under stochastic arc disruption. Model 1 focuses on a single network with small-scale failures, and repairs arcs for quick service restoration. Model 2 considers multiple interdependent infrastructures under large-scale disruptions, and mitigates cascading failures by selectively disconnecting failed components. We formulate both models as scenario-based stochastic mixed-integer programs, in which the first-stage problem builds arcs, and the second-stage problem optimizes recourse operations for restoring service or mitigating losses. The goal is to minimize the total cost of infrastructure design and recovery operations. We develop cutting-plane algorithms and several heuristic approaches for solving the two models. Model 1 is tested on an IEEE 118-bus system. Model 2 is tested on systems consisting of the 118-bus system, a 20-node network, and/or a 50-node network, with randomly generated interdependency sets in three different topological forms (i.e., chain, tree, and cycle). The computational results demonstrate that (i) decomposition and cutting-plane algorithms effectively solve Model 1, and (ii) heuristic approaches dramatically decrease the CPU time for Model 2, but yield worse bounds when cardinalities of interdependency sets increase. Future research includes developing special algorithms for optimizing Model 2 for complex multiple infrastructures with special topological forms of system interdependency.

Risk and Energy Consumption Tradeoffs in Cloud Computing Service via Stochastic Optimization Models

Energy efficiency and computational reliability are two key concerns associated with modern computations that involve computational resource sharing and highly uncertain job arrivals from various sources (customers). In 2010, large-scale data center operations consume around 2% of the total energy use in the US. Meanwhile, the development of cloud computing in the IT industry possesses great potential for lowering energy consumption by partitioning and scheduling job requests among multiple computational servers. In this paper, we formulate stochastic integer programming models to minimize energy consumption of cloud computing servers over finite time periods, while maintaining a pre-specified quality of service (QoS) level for satisfying uncertain computational requests. The models dynamically monitor and predict customer requests for each period, and proactively switch servers on/off according to estimated customer requests. QoS levels are maintained by either enforcing zero unsatisfied requests, or imposing a joint chance constraint to bound possible failures in a backlogging model. When uncertain requests follow continuous distributions, we employ the Sampling Average Approximation for generating scenario-based requests. Such an approach transforms the original probabilistic model into deterministic mixed-integer linear programs. We further demonstrate computational results of all models by testing instances with different parameter combinations, and investigate how backlogging, unit penalty cost and QoS levels influence computational performances and optimal solutions.

Loss-Constrained Minimum Cost Flow under Arc Failure Uncertainty with Applications in Risk-Aware Kidney Exchange

In this article, we study a Stochastic Minimum Cost Flow (SMCF) problem under arc failure uncertainty, where an arc flow solution may correspond to multiple path flow representations. We assume that the failure of an arc will cause flow losses on all paths using that arc, and for any path carrying positive flows, the failure of any arc on the path will lose all flows carried by the path. We formulate two SMCF variants to minimize the cost of arc flows, while respectively restricting the Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) of random path flow losses due to uncertain arc failure (reflected as network topological changes). We formulate a linear program to compute possible losses, yielding a mixed-integer programming formulation of SMCF-VaR and a linear programming formulation of SMCF-CVaR. We present a kidney exchange problem under uncertain match failure as an application and use the two SMCF models to maximize the utility/social welfare of pairing kidneys subject to constrained risk of utility losses. Our results show the efficacy of our approaches, the conservatism of using CVaR, and optimal flow patterns given by VaR and CVaR models on diverse instances.

Mitigating hard capacity constraints with inventory in facility location modeling

ABSTRACT Although the traditional capacitated facility location model uses inflexible, limited capacities, facility managers often have many operational tools to extend capacity or to allow a facility to accept demands in excess of the capacity constraint for short periods of time. We present a mixed-integer program that captures these operational extensions. In particular, demands are not restricted by the capacity constraint, as we allow for unprocessed materials from one day to be held over in inventory and processed on a following day. We also consider demands at a daily level, which allows us to explicitly incorporate the daily variation in, and possibly correlated nature of, demands. Large problem instances, in terms of the number of demand nodes, candidate nodes, and number of days in the time horizon, are generated from United States census population data. We demonstrate that, in some instances, optimal locations identified by the new model differ from those of the traditional capacitated facility location problem and result in significant cost savings.

A decomposition approach for solving a broadcast domination network design problem

We consider an optimization problem that integrates network design and broadcast domination decisions. Given an undirected graph, a feasible broadcast domination is a set of nonnegative integer powers fi assigned to each node i, such that for any node j in the graph, there exists some node k having a positive fk-value whose shortest distance to node j is no more than fk. The cost of a broadcast domination solution is the sum of all node power values. The network design problem constructs edges that decrease the minimum broadcast domination cost on the graph. The overall problem we consider minimizes the sum of edge construction costs and broadcast domination costs. We show that this problem is NP-hard in the strong sense, even on unweighted graphs. We then propose a decomposition strategy, which iteratively adds valid inequalities based on optimal broadcast domination solutions corresponding to the first-stage network design solutions. We demonstrate that our decomposition approach is computationally far superior to the solution of a single large-scale mixed-integer programming formulation.

Multi-objective probabilistically constrained programs with variable risk: Models for multi-portfolio financial optimization

We consider a class of multi-objective probabilistically constrained programs (MOPCP) with a joint probabilistic constraint and a variable risk level. We consider two cases with only a random right-hand side vector or a multi-row random technology matrix, and propose a Boolean modeling framework to derive new mixed-integer linear programs (MILP) that are either equivalent reformulations or inner approximations of MOPCP, respectively. Via testing randomly generated MOPCP instances, we demonstrate modeling insights pertaining to the most suitable MILP, to the trade-offs between conflicting objectives of cost/revenue and reliability, and to the parameter scalarization determining relative importance of each objective. We then focus on several MOPCP variants of a multi-portfolio financial optimization problem to implement a downside risk measure, which can be used in a centralized or decentralized investment context. We study the impact of modeling parameters on the portfolios, show, via a cross-validation study, robustness of MOPCP, and perform a comparative analysis of the optimal investment decisions.