Robust Transmission Network Expansion Planning Under Correlated Uncertainty

Robust Optimization for the Day-Ahead Scheduling of Cascaded Hydroelectric Systems

The robust optimization method has progressively become a research hot spot as a valuable means for dealing with parameter uncertainty in optimization problems. Based on the asymmetric cost consensus model, this paper considers the uncertainties of the experts’ unit adjustment costs under the background of group decision making. At the same time, four uncertain level parameters are introduced. For three types of minimum cost consensus models with direction restrictions, including MCCM-DC, $$\varepsilon $$ -MCCM-DC and threshold-based (TB)-MCCM-DC, the robust cost consensus models corresponding to four types of uncertainty sets (Box set, Ellipsoid set, Polyhedron set and Interval-Polyhedron set) are established. Sensitivity analysis is carried out under different parameter conditions to determine the robustness of the solutions obtained from robust optimization models. The robust optimization models are then compared to the minimum cost models for consensus. The example results show that the Interval-Polyhedron set’s robust models have the smallest total costs and strongest robustness. Decision makers can choose the combination of uncertainty sets and uncertain levels according to their risk preferences to minimize the total cost. Finally, in order to reduce the conservatism of the classical robust optimization method, the pricing information of the new product MACUBE 550 is used to build a data-driven robust optimization model. Ellipsoid uncertainty set is proved to better trade-off the average performance and robust performance through different measurement indicators. Therefore, the uncertainty set can be selected according to the needs of the group.

Robust optimization considers optimization problems with uncertainty in the data. The common data model assumes that the uncertainty can be represented by an uncertainty set. Classic robust optimization considers the solution under the worst case scenario. The resulting solutions are often too conservative, e.g. they have high costs compared to non-robust solutions. This is a reason for the development of less conservative robust models. In this paper we extract the basic idea of the concept of light robustness originally developed in Fischetti and Monaci (Robust and online large-scale optimization, volume 5868 of lecture note on computer science. Springer, Berlin, pp 61–84, 2009) for interval-based uncertainty sets and linear programs: fix a quality standard for the nominal solution and among all solutions satisfying this standard choose the most reliable one. We then use this idea in order to formulate the concept of light robustness for arbitrary optimization problems and arbitrary uncertainty sets. We call the resulting concept generalized light robustness. We analyze the concept and discuss its relation to other well-known robustness concepts such as strict robustness (Ben-Tal et al. in Robust optimization. Princeton University Press, Princeton, 2009), reliability (Ben-Tal and Nemirovski in Math Program A 88:411–424, 2000) or the approach of Bertsimas and Sim (Oper Res 52(1):35–53, 2004). We show that the light robust counterpart is computationally tractable for many different types of uncertainty sets, among them polyhedral or ellipsoidal uncertainty sets. We furthermore discuss the trade-off between robustness and nominal quality and show that non-dominated solutions with respect to nominal quality and robustness can be computed by the generalized light robustness approach.

Robust optimization (RO) is a tractable method to address uncertainty in optimization problems where uncertain parameters are modeled as belonging to uncertainty sets that are commonly polyhedral or ellipsoidal. The two most frequently described methods in the literature for solving RO problems are reformulation to a deterministic optimization problem or an iterative cutting-plane method. There has been limited comparison of the two methods in the literature, and there is no guidance for when one method should be selected over the other. In this paper we perform a comprehensive computational study on a variety of problem instances for both robust linear optimization (RLO) and robust mixed-integer optimization (RMIO) problems using both methods and both polyhedral and ellipsoidal uncertainty sets. We consider multiple variants of the methods and characterize the various implementation decisions that must be made. We measure performance with multiple metrics and use statistical techniques to quantify certainty in the results. We find for polyhedral uncertainty sets that neither method dominates the other, in contrast to previous results in the literature. For ellipsoidal uncertainty sets we find that the reformulation is better for RLO problems, but there is no dominant method for RMIO problems. Given that there is no clearly dominant method, we describe a hybrid method that solves, in parallel, an instance with both the reformulation method and the cutting-plane method. We find that this hybrid approach can reduce runtimes to 50–75 % of the runtime for any one method and suggest ways that this result can be achieved and further improved on.

This paper is concerned with the constrained shortest path (CSP) problem, where in addition to the arc cost, a transit time is associated to each arc. The presence of uncertainty in transit times is a critical issue in a wide variety of world applications, such as telecommunication, traffic, and transportation. To capture this issue, we present tractable approaches for solving the CSP problem with uncertain transit times from the viewpoint of robust and stochastic optimization. To study robust CSP problem, two different uncertainty sets, $${\varGamma }$$Γ-scenario and ellipsoidal, are considered. We show that the robust counterpart of the CSP problem under both uncertainty sets, can be efficiently solved. We further consider the CSP problem with random transit times and show that the problem can be solved by solving robust constrained shortest path problem under ellipsoidal uncertainty set. We present extensive computational results on a set of randomly generated networks. Our results demonstrate that with a reasonable extra cost, the robust optimal path preserves feasibility, in almost all scenarios under $${\varGamma }$$Γ-scenario uncertainty set. The results also show that, in the most cases, the robust CSP problem under ellipsoidal uncertainty set is feasible.

Optimization is a key analytical technique used for quantitative decision-making in real-world problems. In practice, many situations call for decision-making in the face of incomplete knowledge and/or dynamic environments. Making high-quality decisions in these settings requires optimization techniques that are designed to account for uncertainty. Furthermore, as new technologies are developed, more complex higher-dimensional optimization models become prevalent. This dissertation examines various models for optimization under uncertainty, as well as efficient algorithms for solving such models that are scalable as the model size grows.We study three models for optimization under uncertainty: robust optimization (RO), joint estimation-optimization (JEO), and joint prediction-optimization (JPO). Robust optimization accounts for inexact information by finding solutions, which remain feasible to all perturbations ofinputs within a given uncertainty set. Joint estimation-optimization considers a dynamic setting where inputs are updated over time as new data is collected and converge to some ideal input that isnot revealed to the modeller. Joint prediction-optimization considers the use of a prediction model to obtain optimization inputs from side information, an approach that is widely used amongst practitioners.The dissertation considers theoretical properties and algorithmic performance guarantees for these three models.We first present a generic framework to derive primal-dual algorithms for both RO and JEO. Previously, algorithms for such models were derived in an ad-hoc manner, and analyzed on a case-by-case basis. Our framework considers both of these optimization under uncertainty modelsthrough a common lens of saddle point problems. By analyzing these, we highlight three quantities which directly bound the performance guarantees for our respective models, and show how regretminimization techniques from online convex optimization can be used to control these three quantities. Thus, our framework allows us to transfer regret bounds for these quantities into performance guarantees for the associated algorithms. Since regret minimization algorithms from online convexoptimization are key to our framework, we also examine these, and in particular derive improved regret bounds for RO and JEO in the presence of favourable structure such as strong convexityand smoothness. We show that a number of previous algorithms for both robust optimization and joint estimation optimization can be derived from our uni ed framework. More importantly, our framework can be used to derive more efficient algorithms for both models in a principled manner. For robustoptimization, our framework is used to derive algorithms that can drastically reduce the cost of iterative methods by replacing nominal oracles with cheaper first-order updates. For joint estimation optimization, we derive algorithms for the non-smooth strongly convex setting, which has not been considered previously.We demonstrate the use of our framework through two examples: robust quadratic programming with ellipsoidal uncertainty sets, and dynamic non-parametric choice model estimation. For robust quadratic programming, we analyze the trust-region subproblem (TRS). The TRS is thewell-studied problem of minimizing a non-convex quadratic function over the unit ball, and it arises naturally in the context of robust quadratic constraints. We give a second-order cone based convexi cation of TRS which, in contrast to previous work, is still in the space of original variables.We then show how to apply this convexication to robust quadratic programming, and derive two efficient algorithms for it using our framework. We carry out a numerical study on robust portfolio optimization problems, and the numerical results show improvement of our approach over previous approaches in the high-dimensional regime. We frame dynamic non-parametric choice model estimation as an instance of JEO. A particular challenge in this setting is the high-dimensionality of the resulting primal problem. Nevertheless, our generic primal-dual framework encompassing JEO applications is quite flexible and allows us to derive algorithms that can bypass this high dimensionality challenge. We test our approach for non-parametric choice estimation computationally, and highlight interesting trade-o s between data updating and convergence rates. Finally, we give a joint analysis of prediction and optimization. A natural performance measure in this setting is the optimality gap. Unfortunately, it is difficult to directly tune prediction models using this performance measure due to its non-convexity. We thus characterize sufficient conditions under which the more common prediction performance measures arising in statistics/machine learning, such as squared error, can be related to the true optimality gap performance measure. We derive conditions on a performance measure that guarantee that the optimality gap will be minimized, and give an explicit relationship between the squared error and the optimality gap. Such conditions allow practitioners to choose prediction methods for obtaining optimization parameters in a more judicious manner.

Robust portfolio selection involving options under a “ marginal+joint ” ellipsoidal uncertainty set

‘Separable’ uncertainty sets have been widely used in robust portfolio selection models (e.g. see [E. Erdoğan, D. Goldfarb, and G. Iyengar, Robust portfolio management, manuscript, Department of Industrial Engineering and Operations Research, Columbia University, New York, 2004; D. Goldfarb and G. Iyengar, Robust portfolio selection problems, Math. Oper. Res. 28 (2003), pp. 1–38; R.H. Tütüncü and M. Koenig, Robust asset allocation, Ann. Oper. Res. 132 (2004), pp. 157–187]). For these uncertainty sets, each type of uncertain parameter (e.g. mean and covariance) has its own uncertainty set. As addressed in [Z. Lu, A new cone programming approach for robust portfolio selection, Tech. Rep., Department of Mathematics, Simon Fraser University, Burnaby, BC, 2006; Z. Lu, A computational study on robust portfolio selection based on a joint ellipsoidal uncertainty set, Math. Program. (2009), DOI: 10.1007/510107-009-0271-z], these ‘separable’ uncertainty sets typically share two common properties: (1) their actual confidence level, namely, the probability of uncertain parameters falling within the uncertainty set, is unknown, and it can be much higher than the desired one; and (2) they are fully or partially box-type. The associated consequences are that the resulting robust portfolios can be too conservative, and moreover, they are usually highly non-diversified, as observed in the computational experiments conducted in [Z. Lu, A new cone programming approach for robust portfolio selection, Tech. Rep., Department of Mathematics, Simon Fraser University, Burnaby, BC, 2006; Z. Lu, A computational study on robust portfolio selection based on a joint ellipsoidal uncertainty set, Math. Program. (2009), DOI: 10.1007/510107-009-0271-Z; R.H.Tütüncü and M. Koenig, Robust asset allocation, Ann. Oper. Res. 132 (2004), pp. 157–187]. To combat these drawbacks, we consider a factor model for random asset returns. For this model, we introduce a ‘joint’ ellipsoidal uncertainty set for the model parameters and show that it can be constructed as a confidence region associated with a statistical procedure applied to estimate the model parameters. We further show that the robust maximum risk-adjusted return (RMRAR) problem with this uncertainty set can be reformulated and solved as a cone programming problem. The computational results reported in [Z. Lu, A new cone programming approach for robust portfolio selection, Tech. Rep., Department of Mathematics, Simon Fraser University, Burnaby, BC, 2006; Z. Lu, A computational study on robust portfolio selection based on a joint ellipsoidal uncertainty set, Math. Program. (2009), DOI: 10.1007/510107-009-0271-Z] demonstrate that the robust portfolio determined by the RMRAR model with our ‘joint’ uncertainty set outperforms that with Goldfarb and Iyengar’s ‘separable’ uncertainty set proposed in the seminal paper [D. Goldfarb and G. Iyengar, Robust portfolio selection problems, Math. Oper. Res. 28 (2003), pp. 1–38] in terms of wealth growth rate and transaction cost; moreover, our robust portfolio is fairly diversified, but Goldfarb and Iyengar’s is surprisingly highly non-diversified.

Robust transmission expansion planning with uncertain generations and loads using full probabilistic information

The “separable” uncertainty sets have been widely used in robust portfolio selection models [e.g., see Erdogan et al. (Robust portfolio management. manuscript, Department of Industrial Engineering and Operations Research, Columbia University, New York, 2004), Goldfarb and Iyengar (Math Oper Res 28:1–38, 2003), Tutuncu and Koenig (Ann Oper Res 132:157–187, 2004)]. For these uncertainty sets, each type of uncertain parameters (e.g., mean and covariance) has its own uncertainty set. As addressed in Lu (A new cone programming approach for robust portfolio selection, technical report, Department of Mathematics, Simon Fraser University, Burnaby, 2006; Robust portfolio selection based on a joint ellipsoidal uncertainty set, manuscript, Department of Mathematics, Simon Fraser University, Burnaby, 2008), these “separable” uncertainty sets typically share two common properties: (i) their actual confidence level, namely, the probability of uncertain parameters falling within the uncertainty set is unknown, and it can be much higher than the desired one; and (ii) they are fully or partially box-type. The associated consequences are that the resulting robust portfolios can be too conservative, and moreover, they are usually highly non-diversified as observed in the computational experiments conducted in this paper and Tutuncu and Koenig (Ann Oper Res 132:157–187, 2004). To combat these drawbacks, the author of this paper introduced a “joint” ellipsoidal uncertainty set (Lu in A new cone programming approach for robust portfolio selection, technical report, Department of Mathematics, Simon Fraser University, Burnaby, 2006; Robust portfolio selection based on a joint ellipsoidal uncertainty set, manuscript, Department of Mathematics, Simon Fraser University, Burnaby, 2008) and showed that it can be constructed as a confidence region associated with a statistical procedure applied to estimate the model parameters. For this uncertainty set, we showed in Lu (A new cone programming approach for robust portfolio selection, technical report, Department of Mathematics, Simon Fraser University, Burnaby, 2006; Robust portfolio selection based on a joint ellipsoidal uncertainty set, manuscript, Department of Mathematics, Simon Fraser University, Burnaby, 2008) that the corresponding robust maximum risk-adjusted return (RMRAR) model can be reformulated and solved as a cone programming problem. In this paper, we conduct computational experiments to compare the performance of the robust portfolios determined by the RMRAR models with our “joint” uncertainty set (Lu in A new cone programming approach for robust portfolio selection, technical report, Department of Mathematics, Simon Fraser University, Burnaby, 2006; Robust portfolio selection based on a joint ellipsoidal uncertainty set, manuscript, Department of Mathematics, Simon Fraser University, Burnaby, 2008) and Goldfarb and Iyengar’s “separable” uncertainty set proposed in the seminal paper (Goldfarb and Iyengar in Math Oper Res 28:1–38, 2003). Our computational results demonstrate that our robust portfolio outperforms Goldfarb and Iyengar’s in terms of wealth growth rate and transaction cost, and moreover, ours is fairly diversified, but Goldfarb and Iyengar’s is surprisingly highly non-diversified.

This paper studies robust solutions and semidefinite linear programming (SDP) relaxations of a class of convex polynomial programs in the face of data uncertainty. The class of convex programs, called robust SOS-convex programs, includes robust quadratically constrained convex programs and robust separable convex polynomial programs. It establishes sums of squares polynomial representations characterizing robust solutions and exact SDP-relaxations of robust SOS-convex programs under various commonly used uncertainty sets. In particular, the results show that the polytopic and ellipsoidal uncertainty sets, that allow second-order cone re-formulations of robust quadratically constrained programs, continue to permit exact SDP-relaxations for a broad class of robust SOS-convex programs. They also yield exact second-order cone relaxation for robust quadratically constrained programs.

This paper presents exact dual semi-definite programs (SDPs) for robust SOS-convex polynomial optimization problems with affinely adjustable variables in the sense that the optimal values of the robust problem and its associated dual SDP are equal with the solution attainment of the dual problem. This class of robust convex optimization problems includes the corresponding quadratically constrained convex quadratic optimization problems and separable convex polynomial optimization problems, and it employs a general bounded spectrahedron uncertainty set that covers the most commonly used uncertainty sets of numerically solvable robust optimization models, such as boxes, balls and ellipsoids. As special cases, it also demonstrates that explicit exact dual SDP and second-order cone programming (SOCP) in terms of original data hold for the robust two-stage convex quadratic programs with quadratic constraints and the robust two-stage separable convex quadratic programs under an ellipsoidal uncertainty set, respectively. Finally, the paper illustrates the results via numerical implementations of the developed SDP duality scheme on adjustable robust lot-sizing problems with nonlinear costs under demand uncertainty.

This paper extends the conventional DEA models to a robust DEA (RDEA) framework by proposing new models for evaluating the efficiency of a set of homogeneous decision-making units (DMUs) under ellipsoidal uncertainty sets. Four main contributions are made: (1) we propose new RDEA models based on two uncertainty sets: an ellipsoidal set that models unbounded and correlated uncertainties and an interval-based ellipsoidal uncertainty set that models bounded and correlated uncertainties, and study the relationship between the RDEA models of these two sets, (2) we provide a robust classification scheme where DMUs can be classified into fully robust efficient, partially robust efficient and robust inefficient, (3) the proposed models are extended to the additive DEA model and its efficacy is analyzed with two imprecise additive DEA models in the literature, and finally, (4) we apply the proposed models to study the performance of banks in the Italian banking industry. We show that few banks which were resilient in their performance can be robustly classified as partially efficient or fully efficient in an uncertain environment.

The performance of adaptive beamforming degrades dramatically in the presence of steering vector uncertainties. In this paper, the robust adaptive beamforming with ellipsoidal steering vector uncertainty set is investigated. It belongs to the class of diagonal loading approaches. In order to select the loading level, the ellipsoidal uncertainty set is transformed into a spherical one first. Then the loading is determined based on worst-case performance optimization. Instead of being solved iteratively, the optimal loading is presented in closed-form here after some approximations. Compared with those iterative methods, the proposed one consumes less computational complexity and reveals how different factors affect the optimal loading. Numerical examples confirm the correctness and effectiveness of the proposed method.

A scenario-based robust distribution expansion planning under ellipsoidal uncertainty set using second-order cone programming