John E. Mitchell

• BA Hons Maths: Cambridge, PhD ORIE: Cornell
• Rensselaer Polytechnic Institute

This article formulates and solves a stochastic optimization model to investigate the impact of crowdsourced platforms (e.g., ridesharing, on‐demand delivery, volunteer food rescue, and carpooling) offering small, personalized menus of requests and incentive offers for drivers to choose from. To circumvent nonlinear variable relationships, we explo...

This paper considers an extension of the shortest path network interdiction problem that incorporates robustness to account for parameter uncertainty. The shortest path interdiction problem is a game of two players with conflicting agendas and capabilities: an evader, who traverses the arcs of a network from a source node to a sink node using a pat...

We consider a new class of multi-period network interdiction problems, where interdiction and restructuring decisions are decided upon before the network is operated and implemented throughout the time horizon. We discuss how we apply this new problem to disrupting domestic sex trafficking networks, and introduce a variant where a second cooperatin...

We consider a new class of max flow network interdiction problems, where the defender is able to introduce new arcs to the network after the attacker has made their interdiction decisions. We prove properties of when this restructuring will not increase the value of the minimum cut, which has important practical interpretations for problems of disr...

We consider a new class of multi-period network interdiction problems, where interdiction and restructuring decisions are decided upon before the network is operated and implemented throughout the time horizon. We discuss how we apply this new problem to disrupting domestic sex trafficking networks, and introduce a variant where a second cooperatin...

We consider two new problems regarding the impact of edge addition or removal on the modularity of partitions (or community structures) in a network. The first problem seeks to add edges to enforce that a desired partition is one partition that maximizes modularity. The second problem seeks to find the sparsest representation of a network that has...

This paper considers a large class of problems where we seek to recover a low rank matrix and/or sparse vector from some set of measurements. While methods based on convex relaxations suffer from a (possibly large) estimator bias, and other nonconvex methods require the rank or sparsity to be known a priori, we use nonconvex regularizers to minimiz...

Peer-to-peer logistics platforms coordinate independent drivers to fulfill requests for last mile delivery and ridesharing. To balance demand-side performance with driver autonomy, a new stochastic methodology provides drivers with a small but personalized menu of requests to choose from. This creates a Stackelberg game, in which the platform leads...

Urban deliveries of goods have become a rising challenge due to the increase of online shopping in addition to the traditional business-to-business (B2B) dynamics. To maintain a satisfactory level of service, carriers must conduct delivery operations under restrictive environments, creating undesirable externalities such as congestion and pollution...

Rank minimization is of interest in machine learning applications such as recommender systems and robust principal component analysis. Minimizing the convex relaxation to the rank minimization problem, the nuclear norm, is an effective technique to solve the problem with strong performance guarantees. However, nonconvex relaxations have less estima...

We consider two new problems regarding the impact of edge addition or removal on the modularity of partitions (or community structures) in a network. The first problem seeks to add edges to enforce that a desired partition is the partition that maximizes modularity. The second problem seeks to find the sparsest representation of a network that has...

We consider a new class of max flow network interdiction problems, where the defender is able to introduce new arcs to the network after the attacker has made their interdiction decisions. We provide an example of when interdiction can result in an increase to the maximum flow, and prove properties of when this restructuring will not increase the v...

The problem of minimizing the rank of a symmetric positive semidefinite matrix subject to constraints can be lifted to give an equivalent semidefinite program with complementarity constraints. The formulation requires two positive semidefinite matrices to be complementary. This is a continuous and nonconvex reformulation of the rank minimization pr...

Today's deep learning models are primarily trained on CPUs and GPUs. Although these models tend to have low error, they consume high power and utilize large amount of memory owing to double precision floating point learning parameters. Beyond the Moore's law, a significant portion of deep learning tasks would run on edge computing systems, which wi...

This work extends the logical benders approach for solving linear programs with complementarity constraints proposed by Hu et al. (SIAM J Optim 19(1):445–471, 2008) and Bai et al. (Comput Optim Appl 54(3):517–554, 2013). We develop a novel interpretation of the logical Benders method as a reversed branch-and-bound search, where the whole exploratio...

Rank minimization is of interest in machine learning applications such as recommender systems and robust principal component analysis. Minimizing the convex relaxation to the rank minimization problem, the nuclear norm, is an effective technique to solve the problem with strong performance guarantees. However, nonconvex relaxations have less estima...

During the development of a suite of computer-aided decision support tools for the restoration of interdependent infrastructures impacted by an extreme natural hazard event, it became apparent that the release of vulnerability data on actual infrastructure systems could raise security concerns. As a result, an artificial and customizable infrastruc...

We formally present the problem of scheduling tasks with effectiveness precedence relationships in order to achieve the minimum total weighted completion time. We provide the problem formulation and define the scope of the problem considered. We present computational complexity results for this problem and an approximation algorithm for it. We prov...

This paper develops a framework to create resilience indices for multi-echelon assembly supply chain (MEASC) networks. Each supplier within this network assembles a component from a series of sub-components received from other suppliers, thus, disruptions at suppliers can cascade and significantly affect the performance of the whole network. The fr...

We consider optimization problems related to the scheduling of multi-echelon assembly supply chain (MEASC) networks that have applications in the recovery from large-scale disruptive events. Each manufacturer within this network assembles a component from a series of sub-components received from other manufacturers and, due to high qualification st...

Civil infrastructure systems (transportation, power, communications, water, and sewer services) are crucial for the operation and wellbeing of a community during normal circumstances certainly, but especially so during and following an extreme natural hazard event. In addition to these basic utility services, social infrastructure such as police an...

The problem of minimizing the rank of a symmetric positive semidefinite matrix subject to constraints can be cast equivalently as a semidefinite program with complementarity constraints (SDCMPCC). The formulation requires two positive semidefinite matrices to be complementary. We investigate calmness of locally optimal solutions to the SDCMPCC form...

This work considers modeling approaches to the problem of restoration decision making by supply chain managers. Specifically, we consider an oil supply chain network that has suffered a cyber attack. As a result of the attack, critical services to the supply chain are lost, and the manager has uncertain information regarding when these services wil...

Organization and efficiency of relief operations are vital following a major disaster, as well as the guarantee that all of the affected population will adequately have their basic needs met. However, in a post-disaster environment, uncertainty often impacts all aspects of the relief efforts. Placement of relief distribution centers, as well as pub...

A linear program with linear complementarity constraints (LPCC) requires the minimization of a linear objective over a set of linear constraints together with additional linear complementarity constraints. This class has emerged as a modeling paradigm for a broad collection of problems, including bilevel programs, Stackelberg games, inverse quadrat...

A linear program with linear complementarity constraints (LPCC) requires the minimization of a linear objective over a set of linear constraints together with additional linear complementarity constraints. This class has emerged as a modeling paradigm for a broad collection of problems, including bilevel programs, Stackelberg games, inverse quadrat...

Modeling emergency evacuation could help reduce losses and damages from disasters. In this paper, based on the system optimum principle, we develop a multimodal evacuation model that considers multiple transportation modes and their interactions, and captures the proper traffic dynamics including the congestion effects, the cooperative behavior of...

Modeling emergency evacuation could help reduce losses and damages from disasters. In this paper, based on the system optimum principle, we develop a multimodal evacuation model that considers multiple transportation modes and their interactions, and captures the proper traffic dynamics including the congestion effects, the cooperative behaviour of...

Semidefinite relaxations of certain combinatorial optimization problems lead to approximation algorithms with performance guarantees. For large-scale problems, it may not be computationally feasible to solve the semidefinite relaxations to optimality. In this paper, we investigate the effect on the performance guarantees of an approximate solution...

In this paper, we consider methods to determine the best single arc mitigation plan for improving rapid recovery of a network with a given level of statistical certainty. This problem is motivated by infrastructure managers interested in increasing the resilience of their systems through costly long-term mitigation procedures. Our problem is two st...

This paper studies several classes of nonconvex optimization problems defined over convex cones, establishing connections between them and demonstrating that they can be equivalently formulated as convex completely positive programs. The problems being studied include: a conic quadratically constrained quadratic program (QCQP), a conic quadratic pr...

This paper introduces the new concept of restoration interdependencies that exist among infrastructures during their restoration efforts after an extreme event. Restoration interdependencies occur whenever a restoration task in one infrastructure is impacted by a restoration task, or lack thereof, in another infrastructure. This work identifies exa...

We consider restoring multiple interdependent infrastructure networks after a disaster damages components in them and disrupts the services provided by them. Our particular focus is on interdependent infrastructure restoration (IIR) where both the operations and the restoration of the infrastructures are linked across systems. We provide new mathem...

This paper addresses the problem of designing supply chains that are resilient to natural or human-induced extreme events. It focuses on the development of efficient restoration strategies that aid the supply chain in recovering from a disruption, thereby limiting the impact on its customers. The proposed restoration model takes into account possib...

Quadratic Convex Reformulation (QCR) is a technique that has been proposed for binary and mixed integer quadratic programs. In this paper, we extend the QCR method to convex quadratic programs with linear complementarity constraints (QPCCs). Due to the complementarity relationship between the nonnegative variables $y$ and $w$ , a term $y^{T}Dw$...

Nonconvex quadratic constraints can be linearized to obtain relaxations in a well-understood manner. We propose to tighten the relaxation by using second-order cone constraints, resulting in a convex quadratic relaxation. Our quadratic approximation to the bilinear term is compared to the linear McCormick bounds. The second-order cone constraints a...

The paper introduces a dynamic spatial price equilibrium model that estimates urban freight-related flows for integrated producer-carrier operations that compete in a market where a generic commodity is traded. The original model obtains the patterns of commodity flows, production levels, and delivery tours that maximize economic welfare. This mode...

Despite extensive efforts to model mammalian cell metabolism in academic research labs, dating back over 30 years and similar levels of bioprocess analysis in industrial research laboratories, much about mammalian cell metabolism in industrial biprocesses remains poorly understood. Recent advances in metabolomics in conjunction with the sequencing...

A linear program with linear complementarity constraints (LPCC) is among the simplest mathematical programs with complementarity constraints. Yet the global solution of the LPCC remains difficult to find and/or verify. In this work we study a specific type of the LPCC which we term a bi-parametric LPCC. Reformulating the bi-parametric LPCC as a non...

Support vector machine regression is a robust data fitting method to minimize the sum of deducted residuals of regression, and thus is less sensitive to changes of data near the regression hyperplane. Two design parameters, the insensitive tube size (\(\varepsilon _\mathrm{e}\)) and the weight assigned to the regression error trading off the normed...

The inclusion of transaction costs is an essential element of any realistic portfolio optimization. We extend the standard portfolio optimization problem to consider convex transaction costs incurred when rebalancing an investment portfolio. Market impact costs measure the effect on the price of a security that result from an effort to buy or sell...

The main objective of this chapter is to provide a comprehensive overview of empirical findings and models that focus on urban freight tours. In doing so, the chapter provides background information; reviews the research literature; classifies the tour models in simulation, hybrid, and analytical models; discusses strengths and weaknesses of a samp...

Park and Ride facilities (P&R) are car parks at which users can transfer to public transportation to reach their final destination. We propose a mixed linear programming formulation to determine the location of a fixed number of P&R facilities so that their usage is maximized. The facilities are modeled as hubs. Commuters can use one of the P&R fac...

A solution of the standard formulation of a linear program with linear complemen- tarity constraints (LPCC) does not satisfy a constraint qualification. A family of relaxations of an LPCC, associated with a probability-one homotopy map, proposed here is shown to have several desirable properties. The homotopy map is nonlinear, replacing all the con...

The paper shows that the global resolution of a general convex quadratic program with complementarity constraints (QPCC), possibly infeasible or unbounded, can be accomplished in finite time. The method constructs a minmax mixed integer formulation by introducing finitely many binary variables, one for each complementarity constraint. Based on the...

We consider the problem faced by managers of critical civil interdependent infrastructure systems of restoring essential public services after a non-routine event causes disruptions to these services. In order to restore the services, we must determine the set of components (or tasks) that will be temporarily installed or repaired, assign these tas...

We consider the problem of restoring services provided by infrastructure systems after an extreme event disrupts them. This research proposes a novel integrated network design and scheduling problem that models these restoration efforts. In this problem, work groups must be allocated to build nodes and arcs into a network in order to maximize the c...

We analyze the properties of an interior point cutting plane algorithm that is based on a semi-infinite linear formulation of the dual semidefinite program. The cutting plane algorithm approximately solves a linear relaxation of the dual semidefinite program in every iteration and relies on a separation oracle that returns linear cutting planes. We...

The class of mathematical programs with complementarity constraints (MPCCs) constitutes a powerful modeling paradigm. In an effort to find a global optimum, it is often useful to examine the relaxation obtained by omitting the complementarity constraints. We discuss various methods to tighten the relaxation by exploiting com-plementarity, with the...

The paper is a manifestation of the fundamental importance of the linear program with linear complementarity constraints (LPCC) in disjunctive and hierarchical programming as well as in some novel paradigms of mathematical programming. In addition to providing a unified framework for bilevel and inverse linear optimization, nonconvex piecewise line...

Research on resilient infrastructures in this paper focuses on getting disrupted infrastructures back to normal as soon as possible, i.e., a timely, efficient recovery from disruptions caused by natural, technological and willful disasters. The models and algorithms are presented for the optimal restoration strategy, with consideration of interdepe...

We consider a new class of integrated network design and scheduling problems, with important applications in the restoration of services provided by civil infrastructure systems after an extreme event. Critical services such as power, waste water, and transportation are provided by these infrastructure systems. The restoration of these services is...

This collaborative project aims at the study of the global resolution of convex programs with complementarity constraints (CPCCs), which form a large subclass of the class of mathematical programs with complementarity constraints (MPCCs). Despite the large literature on the local properties of an MPCC, there is a lack of systematic investigation on...

The performance of branch-and-bound methods for integer programming problems has been dramatically improved by incorporating cutting planes. The resulting technique is known as branch-and-cut. Cutting planes are inequalities that can be used to improve the linear programming relaxation of an integer programming problem. They are added as required,...

Interior point methods have proven very successful at solving linear programming problems. When an explicit linear programming formulation is either not available or is too large to employ directly, a column generation approach can be used. Examples of column generation approaches include cutting plane methods for integer programming and decomposit...

We present an analytic center cutting surface algorithm that uses mixed linear and multiple second-order cone cuts. Theoretical issues and applications of this technique are discussed. From the theoretical viewpoint, we derive two complexity results. We show that an approximate analytic center can be recovered after simultaneously adding p second-o...

After a disruption in an interconnected set of systems, it is necessary to restore service. This requires the determination of the tasks that need to be undertaken to restore service, and then scheduling those tasks using the available resources. This chapter discusses combining mathematical programming and constraint programming into multiple obje...

Modern society relies upon the complex interaction of the civil infrastructure systems, such as transportation, power, telecommunications and water. These systems are highly dependent on each other to provide service. The reliance on any of them on power is obvious. Failures in one system can have far-reaching effects. This paper will present an ov...

We analyze the problem of finding a point strictly interior to a bounded, convex, and fully dimensional set from a finite dimensional Hilbert space. We generalize the results obtained for the linear programming (LP), semidefinite programming (SDP), and second-order core programming (SOCP) cases. The cuts added by our algorithm are central and conic...

This paper presents a general method for exact distance computation between convex objects represented as intersections of implicit surfaces. Exact distance computation algorithms are particularly important for applications involving objects that make intermittent contact, such as in dynamic simulations and in haptic interactions. They can also be...

It is not straightforward to find a new feasible solution when several conic constraints are added to a conic optimization problem. Examples of conic constraints include semidefinite constraints and second order cone constraints. In this paper, a method to slightly modify the constraints is proposed. Because of this modification, a simple procedure...

This paper presents a parameter-free integer-programming based algorithm for the global resolution of a linear program with linear complementarity constraints (LPEC). The cornerstone of the algorithm is a minimax integer program formulation that characterizes and provides certificates for the three outcomes—infeasibility, unboundedness, or solvabil...

Autonomous wireless devices such as sensor nodes and stereo cameras, due to their low cost of operation coupled with the potential for remote deploy- ment, have found a plethora of applications ranging from monitoring air, soil and water to seismic detection and military surveillance. Typically, such a net- work spans a region of interest with the...

Filling a gap in nonconvex quadratic programming, this paper shows that the global resolution of a feasible quadratic program (QP), which is not known a priori to be bounded or unbounded below, can be accomplished in finite time by solving two linear programs with linear complementarity constraints, i.e., LPCCs. Specifically, this task can be divid...

Keywords A Standard Form Primal Heuristics Preprocessing Families of Cutting Planes When to Add Cutting Planes Lifting Cuts Implementation Details Solving Large Problems Conclusions See also References

Keywords Synonyms Overview Partitioning Strategies Branching Variable Selection Node Selection Preprocessing and Reformulation Heuristics Continuous Reduced Cost Implications Subproblem Solver See also References

Modern society depends on the operations of civil infrastructure systems, such as transportation, energy, telecommunications, and water. These systems have become so interconnected, one relying on another, that disruption of one may lead to disruptions in all. The approach taken in this research is to model these systems by explicitly identifying t...

Given a complete graph Kn = (V;E) with edge weight ce on each edge, we con- sider the problem of partitioning the vertices of graph Kn into subcliques that have at least S vertices, so as to minimize the total weight of the edges that have both end- points in the same subclique. In this paper, we consider using the branch-and-price method to solve...

We investigate solution of the maximum cut problem using a polyhedral cut and price approach. The dual of the well-known SDP relaxation of maxcut is formulated as a semi-infinite linear programming problem, which is solved within an interior point cutting plane algorithm in a dual setting; this constitutes the pricing (column generation) phase of t...

In this paper, we present an interior point approach to exact distance computation between convex objects represented as intersections of implicit surfaces. The implicit surfaces considered include planes (polyhedra), quadrics, and generalizations of quadrics including superquadrics and hyperquadrics, as well as intersections of these surfaces. Exa...

Cutting plane methods provide the means to solve large scale semidefinite programs (SDP) cheaply and quickly. They can also conceivably be employed for the purposes of re-optimization after branching or the addition of cutting planes. We give a survey of various cutting plane approaches for SDP in this paper. These cutting plane approaches arise fr...

We discuss interior point methods for large-scale linear programming, with an emphasis on methods that are useful for problems arising in telecommunications. We give the basic framework of a primal-dual interior point method, and consider the numerical issues involved in calculating the search direction in each iteration, including the use of facto...

We study the issue of updating the analytic center after multiple cutting planes have been added through the analytic center of the current polytope. This is an important issue that arises at every stage of cutting-plane algorithms. If q n cuts are to be added, we show that we can use a selective orthonormalization procedure to modify the cuts befo...

The presence of complementarity constraints brings a combinatorial flavour to an optimization problem. A quadratic programming problem with complementarity constraints can be relaxed to give a semidefinite programming problem. The solution to this relaxation can be used to generate feasible solutions to the complementarity constraints. A quadratic...

The sports team realignment problem can be modelled as k-way equipartition: given a complete graph Kn = (V;E), with edge weight ce on each edge, partition the vertices V into k divisions that have exactly S vertices, so as to minimize the total weight of the edges that have both endpoints in the same division. In this paper, we discuss solving k-wa...

Recent events have heightened our awareness of the vulnerability of civil infrastructure systems. Most of the research on this topic has focused on individual systems, while more recent efforts have recognized the interconnectedness of systems. Infrastructure systems have become so highly interconnected that a failure in one system can propagate th...

ABSTRACT

An Abstract of a Thesis Submitted to the Graduate Faculty of Rensselaer Polytechnic Institute in Partial Fulflllment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Subject: Mathematics The original of the complete thesis is on flle

The National Football League (NFL) in the United States expanded to 32 teams in 2002 with the addition of a team in Houston. At that point, the league was realigned into eight divisions, each containing four teams. We describe a branch-and-cut algorithm for minimizing the sum of intradivisional travel distances. We consider first the case where any...

Interior point methods, the traditional methods for the SDP, are fairly limited in the size of problems they can handle. This paper deals with an LP approach to overcome some of these shortcomings. We begin with a semi-infinite linear programming formulation of the SDP and discuss the issue of its discretization in some detail. We further show that...

Critical infrastructure systems provide services that are essential to both the economy and wellbeing of nations and their citizens. As documented in a recent report to the U.S. Congress (U.S. General Accounting Office, 2001), it is of vital importance that these services not be degraded, whether by willful acts such as terrorism or by natural or r...

Branch-and-cut methods are very successful techniques for solving a wide variety of integer programming problems, and they can provide a guarantee of optimality. We describe how a branch-and-cut method can be tailored to a specic integer programming problem, and how families of general cutting planes can be used to solve a wide variety of problems....

Until recently, the study of interior point methods has dominated algorithmic research in semidefinite programming (SDP). From a theoretical point of view, these interior point methods o#er everything one can hope for; they apply to all SDP's, exploit second order information and o#er polynomial time complexity. Still for practical applications wit...

Interior point methods, the traditional methods for the SDP , are fairly limited in the size of problems they can handle. This paper deals with an LP approach to overcome some of these shortcomings. We begin with a semi-infinite linear programming formulation of the SDP and discuss the issue of its discretization in some detail. We further show tha...

Polynomial cutting plane methods based on the logarithmic barrier function and on the volumetric center are surveyed. These algorithms construct a linear programming relaxation of the feasible region, find an appropriate approximate center of the region, and call a separation oracle at this approximate center to determine whether additional constra...

A semidefinite programming problem can be regarded as a convex nonsmooth optimization problem, so it can be represented as a semi-infinite linear programming problem. Thus, in principle, it can be solved using a cutting plane approach; we describe such a method. The cutting plane method uses an interior point algorithm to solve the linear programmi...

Many combinatorial optimization problems have relaxations that are semidefinite programming problems. In principle, the combinatorial optimization problem can then be solved by using a branch-and-cut procedure, where the problems to be solved at the nodes of the tree are semidefinite programs. It is desirable that the solution to one node of the tr...

A cutting plane method for linear programming is described. This method is an extension of Atkinson and Vaidya's algorithm, and uses the central trajectory. The logarithmic barrier function is used explicitly, motivated partly by the successful implementation of such algorithms. This makes it possible to maintain primal and dual iterates, thus allo...