DECENTRALIZED ALGORITHMS FOR NASH EQUILIBRIUM PROBLEMS – APPLICATIONS TO MULTI-AGENT NETWORK INTERDICTION GAMES AND BEYOND

Nash equilibrium problems (NEPs) have gained popularity in recent years in the engineering community due to their ready applicability to a wide variety of practical problems ranging from communication network design to power market analysis. There are strong links between the tools used to analyze NEPs and the classical techniques of nonlinear and combinatorial optimization. However, there remain significant challenges in both the theoretical and algorithmic analysis of NEPs. This dissertation studies certain special classes of NEPs, with the overall purpose of analyzing theoretical properties such as existence and uniqueness, while at the same time proposing decentralized algorithms that provably converge to solutions. The subclasses are motivated by relevant application examples

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Complementarity and related problems

In this thesis, we present results related to complementarity problems. We study the linear complementarity problems on extended second order cones. We convert a linear complementarity problem on an extended second order cone into a mixed complementarity problem on the non-negative orthant. We present algorithms for this problem, and exemplify it by a numerical example. Following this result, we explore the stochastic version of this linear complementarity problem. Finally, we apply complementarity problems on extended second order cones in a portfolio optimisation problem. In this application, we exploit our theoretical results to find an analytical solution to a new portfolio optimisation model. We also study the spherical quasi-convexity of quadratic functions on spherically self-dual convex sets. We start this study by exploring the characterisations and conditions for the spherical positive orthant. We present several conditions characterising the spherical quasi-convexity of quadratic functions. Then we generalise the conditions to the spherical quasi-convexity on spherically self-dual convex sets. In particular, we highlight the case of spherical second order cones

Lifted Stationary Points of Sparse Optimization with Complementarity Constraints

We aim to compute lifted stationary points of a sparse optimization problem (P0) with complementarity constraints. We define a continuous relaxation problem (Rv) that has the same global minimizers and optimal value with problem (P0). Problem (Rv) is a mathematical program with complementarity constraints (MPCC) and a difference-of-convex (DC) objective function. We define MPCC lifted-stationarity of (Rv) and show that it is weaker than directional stationarity, but stronger than Clarke stationarity for local optimality. Moreover, we propose an approximation method to solve (Rv) and an augmented Lagrangian method to solve its subproblem, which relaxes the equality constraint in (Rv) with a tolerance. We prove the convergence of our algorithm to an MPCC lifted-stationary point of problem (Rv) and use a sparse optimization problem with vertical linear complementarity constraints to demonstrate the efficiency of our algorithm on finding sparse solutions in practice.Comment: 29 pages, 3 figure

Applications of Machine Learning in Supply Chains

Advances in new technologies have resulted in increasing the speed of data generation and accessing larger data storage. The availability of huge datasets and massive computational power have resulted in the emergence of new algorithms in artificial intelligence and specifically machine learning, with significant research done in fields like computer vision. Although the same amount of data exists in most components of supply chains, there is not much research to utilize the power of raw data to improve efficiency in supply chains.In this dissertation our objective is to propose data-driven non-parametric machine learning algorithms to solve different supply chain problems in data-rich environments.Among wide range of supply chain problems, inventory management has been one of the main challenges in every supply chain. The ability to manage inventories to maximize the service level while minimizing holding costs is a goal of many company. An unbalanced inventory system can easily result in a stopped production line, back-ordered demands, lost sales, and huge extra costs. This dissertation studies three problems and proposes machine learning algorithms to help inventory managers reduce their inventory costs.In the first problem, we consider the newsvendor problem in which an inventory manager needs to determine the order quantity of a perishable product to minimize the sum of shortage and holding costs, while some feature information is available for each product. We propose a neural network approach with a specialized loss function to solve this problem. The neural network gets historical data and is trained to provide the order quantity. We show that our approach works better than the classical separated estimation and optimization approaches as well as other machine learning based algorithms. Especially when the historical data is noisy, and there is little data for each combination of features, our approach works much better than other approaches. Also, to show how this approach can be used in other common inventory policies, we apply it on an $(r,Q)$ policy and provide the results.This algorithm allows inventory managers to quickly determine an order quantity without obtaining the underling demand distribution.Now, assume the order quantities or safety stock levels are obtained for a single or multi-echelon system. Classical inventory optimization models work well in normal conditions, or in other words when all underlying assumptions are valid. Once one of the assumptions or the normal working conditions is violated, unplanned stock-outs or excess inventories arise.To address this issue, in the second problem, a multi-echelon supply network is considered, and the goal is to determine the nodes that might face a stock-out in the next period. Stock-outs are usually expensive and inventory managers try to avoid them, so stock-out prediction might results in averting stock-outs and the corresponding costs.In order to provide such predictions, we propose a neural network model and additionally three naive algorithms. We analyze the performance of the proposed algorithms by comparing them with classical forecasting algorithms and a linear regression model, over five network topologies. Numerical results show that the neural network model is quite accurate and obtains accuracies in $[0.92, 0.99]$ for the hardest to easiest network topologies, with average of 0.950 and standard deviation of 0.023, while the closest competitor, i.e., one of the proposed naive algorithms, obtains accuracies in $[0.91, 0.95]$ with average of 9.26 and standard deviation of .0136. Additionally, we suggest conditions under which each algorithm is the most reliable and additionally apply all algorithms to threshold and multi-period predictions.Although stock-out prediction can be very useful, any inventory manager would like to have a powerful model to optimize the inventory system and balance the holding and shortage costs. The literature on multi-echelon inventory models is quite rich, though it mostly relies on the assumption of accessing a known demand distribution. The demand distribution can be approximated, but even so, in some cases a globally optimal model is not available.In the third problem, we develop a machine learning algorithm to address this issue for multi-period inventory optimization problems in multi-echelon networks. We consider the well-known beer game problem and propose a reinforcement learning algorithm to efficiently learn ordering policies from data.The beer game is a serial supply chain with four agents, i.e. retailer, wholesaler, distributor, and manufacturer, in which each agent replenishes its stock by ordering beer from its predecessor. The retailer satisfies the demand of external customers, and the manufacturer orders from external suppliers. Each of the agents must decide its own order quantity to minimize the summation of holding and shortage cost of the system, while they are not allowed to share any information with other agents. For this setting, a base-stock policy is optimal, if the retailer is the only node with a positive shortage cost and a known demand distribution is available. Outside of this narrow condition, there is not a known optimal policy for this game. Also, from the game theory point of view, the beer game can be modeled as a decentralized multi-agent cooperative problem with partial observability, which is known as a NEXP-complete problem.We propose an extension of deep Q-network for making decisions about order quantities in a single node of the beer game. When the co-players follow a rational policy, it obtains a close-to-optimal solution, and it works much better than a base-stock policy if the other agents play irrationally. Additionally, to reduce the training time of the algorithm, we propose using transfer learning, which reduces the training time by one order of magnitude. This approach can be extended to other inventory optimization and supply chain problems