Theory and Applications of Robust Optimization

In this paper we survey the primary research, both theoretical and applied, in the area of Robust Optimization (RO). Our focus is on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying prominent theoretical results of RO, we also present some recent results linking RO to adaptable models for multi-stage decision-making problems. Finally, we highlight applications of RO across a wide spectrum of domains, including finance, statistics, learning, and various areas of engineering.Comment: 50 page

Investment under ambiguity with the best and worst in mind

Recent literature on optimal investment has stressed the difference between the impact of risk and the impact of ambiguity - also called Knightian uncertainty - on investors' decisions. In this paper, we show that a decision maker's attitude towards ambiguity is similarly crucial for investment decisions. We capture the investor's individual ambiguity attitude by applying alpha-MEU preferences to a standard investment problem. We show that the presence of ambiguity often leads to an increase in the subjective project value, and entrepreneurs are more eager to invest. Thereby, our investment model helps to explain differences in investment behavior in situations which are objectively identical

On the approximability of adjustable robust convex optimization under uncertainty

In this paper, we consider adjustable robust versions of convex optimization problems with uncertain constraints and objectives and show that under fairly general assumptions, a static robust solution provides a good approximation for these adjustable robust problems. An adjustable robust optimization problem is usually intractable since it requires to compute a solution for all possible realizations of uncertain parameters, while an optimal static solution can be computed efficiently in most cases if the corresponding deterministic problem is tractable. The performance of the optimal static robust solution is related to a fundamental geometric property, namely, the symmetry of the uncertainty set. Our work allows for the constraint and objective function coefficients to be uncertain and for the constraints and objective functions to be convex, thereby providing significant extensions of the results in Bertsimas and Goyal (Math Oper Res 35:284–305, 2010) and Bertsimas et al. (Math Oper Res 36: 24–54, 2011b) where only linear objective and linear constraints were considered. The models in this paper encompass a wide variety of problems in revenue management, resource allocation under uncertainty, scheduling problems with uncertain processing times, semidefinite optimization among many others. To the best of our knowledge, these are the first approximation bounds for adjustable robust convex optimization problems in such generality.National Science Foundation (U.S.) (NSF Grant CMMI-1201116

Essays in Robust and Data-Driven Risk Management

Risk defined as the chance that the outcome of an uncertain event is different than expected. In practice, the risk reveals itself in different ways in various applications such as unexpected stock movements in the area of portfolio management and unforeseen demand in the field of new product development. In this dissertation, we present four essays on data-driven risk management to address the uncertainty in portfolio management and capacity expansion problems via stochastic and robust optimization techniques.The third chapter of the dissertation (Portfolio Management with Quantile Constraints) introduces an iterative, data-driven approximation to a problem where the investor seeks to maximize the expected return of his/her portfolio subject to a quantile constraint, given historical realizations of the stock returns. Our approach involves solving a series of linear programming problems (thus) quickly solves the large scale problems. We compare its performance to that of methods commonly used in finance literature, such as fitting a Gaussian distribution to the returns. We also analyze the resulting efficient frontier and extend our approach to the case where portfolio risk is measured by the inter-quartile range of its return. Furthermore, we extend our modeling framework so that the solution calculates the corresponding conditional value at risk CVaR) value for the given quantile level.The fourth chapter (Portfolio Management with Moment Matching Approach) focuses on the problem where a manager, given a set of stocks to invest in, aims to minimize the probability of his/her portfolio return falling below a threshold while keeping the expected portfolio returnno worse than a target, when the stock returns are assumed to be Log-Normally distributed. This assumption, common in finance literature, creates computational difficulties. Because the portfolio return itself is difficult to estimate precisely. We thus approximate the portfolio re-turn distribution with a single Log-Normal random variable by the Fenton-Wilkinson method and investigate an iterative, data-driven approximation to the problem. We propose a two-stage solution approach, where the first stage requires solving a classic mean-variance optimization model, and the second step involves solving an unconstrained nonlinear problem with a smooth objective function. We test the performance of this approximation method and suggest an iterative calibration method to improve its accuracy. In addition, we compare the performance of the proposed method to that obtained by approximating the tail empirical distribution function to a Generalized Pareto Distribution, and extend our results to the design of basket options.The fifth chapter (New Product Launching Decisions with Robust Optimization) addresses the uncertainty that an innovative firm faces when a set of innovative products are planned to be launched a national market by help of a partner company for each innovative product. Theinnovative company investigates the optimal period to launch each product in the presence of the demand and partner offer response function uncertainties. The demand for the new product is modeled with the Bass Diffusion Model and the partner companies\u27 offer response functions are modeled with the logit choice model. The uncertainty on the parameters of the Bass Diffusion Model and the logic choice model are handled by robust optimization. We provide a tractable robust optimization framework to the problem which includes integer variables. In addition, weprovide an extension of the proposed approach where the innovative company has an option to reduce the size of the contract signed by the innovative firm and the partner firm for each product.In the sixth chapter (Log-Robust Portfolio Management with Factor Model), we investigate robust optimization models that address uncertainty for asset pricing and portfolio management. We use factor model to predict asset returns and treat randomness by a budget of uncertainty. We obtain a tractable robust model to maximize the wealth and gain theoretical insights into the optimal investment strategies

Robust Strategies for Optimal Order Execution in the Almgren-Chriss Framework

Assuming geometric Brownian motion as unaffected price process $S^0$, Gatheral & Schied (2011) derived a strategy for optimal order execution that reacts in a sensible manner on market changes but can still be computed in closed form. Here we will investigate the robustness of this strategy with respect to misspecification of the law of $S^0$. We prove the surprising result that the strategy remains optimal whenever $S^0$ is a square-integrable martingale. We then analyze the optimization criterion of Gatheral & Schied (2011) in the case in which $S^0$ is any square-integrable semimartingale and we give a closed-form solution to this problem. As a corollary, we find an explicit solution to the problem of minimizing the expected liquidation costs when the unaffected price process is a square-integrable semimartingale. The solutions to our problems are found by stochastically solving a finite-fuel control problem without assumptions of Markovianity