A robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended. Under this approach, we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes. A concern with such an approach is that it might be too conservative. In this paper, we propose an approach that attempts to make this trade-off more attractive; that is, we investigate ways to decrease what we call the price of robustness. In particular, we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations. An attractive aspect of our method is that the new robust formulation is also a linear optimization problem. Thus we naturally extend our methods to discrete optimization problems in a tractable way. We report numerical results for a portfolio optimization problem, a knapsack problem, and a problem from the Net Lib library.

In this paper we survey the primary research, both theoretical and applied, in the field of Robust Optimization (RO). Our focus will be on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying the most prominent theoretical results of RO over the past decade, we will also present some recent results linking RO to adaptable models for multi-stage decision-making problems. Finally, we will highlight successful applications of RO across a wide spectrum of domains, including, but not limited to, finance, statistics, learning, and engineering.

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Abstract-Unit commitment, one of the most critical tasks in electric power system operations, faces new challenges as the supply and demand uncertainty increases dramatically due to the integration of variable generation resources such as wind power and price responsive demand. To meet these challenges, we propose a two-stage adaptive robust unit commitment model for the security constrained unit commitment problem in the presence of nodal net injection uncertainty. Compared to the conventional stochastic programming approach, the proposed model is more practical in that it only requires a deterministic uncertainty set, rather than a hard-to-obtain probability distribution on the uncertain data. The unit commitment solutions of the proposed model are robust against all possible realizations of the modeled uncertainty. We develop a practical solution methodology based on a combination of Benders decomposition type algorithm and the outer approximation technique. We present an extensive numerical study on the real-world large scale power system operated by the ISO New England. Computational results demonstrate the economic and operational advantages of our model over the traditional reserve adjustment approach.Index Terms-Bilevel mixed-integer optimization, power system control and reliability, robust and adaptive optimization, security constrained unit commitment.

We derive dynamic optimal trading strategies that minimize the expected cost of trading a large block of equity over a fixed time horizon. Specifically, given a fixed block SM of shares to be executed within a fixed finite number of periods ¹, and given a price-impact function that yields the execution price of an individual trade as a function of the shares traded and market conditions, we obtain the optimal sequence of trades as a function of market conditions -closed-form expressions in some casesthat minimizes the expected cost of executing SM within ¹ periods. Our analysis is extended to the portfolio case in which price impact across stocks can have an important effect on the total cost of trading a portfolio.1998 Elsevier Science B.V. All rights reserved. JEL classification: G23

In the last twenty-five years , algorithmic advances in integer optimization combined with hardware improvements have resulted in an astonishing 200 billion factor speedup in solving Mixed Integer Optimization (MIO) problems. We present a MIO approach for solving the classical best subset selection problem of choosing k out of p features in linear regression given n observations. We develop a discrete extension of modern first order continuous optimization methods to find high quality feasible solutions that we use as warm starts to a MIO solver that finds provably optimal solutions. The resulting algorithm (a) provides a solution with a guarantee on its suboptimality even if we terminate the algorithm early, (b) can accommodate side constraints on the coefficients of the linear regression and (c) extends to finding best subset solutions for the least absolute deviation loss function. Using a wide variety of synthetic and real datasets, we demonstrate that our approach solves problems with n in the 1000s and p in the 100s in minutes to provable optimality, and finds near optimal solutions for n in the 100s and p in the 1000s in minutes. We also establish via numerical experiments that the MIO approach performs better than Lasso and other popularly used sparse learning procedures, in terms of achieving sparse solutions with good predictive power.

State-of-the-art decision tree methods apply heuristics recursively to create each split in isolation, which may not capture well the underlying characteristics of the dataset. The optimal decision tree problem attempts to resolve this by creating the entire decision tree at once to achieve global optimality. In the last 25 years, algorithmic advances in integer optimization coupled with hardware improvements have resulted in an astonishing 800 billion factor speedup in mixed-integer optimization (MIO). Motivated by this speedup, we present optimal classification trees, a novel formulation of the decision tree problem using modern MIO techniques that yields the optimal decision tree for axes-aligned splits. We also show the richness of this MIO formulation by adapting it to give optimal classification trees with hyperplanes that generates optimal decision trees with multivariate splits. Synthetic tests demonstrate that these methods recover the true decision tree more closely than heuristics, refuting the notion that optimal methods overfit the training data. We comprehensively benchmark these methods on a sample of 53 datasets from the UCI machine learning repository. We establish that these MIO methods are practically solvable on real-world datasets with sizes in the 1000s, and give average absolute improvements in out-of-sample accuracy over CART of 1-2 and 3-5% for the univariate and multivariate cases, respectively. Furthermore, we identify that optimal classification trees are likely to outperform CART by 1.2-1.3% in situations where the CART accuracy is high and we have sufficient training data, while the multivariate version outperforms CART by 4-7% when the CART accuracy or dimension of the dataset is low.

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