Research

Research Interest

Research Methodology

In many real-life decision-making problems, practitioners are often confronted with uncertainty in the underlying system parameters. Moreover, in solving problems under uncertainty, analysis of historical/survey data can inform the decision-makers about uncertainty such as decision-makers can estimate the probability distributions of the uncertain parameters analyzing data. Also, decision-making problems often involve multiple competing players, requiring the decision-makers to account for other competing players' objectives in best accomplishing their objectives. My academic interest is to develop mathematical optimization frameworks for problems having uncertainty and capturing the interaction of multiple competing players. Specifically, I develop data-driven stochastic programs and bi-level programs to model these problems as well as develop customized decomposition and approximation algorithms utilizing the problem structure to solve those optimization problems.

Besides optimization, practitioners often deal with problems that require analyzing big data to identify hidden patterns such as detecting malicious cyber nodes in a large computer network. My research interest also lies in developing/implementing supervised and unsupervised machine learning algorithms for big data analytics utilizing high-performance computing environments. 

We can utilize machine learning models in optimizing problems where the traditional derivative-based optimization methods are not applicable. I also utilize statistical-learning-based optimization methods, such as Bayesian optimization, to solve decision-making problems where derivative information is not available or it is computationally very challenging to estimate.

Ph.D. Dissertation

My Ph.D. dissertation is focused on developing new data-driven stochastic programming models, mathematical reformulations to linearize the original nonconvex models, and designing customized decomposition and approximation algorithms for solving optimization problems under endogenous (decision-dependent) uncertainty. Unlike traditional stochastic programs, where uncertainty is usually modeled using scenarios governed by a known probability distribution, in some application areas, such as retrofitting systems under imperfect and multi-level effects of protection resources on component reliability, it is beneficial to model the probability distributions of the uncertain parameters as a function of the decision variables. The stochastic program modeling this decision–dependent uncertainty is known as stochastic programming with endogenous uncertainty (SPEU). 

In a two-stage stochastic program (as shown in Figure 1), scenarios are generated to represent the realizations of uncertain parameters following a probability distribution, where each scenario represents a particular realization of the uncertain parameter. The optimization problem is decomposed into two-stage, where the first-stage decisions (x) are made without realizing the uncertainty. But the second-stage decisions (a.k.a., recourse decisions) are made after realizing the uncertainty, meaning the second-stage model is solved for each particular scenario after realizing the uncertainty. In optimization under endogenous uncertainty, decisions alter the probability distributions of uncertain parameters. For example, if a decision-maker invests more resources in a network component (e.g., better quality material) then the survival probability of that component will be higher when exposed to disruption, making the survival probability distribution to be a function of investment decision.   

Figure 1: Two-stage stochastic program with exogenous (Left side) and endogenous (Right side) uncertainty 

Therefore, as the probability distribution of uncertain parameters is a function of the decision variables in an SPEU program (Right side in Figure 1), the scenario probability is also a function of the decision variables (x). Therefore, in the first-stage objective function (Right side model in Figure 1), scenario probability is a function of the first-stage decision variables (x). But, this is not the case in the stochastic program with exogenous uncertainty (Left side model in Figure 1), where scenario probabilities are fixed, as the realizations (as well as the corresponding probability distributions) of uncertain parameters do not depend on the decisions (i.e., exogenous to the model). However, in this stochastic program with exogenous uncertainty, the first-stage decisions affect the recourse decisions by appearing in the second-stage model as a parameter (Left side model in Figure 1).

This SPEU framework has many application areas including but not limited to critical infrastructure protection, disaster preparedness, and mitigation, supply chain risk mitigation, power generation capacity expansion planning. Solving problems having a natural decision-dependent uncertainty structure using a traditional (exogenous) stochastic program could be computationally very difficult as the exogenous stochastic program would need a larger number of scenarios than the SPEU framework to capture the decision-dependent uncertainty property. Moreover, modeling and solving problems as SPEU can exploit the underlying decision-dependent uncertainty property of the problems and can yield better quality solutions and more insights into the problem. But, this SPEU modeling framework has not been utilized in many application areas to date as well as most of the studies on SPEU approached the problem with problem-specific heuristic methods and commercial nonlinear solvers. But, heuristic methods cannot provide optimal solutions, and commercial optimization solvers/software are not available to industry decision-makers always. Moreover, directly using optimization solvers/software is not computationally efficient for solving mathematical models with difficult structures (e.g., nonconvex stochastic mixed-integer nonlinear). Also, very few studies analyzed theoretical properties (e.g., convexity, submodularity, etc.) of the SPEU framework and limited them to only binomial probability distributions. But, there are other widely used probability distributions in many areas. It is possible to provide good quality approximate solutions to large-scale problems by taking advantage of some theoretical properties. Therefore, there is a need to make advances in the SPEU modeling and solution approaches. Until this need is met, decision-makers in various application areas cannot obtain good quality solutions and valuable insights into their decision-making problems, resulting in systems that underperform and have excessive cost.  

Therefore, in my Ph.D. research, I have developed novel data-driven stochastic programming with endogenous uncertainty (SPEU) models for solving new problems having endogenous (decision-dependent) uncertainty that has not been studied to date. These developed mathematical models incorporate supervised machine learning models into the stochastic programming framework to estimate the probability distributions of uncertain parameters by analyzing historical/survey data. As in the SPEU models, scenario probabilities are a function of the decision variables of the optimization routine, the models are naturally nonconvex. This nonconvexity makes it difficult to solve these SPEU models compared to the traditional stochastic programming models. I developed new mathematical reformulations to linearize the developed nonconvex nonlinear models to reduce computational complexity. I designed customized decomposition algorithms to solve the reformulated models for realistic-sized problem instances to optimality. I also incorporated some new valid inequalities to improve the computational efficiency of the developed algorithm to outperform the existing solvers in computation time. 

In my dissertation, I studied the submodularity property of the SPEU modeling framework and mathematically proved that the objective function of the stochastic program is submodular when the uncertain parameters are exponentially distributed. Taking advantage of this submodularity property, I implemented an efficient approximation algorithm capable of solving large-scale decision-making problems under endogenous uncertainty much faster than the exact algorithms with an approximation guarantee to the solution quality. Unlike the heuristic algorithms, approximation algorithms provide a worst-case performance guarantee to the solutions.  In my Ph.D. research, I demonstrated the benefits of the developed methodologies in the application areas of infrastructure protection and reliable transportation network design, disaster mitigation, reliability optimization of complex systems, and supply chain risk mitigation. 

Research Thrusts

Infrastructure Protection and Reliable Transportation Network Design

In my Ph.D. dissertation, I developed a data-driven SPEU modeling framework where both exogenous and endogenous uncertainties coexist to model an integrated facility protection and transportation network design problem under random disruptions with the following realistic assumptions: (1) effect of protection resources is imperfect, (2) protection is multi-level, and (3) facilities have multiple post-disruption capacities. The proposed two-stage stochastic program integrates supervised machine learning models to estimate the conditional probability distributions of the facilities’ post-disruption capacities by analyzing historical data. To linearize the non-convex, nonlinear stochastic program, we developed a new mathematical reformulation technique. Utilizing the structure of the model, we developed a customized L-shaped decomposition algorithm to solve the problem as well as introduced several new valid inequalities to improve the computational efficiency of the algorithm. 

Figure 2: A distribution network subject to disruptions based on the southeastern United States 

The numerical results based on a distribution network in the southeastern United States (as shown in Figure 2) demonstrate that the optimal solution from the proposed methods provides an average reduction of 18.71% in expected transportation cost than the existing methods assuming perfect and binary protection and disruption. Also, the proposed methodology provides substantially (on average 600%) lower expected transportation costs than the methods ignoring the endogenous uncertainty in the facilities’ post-disruption capacities. This research provides several key managerial insights about the effect of budget variation on the expected post-disruption transportation cost and the effect of the estimation error of the supervised machine learning algorithms on the optimal solution of the optimization model. These insights are beneficial for the network managers in deciding on the investment amount in reliable network design and critical infrastructure protection as well as in choosing the machine learning algorithms that have the least impact on the optimization model.

Relevant Publications

Disaster (Wildfire Risk) Mitigation

Centralized government (land and fire management) agencies often face challenges in implementing wildfire prevention measures (hazardous fuel reduction treatments, such as prescribed burning, mechanical or chemical vegetation control) on nonindustrial private forest lands. These agencies can only incentivize the private landowners to encourage them to implement wildfire prevention measures on their lands through a cost-share program by sharing some portion of the implementation cost. The key challenges for the agencies in designing a cost-share program with their limited budget are: (1) the agency does not know whether a landowner will accept or reject the cost-share program for a given financial assistance amount, (2) different landowners’ land have different biophysical characteristics (e.g., vegetation density, moisture content, elevation, slope, etc.) and pose different amount of wildfire risk to the surrounding landscape. 

In my Ph.D. dissertation, I proposed a novel risk-based incentive structure design program to incentivize private landowners in implementing fuel reduction treatments on their lands, where (1) the landscape is divided into a number of raster cells (shown in Figure 3) based on the biophysical characteristics, and (2) different landowners are offered different incentive amounts by the budget-constrained government agency depending on the wildfire risks each landowner's land pose to the surrounding landscape. Another challenge the government agency faces is that the agency does not know whether a landowner will accept or reject the cost-share program for a given incentive amount. However, based on our landowner survey data, we found that the landowners are more likely to accept the cost-share program as the incentive amount increases, which makes the landowners' behavioral uncertainty endogenous to the incentive offer decision.  

Figure 3: A small rasterized landscape

Figure 4: Landscape modeled as a directed network

We modeled the risk-based incentive structure design problem in reducing the risk of catastrophic future wildfires as a data-driven SPEU model. The proposed SPEU model accounts for the endogenous uncertainty in the landowners’ accept/reject decisions of the incentive. The proposed SPEU model utilizes a supervised machine learning model to estimate the landowners' likelihood of accepting the cost-share program by analyzing landowner survey data. Due to a large number of scenarios, the second-stage value of the two-stage SPEU model is computed using a simulation program that models the wildfire spread in the landscape accounting for the information on fuel reduction treatment, weather, and landscape characteristics. To model the wildfire spread as a network flow problem, the landscape is represented as a directed network (Figure 4), where each node represents a land parcel of the landscape and the arc weights represent the fire travel time between the land parcels. This work falls in the class of stochastic network interdiction problems, as the spatial optimization of fuel treatment resources prevents future catastrophic wildfires. 

Results based on real data from Santa Fe National Forest, New Mexico show that the proposed method can provide decision support to land management agencies for a realistic-sized landscape. This novel risk-based approach provides up to 37.3% more reduction in expected wildfire damage than the currently used methods. 

Relevant Publications

Supply Chain Resiliency and Risk Mitigation

I applied the SPEU solution approaches developed in my Ph.D. research in mitigating the supplier risks—poor quality and delivery lateness—for manufacturing companies. We studied the problem of selecting supplier development programs (SDPs) to proactively mitigate supplier risks, where the effects of SDPs on supplier performance improvement are uncertain and depend probabilistically on the SDP selection decision. We proposed a new SPEU model that selects optimal SDPs accounting for multiple supplier risks [1]. We implemented an accelerated L-shaped decomposition algorithm and proposed a sample-based greedy approximation algorithm to solve the model. While the L-shaped decomposition algorithm provides the exact solution for moderately-sized problems, the sample-based greedy approximation algorithm provides very good-quality approximate solutions for large-scale problem instances. Our case study based on four manufacturing companies shows that the proposed model and solution approaches are beneficial for low-volume, high-value manufacturing industries (e.g., nuclear power plant construction, shipbuilding). 

Currently, I am working on a supply chain resiliency optimization problem.  

Relevant Publications

Cyber and Information Network Security

My foray into cyber-security research was through my M.S. research at Mississippi State University, (1)  cyber network security problems from the perspective of a cyber network owner (defender) who seeks to efficiently utilize the limited defense resources to deploy security countermeasures (e.g., firewalls, intrusion detection/prevention systems) to protect critical assets from cyber-attacks. In reality, cyber systems are vulnerable to multiple types of adversaries (attackers), each with different capabilities (e.g., skills, resources, etc.). Additionally, network owner faces uncertainty about the attackers’ capabilities, resources, and probabilities of attack success while making strategic decisions against them. In the presence of these uncertainties about attackers, it is crucial to find an optimal security countermeasures deployment decision that performs well under any particular attacker’s attributes (e.g., capabilities, resources, attack success probabilities).

To provide decision support to the cyber network owners, I used the concept of an attack graph that maps an organization’s cyber network topology to potential attack paths through which attackers can breach the critical assets of the organization. A node in the attack graph represents a security condition or an asset, whereas an arc represents an attacker’s action (attack). The goal of the network owners is to deploy countermeasures in the network that interdict (remove) the corresponding attack paths from the network, meaning that the attacker cannot use that path. Figure 5 demonstrates how attack graphs can be used to represent the potential attack paths for an example cyber system.

Figure 5: Attack graph representation of an example cyber system

In one of the works, I developed a two-stage stochastic mixed-integer programming model based on the attack graph concept to minimize the expected loss from cyber attacks by optimally deploying security countermeasures under uncertainty in attackers’ attack success probabilities [1]. We implemented a Sample Average Approximation (SAA) algorithm—a sampling strategy— and an L-shaped algorithm to solve the stochastic programming model for a large number of scenarios. 


In reality, the cyber network security against attackers can be viewed as a two-player Stackelberg game, where the network owner tries to minimize the loss, whereas the attacker seeks to maximize the loss to the defender. Therefore, to obtain a robust countermeasure deployment decision, the network owner needs to account for the attacker’s objective, making the problem a bi-level optimization. As the network owner has uncertainty about the attackers’ capabilities, some attackers having large capabilities (e.g., resources) can cause severe cyber-attacks, resulting in substantially large financial and reputation loss. Risk-neutral optimization models ignore the risk of these severe attacks. But, these severe cyber-attacks can cripple an organization. Because it is impossible to recover from such consequences in a short time; if recovery is possible at all. Therefore, it is crucial to account for the minimization of the risks of substantially large losses in the presence of uncertainty through a risk-averse modeling approach.


To provide decision-support to the risk-averse network managers, we developed a new risk-averse bi-level stochastic integer programming model that explicitly models (accounts for) multiple attackers with uncertainty in attackers’ capabilities and the risks of substantially large losses from severe attacks while computing the optimal decision [2]. In the stochastic programming model, we incorporated a risk-measure, conditional-value-at-risk (Figure 6), to model the risk of substantially large losses. We developed a customized constraint-and-column generation algorithm as well as several novel speed-up techniques in solving the risk-averse bi-level model. The proposed method provides optimal countermeasure deployment decisions that minimize the expected loss as well as the risk of substantially large losses from severe attacks. Results show that the stochastic risk-averse model provides substantially better network interdiction (countermeasure deployment) decisions (yielding up to 44.78% less expected loss from cyber-attacks)  than the existing models that ignore uncertainty in attackers’ capabilities and the risk of severe attacks.  


These countermeasure deployment decisions obtained from the optimization methods can then be mapped back in the cyber-network topology to determine on which network devices or links of an organization a security countermeasure should be deployed to protect critical assets. These methods can be easily adapted to secure different smart systems, such as smart manufacturing and smart power grids, from cyber-physical attacks.   

Figure 6: Conditional-value-at-risk measure

Figure 7: Effect of attackers' budget distribution on risk measure

Besides the mathematical optimization methods, I worked on developing/implementing supervised machine learning algorithms to detect malicious cyber nodes in large (millions of nodes graph), real-life, and highly class-imbalanced networks utilizing high-performance computing [3]. Malicious entities in the internet network, commonly known as bots, are a serious threat to internet network security, as these bots can perform a wide range of malicious activities such as sending spam emails, conducting distributed denial of service (DDoS), and click-fraud scams. It is challenging to detect bots in large, highly class-imbalanced network data, where the bot proportion is very small compared to the normal traffic. Most of the existing machine learning algorithms fail to detect bots (minority class) in highly class-imbalanced data, as the algorithms are biased towards the majority class. Moreover, most of the existing methods can only detect specific types of bots. Various types of bots can exist in a network, necessitating a generalized bot detection method capable of detecting various types of bots.


Therefore, we developed a novel bot detection method for large (millions of nodes), highly class-imbalanced (bot proportion is very low: 0.00583%), real data. We preprocessed the data to remove noisy observations. By analyzing the data, we developed and computed three novel features that can distinguish bots from normal traffic. Using these features, we developed the following supervised machine learning algorithms: Quadratic discriminant analysis, Gaussian Naïve Bayes, Support Vector Machine, K-Nearest Neighbor, and Random Forest. We trained these algorithms and tuned their parameters to reduce their bias towards the majority class. Due to large-scale (millions of node graphs) data, we used parallel computing to implement our methods in a high-performance computing environment. Results show that the proposed method substantially outperforms the existing methods in detecting bots from large, real-world, highly class-imbalanced data.

In another work, we developed a scalable distributed graph simulation algorithm using the concept of role mining, transition probability matrix, and parallel computing to generate very large-scale graphs (in the scale of billions of nodes) containing malicious nodes [4]. This distributed algorithm is capable of generating very large-scale graphs with accurate simulation of the behavior of malicious nodes from a given real-life input network. Figures 8, 9, and 10 demonstrate the original, simulated, and scaled-up simulated graphs, respectively. We also developed a hypergraph generation algorithm for a parallel computing environment to generate large hypergraphs [5].

Figure 8: Original graph

Figure 9: Simulated graph

Figure 10: Scaled-up simulated graph

Relevant Publications

Aerial Drone and Ground-only Vehicle Routing

I am studying aerial drone deployment optimization problems for more efficient goods delivery. Aerial drones offer a distinct potential to improve the delivery of goods in the last mile (often with other vehicles), especially for time-sensitive (e.g., medicine, prepared food, groceries), small, and localized deliveries. But, there is little understanding among interested industry parties on the following questions: (1) how drones may be efficiently deployed; (2) how do different delivery methods,  operating, and environmental conditions affect the mission capability and energy efficiency of drones; and (3) how suitable and energy-efficient are the drones compared to ground-only vehicles?  

Figure 11: Drone deployment for direct delivery of goods to customers

Figure 12: Drone deployment for pickup and delivery services

To answer the above questions, I have been developing data-driven mixed-integer programming models accounting for delivery time windows and battery swapping for drone and ground-only vehicle routing problems under different delivery scenarios (e.g., direct delivery from a centralized depot to retail consumers, delivery services like UberEats, and parcel delivery services with other vehicles), environmental (e.g., wind, temperature), and operating conditions (e.g., drone speed, flight path, package weight). These optimization models integrate energy consumption prediction models based on actual drone flight test data. I am also developing efficient solution approaches to solve the developed models for a 24-hour planning horizon. The developed models and solution approaches provide insights into the following questions: (1) required drone and vehicle fleet sizing under different delivery methods (e.g., speed, landing vs. package dropping, straight path vs. waypoints), environmental conditions (e.g., wind, temperature); (2) number of additional batteries and charging infrastructure needed under different delivery methods and environmental conditions; (3) energy and cost efficiency as well as mission capability of drones compared to ground vehicles under different delivery scenarios, operating, and environmental conditions.  

Relevant Publications

Wireless Network Security

I am studying the vulnerability of multi-hop wireless networks to jamming attacks. Gaining insights into network vulnerability is the first step in protecting wireless networks from adversaries. Unlike wired networks (e.g., transportation networks as shown in Figure 13), wireless networks (Figure 14) have a special property called radio interference that significantly affects the network throughput and makes the modeling of wireless networks to be different than wired networks. Due to this radio interference property, communications interfere with each other if they are within their range of interference and thus all communications cannot be active simultaneously, which substantially affects the network throughput. Figure 14 shows a connectivity graph of a small wireless communication network, where the same colored arcs can communicate (be active) simultaneously, which forms a schedulable set, a.k.a., independent set in graph theory. Based on this interference among communications, we can construct a conflict graph (or interference graph) to represent the radio interference. Figure 15 demonstrates a partial conflict graph corresponding to the connectivity graph shown in Figure 14. An independent set of nodes (equivalently a set of arcs in the connectivity graph) in the conflict graph can transmit data simultaneously, and scheduling maximal independent sets ensures maximum flow in the network.           

Figure 13: Wired network 

Figure 14: Wireless network (connectivity graph)

Figure 15: Conflict graph

There exist several models of this radio interference, such as capture, protocol, and interference range. However, However, a physical model of radio interference in wireless networks is more realistic; but, the jammer placement problem under the physical model of interference has not been analyzed. We have developed attacker-defender bi-level mixed-integer programming models under the physical (i.e., additive) and other non-additive models (e.g., capture, protocol, interference range) of radio interference to find optimal locations of jamming devices that minimize the maximum network throughput. We have developed an improved branch-and-cut and a hybrid Tabu search algorithm to solve the bi-level models. This research provides key insights and helps in finding the most beneficial strategies for designing a network to be jamming-resistant and can lead to new tools for designing wireless networks to mitigate the effect of jamming attacks.

Relevant Publications

Power Systems Resiliency and Operations with Renewable Energy Resources

My first hands-on experience on power system research was through my internship at Argonne National Laboratory, where I worked on solving a power generation capacity expansion problem integrating unit commitment and economic dispatch decisions for both hourly and sub-hourly (5-minutes) resolutions. We formulated the problem as a mixed-integer program to find the minimum cost capacity expansion and unit commitment decisions with variable renewable energy sources and storage. The problem considers several different power generation technologies including thermal, renewable energy sources (wind, solar), and storage, where the goal is to find the minimum cost capacity expansion mix (number of units of different technologies). We formulated the problem as a mixed-integer program. The capacity expansion problem is solved over a year whereas the operational mode (unit commitment and economic dispatch) is solved for each day. As solving the capacity expansion problem along with unit commitment and economic dispatch is computationally infeasible, I worked on implementing a scenario reduction algorithm that approximates the yearly planning horizon by a user-specified smaller number of representative days. This scenario reduction algorithm that I implemented takes the temporal data (load profile, wind, and solar shapes) over the year and then reduces the 365 days into a given number of representative days based on the similarity of the temporal data of the days. This research provides insight into the following questions: (1) how sensitive are the capacity expansion decisions to changes in the representative days, (2) whether the expansion decisions change from hourly to 5-min resolutions, (3) how the cost parameters of renewables (wind, solar) and storage affect the expansion decisions, and (3) whether the change in storage cost parameters causes the wind/solar decisions to change as well as whether this effect is different for hourly and 5-min resolutions?  

I am currently studying a distributed local market energy transaction problem incorporating electrical vehicle (EV) charging stations and renewable energy resources under uncertainties as a Stackelberg game. In this game setting, a local energy transaction agent (center) acts as a leader (upper-level), whereas multiple prosumers (lower-level) equipped with distributed energy resources can purchase/sell energy through the local energy transaction center as well as via peer-to-peer energy transaction [1]. In this bi-level game-theoretic trading framework, the local energy transaction agent, equipped with energy storage,  seeks to maximize its profit by coordinating the local power balance considering network constraints. Realizing the electricity price from the upper-level agent, the prosumers, EV charging station, commercial buildings, and renewable energy generation firms, decide on the transaction amount to minimize their operational cost. The lower-level agents (i.e., prosumers) consider the uncertainties in the operational level, such as the EV arrival probability at the charging station, the uncertainty in the load profile of commercial buildings, and the uncertainty in renewable energy generation. We are working on developing efficient algorithms in efficiently solving this decision-making problem.       

Relevant Publications