Adjustable Robust Optimization Reformulations of Two-Stage Worst-Case Regret Minimization Problems

2021 ◽  
Author(s):  
Mehran Poursoltani ◽  
Erick Delage
Keyword(s):  
Robust Optimization ◽  
Decision Rules ◽  
Case Scenario ◽  
Two Stage ◽  
Quality Performance ◽  
Worst Case ◽  
Regret Minimization ◽  
Minimization Problems ◽  
Adjustable Robust Optimization ◽  
Optimization Reformulations

Although the stochastic optimization paradigm exploits probability theory to optimize the tradeoff between risk and returns, robust optimization has gained significant popularity by reducing computation requirements through the optimization of the worst-case scenario in a set. An appealing alternative to stochastic and robust optimization consists in optimizing decisions using the notion of regret. Although regret minimization models are generally perceived as leading to less conservative decisions than those produced by robust optimization, their numerical optimization is a real challenge in general. In “Adjustable Robust Optimization Reformulations of Two-Stage Worst-case Regret Minimization Problems,” M. Poursoltani and E. Delage show how to reduce a two-stage worst-case absolute/relative regret minimization problem to a two-stage robust optimization one. This opens the way for taking advantage of recent advanced approximate and exact solution schemes for these hard problems. Their experiments corroborate the high-quality performance of affine decision rules as a popular polynomial-time approximation scheme, from which, under mild conditions, one can even expect exact regret-averse decisions.

scholarly journals Adjustable Robust Optimization Algorithm for Residential Microgrid Multi-Dispatch Strategy with Consideration of Wind Power and Electric Vehicles

Energies ◽  
2018 ◽  
Vol 11 (8) ◽  
pp. 2050 ◽  
Author(s):  
Ruifeng Shi ◽  
Shaopeng Li ◽  
Changhao Sun ◽  
Kwang Lee
Keyword(s):  
Robust Optimization ◽  
Wind Power ◽  
Distributed Generation ◽  
Electric Vehicles ◽  
Case Scenario ◽  
Worst Case ◽  
Adjustable Robust Optimization ◽  
Renewable Energy Generation ◽  
The Relationship ◽  
Ev Charging

A prospect of increasing penetration of uncoordinated electric vehicles (EVs) together with intermittent renewable energy generation in microgrid systems has motivated us to explore an effective strategy for safe and economic operation of such distributed generation systems. This paper presents a robust economic dispatch strategy for grid-connected microgrids. Uncertainty from wind power and EV charging loads is modeled as an uncertain set of interval predictions. Considering the worst case scenario, the proposed strategy can help to regulate the EV charging behaviors, and distributed generation in order to reduce operation cost under practical constraints. To address the issue of over-conservatism of robust optimization, a dispatch interval coefficient is introduced to adjust the level of robustness with probabilistic bounds on constraints, which gradually improves the system's economic efficiency. In addition, in order to facilitate the decision-making strategies from an economic perspective, this paper explores the relationship between the volatility of uncertain parameters and the economy based on the theory of interval forecast. Numerical case studies demonstrate the feasibility and robustness of the proposed dispatch strategy.

Risk-Based Robust Bidding Strategies for EVs’ Aggregators in Day-ahead Markets with Uncertainty

2020 ◽  
Author(s):  
Ahmed Abdelmoaty ◽  
Wessam Mesbah ◽  
Mohammad A. M. Abdel-Aal ◽  
Ali T. Alawami
Keyword(s):  
Robust Optimization ◽  
Electricity Market ◽  
Real Life ◽  
Optimization Approach ◽  
Case Scenario ◽  
Worst Case ◽  
Bidding Strategies ◽  
Wind Generators ◽  
Proposed Model ◽  
Optimal Bidding

In the recent electricity market framework, the profit of the generation companies depends on the decision of the operator on the schedule of its units, the energy price, and the optimal bidding strategies. Due to the expanded integration of uncertain renewable generators which is highly intermittent such as wind plants, the coordination with other facilities to mitigate the risks of imbalances is mandatory. Accordingly, coordination of wind generators with the evolutionary Electric Vehicles (EVs) is expected to boost the performance of the grid. In this paper, we propose a robust optimization approach for the coordination between the wind-thermal generators and the EVs in a virtual<br>power plant (VPP) environment. The objective of maximizing the profit of the VPP Operator (VPPO) is studied. The optimal bidding strategy of the VPPO in the day-ahead market under uncertainties of wind power, energy<br>prices, imbalance prices, and demand is obtained for the worst case scenario. A case study is conducted to assess the e?effectiveness of the proposed model in terms of the VPPO's profit. A comparison between the proposed model and the scenario-based optimization was introduced. Our results confirmed that, although the conservative behavior of the worst-case robust optimization model, it helps the decision maker from the fluctuations of the uncertain parameters involved in the production and bidding processes. In addition, robust optimization is a more tractable problem and does not suffer from<br>the high computation burden associated with scenario-based stochastic programming. This makes it more practical for real-life scenarios.<br>

scholarly journals On the Optimality of Affine Policies for Budgeted Uncertainty Sets

2021 ◽  
Author(s):  
Omar El Housni ◽  
Vineet Goyal
Keyword(s):  
Robust Optimization ◽  
Structural Properties ◽  
Optimization Problem ◽  
Hardness Of Approximation ◽  
Uncertain Parameters ◽  
Two Stage ◽  
Uncertainty Sets ◽  
Adjustable Robust Optimization ◽  
Robust Optimization Problem ◽  
Budgeted Uncertainty

In this paper, we study the performance of affine policies for a two-stage, adjustable, robust optimization problem with a fixed recourse and an uncertain right-hand side belonging to a budgeted uncertainty set. This is an important class of uncertainty sets, widely used in practice, in which we can specify a budget on the adversarial deviations of the uncertain parameters from the nominal values to adjust the level of conservatism. The two-stage adjustable robust optimization problem is hard to approximate within a factor better than [Formula: see text] even for budget of uncertainty sets in which [Formula: see text] is the number of decision variables. Affine policies, in which the second-stage decisions are constrained to be an affine function of the uncertain parameters provide a tractable approximation for the problem and have been observed to exhibit good empirical performance. We show that affine policies give an [Formula: see text]-approximation for the two-stage, adjustable, robust problem with fixed nonnegative recourse for budgeted uncertainty sets. This matches the hardness of approximation, and therefore, surprisingly, affine policies provide an optimal approximation for the problem (up to a constant factor). We also show strong theoretical performance bounds for affine policy for a significantly more general class of intersection of budgeted sets, including disjoint constrained budgeted sets, permutation invariant sets, and general intersection of budgeted sets. Our analysis relies on showing the existence of a near-optimal, feasible affine policy that satisfies certain nice structural properties. Based on these structural properties, we also present an alternate algorithm to compute a near-optimal affine solution that is significantly faster than computing the optimal affine policy by solving a large linear program.

Adjustable robust optimization approach for two-stage operation of energy hub-based microgrids

Energy ◽  
2021 ◽  
Vol 222 ◽  
pp. 119894
Author(s):  
Mohammad H. Shams ◽  
Majid Shahabi ◽  
Mohammad MansourLakouraj ◽  
Miadreza Shafie-khah ◽  
João P.S. Catalão
Keyword(s):  
Robust Optimization ◽  
Optimization Approach ◽  
Two Stage ◽  
Energy Hub ◽  
Adjustable Robust Optimization

scholarly journals Optimal Transport-Based Distributionally Robust Optimization: Structural Properties and Iterative Schemes

2021 ◽  
Author(s):  
Jose Blanchet ◽  
Karthyek Murthy ◽  
Fan Zhang
Keyword(s):  
Robust Optimization ◽  
Optimal Transport ◽  
Decision Rules ◽  
Stochastic Gradient Descent ◽  
Transport Cost ◽  
Distributionally Robust Optimization ◽  
Worst Case ◽  
Strongly Convex ◽  
Distributionally Robust ◽  
Convexity Assumptions

We consider optimal transport-based distributionally robust optimization (DRO) problems with locally strongly convex transport cost functions and affine decision rules. Under conventional convexity assumptions on the underlying loss function, we obtain structural results about the value function, the optimal policy, and the worst-case optimal transport adversarial model. These results expose a rich structure embedded in the DRO problem (e.g., strong convexity even if the non-DRO problem is not strongly convex, a suitable scaling of the Lagrangian for the DRO constraint, etc., which are crucial for the design of efficient algorithms). As a consequence of these results, one can develop efficient optimization procedures that have the same sample and iteration complexity as a natural non-DRO benchmark algorithm, such as stochastic gradient descent.

Technical Note—Two-Stage Sample Robust Optimization

2021 ◽  
Author(s):  
Dimitris Bertsimas ◽  
Shimrit Shtern ◽  
Bradley Sturt
Keyword(s):  
Robust Optimization ◽  
Approximation Scheme ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Decision Rules ◽  
Technical Note ◽  
Data Driven ◽  
Two Stage ◽  
Linear Optimization Problems ◽  
Stochastic Linear Optimization

In “Two-Stage Sample Robust Optimization,” Bertsimas, Shtern, and Sturt investigate a simple approximation scheme, based on overlapping linear decision rules, for solving data-driven two-stage distributionally robust optimization problems with the type-infinity Wasserstein ambiguity set. Their main result establishes that this approximation scheme is asymptotically optimal for two-stage stochastic linear optimization problems; that is, under mild assumptions, the optimal cost and optimal first-stage decisions obtained by approximating the robust optimization problem converge to those of the underlying stochastic problem as the number of data points grows to infinity. These guarantees notably apply to two-stage stochastic problems that do not have relatively complete recourse, which arise frequently in applications. In this context, the authors show through numerical experiments that the approximation scheme is practically tractable and produces decisions that significantly outperform those obtained from state-of-the-art data-driven alternatives.

One-stage procedure to establish osseointegration: a zero to five years follow-up report

1995 ◽  
Vol 109 (7) ◽  
pp. 593-598 ◽  
Author(s):  
Anders Tjellström ◽  
Gösta Granström
Keyword(s):  
Success Rate ◽  
Hearing Aids ◽  
Mastoid Process ◽  
Case Scenario ◽  
Two Stage ◽  
Worst Case ◽  
One Stage ◽  
The One ◽  
Lost To Follow Up

AbstractA cohort of 214 patients, who were operated on to insert implants in the mastoid process for the retention of bone-anchored hearing aids and auricular prostheses, was followed-up over a five-year period. About half the group were operated on using the conventional two-stage procedure allowing three to four months for osseointegration. In the second group (one-stage group) the skin penetrating coupling was connected at the time of the implant insertion. The success rate for stable implants was found to be the same in both groups. In the one-stage group four out of 161 implants inserted were lost and in the two stage group three out of 120. The cumulative success rate was also found to be the same. A ‘worst case’ table where patients lost to follow-up, patients who died during the study period, and patients who for some reason left the study is also included. The importance of this ‘worst case’ scenario when follow-up data are presented is discussed.

Person-Based Optimization of Signal Timing

2017 ◽  
Vol 2620 (1) ◽  
pp. 31-42 ◽  
Author(s):  
Zhengyao Yu ◽  
Vikash V. Gayah ◽  
Eleni Christofa
Keyword(s):  
Robust Optimization ◽  
Signal Timing ◽  
Optimization Approach ◽  
Optimization Strategy ◽  
Passenger Cars ◽  
Case Scenario ◽  
Worst Case ◽  
Arrival Times ◽  
Cycle Lengths ◽  
Traffic Signal Timing

Recent studies have proposed the use of person-based frameworks for the optimization of traffic signal timing to minimize the total passenger delay experienced by passenger cars and buses at signalized intersections. The efficiency and applicability of existing efforts, however, have been limited by an assumption of fixed cycle lengths and deterministic bus arrival times. An existing algorithm for person-based optimization of signal timing for isolated intersections was extended to accommodate flexible cycle lengths and uncertain bus arrival times. To accommodate flexible cycle lengths, the mathematical program was redefined to minimize total passenger delay within a fixed planning horizon that allowed cycle lengths to vary within a feasible range. Two strategies were proposed to accommodate uncertain bus arrival times: ( a) a robust optimization approach that conservatively minimized delays experienced in a worst-case scenario and ( b) a blended strategy that combined deterministic optimization and rule-based green extensions. The proposed strategies were tested with numerical simulations of an intersection in State College, Pennsylvania. Results revealed that the flexible cycle length algorithm could significantly reduce bus passenger delay and total passenger delay, with negligible increases in car passenger delay. These results were robust to changes in both bus and car flows. For bus arrival times, the robust optimization strategy seemed to be more effective at low levels of uncertainty and the blended strategy at higher levels of uncertainty. The anticipated benefits decreased with increases in the intersection flow ratio because of the lower flexibility of signal timing at the intersection.

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