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.

scholarly journals Evaluation of Wind Energy Accommodation Based on Two-Stage Robust Optimization

2020 ◽  
Vol 26 (3) ◽  
pp. 61-68
Author(s):  
Kunpeng Tian ◽  
Weiqing Sun ◽  
Dong Han ◽  
Ce Yang
Keyword(s):  
Renewable Energy ◽  
Power System ◽  
Wind Energy ◽  
Robust Optimization ◽  
Optimization Problem ◽  
Mixed Integer ◽  
Master Problem ◽  
Two Stage ◽  
Energy Accommodation ◽  
Robust Optimization Problem

Large-scale renewable energy integration brings unprecedented challenges to electric power system planning and operation. The paper aims at economic dispatch and the safe operation of high penetration renewable energy power systems. According to the principle of power system dispatchability, the assessment of wind energy accommodation is formulated into a two-stage robust optimization problem with a min-max-min structure. Based on the benders algorithm, the intractable robust optimization problem is transformed into the form of sub-problem and master problem. Strong duality theory and big-M method are used to recast the sub problem into a mixed integer linear programming. The envelope of wind energy accommodation can be obtained by using commercial software to solve the master problem and sub problem alternately. For the realization of arbitrary wind power within the envelope, the amount of wind energy leakage and load shedding in power system operation are acceptable. An example of modified IEEE 39-bus test systems is used to verify the effectiveness and practicability of the evaluation method.

Toward the Use of Pareto Performance Solutions and Pareto Robustness Solutions for Multi-Objective Robust Optimization Problems

2012 ◽  
Author(s):  
Weijun Wang ◽  
Stéphane Caro ◽  
Fouad Bennis ◽  
Oscar Brito Augusto
Keyword(s):  
Robust Optimization ◽  
Pareto Front ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Multi Objective Optimization ◽  
Robust Solutions ◽  
Multi Objective ◽  
Robust Optimization Problem ◽  
Design Solutions ◽  
Further Development

For Multi-Objective Robust Optimization Problem (MOROP), it is important to obtain design solutions that are both optimal and robust. To find these solutions, usually, the designer need to set a threshold of the variation of Performance Functions (PFs) before optimization, or add the effects of uncertainties on the original PFs to generate a new Pareto robust front. In this paper, we divide a MOROP into two Multi-Objective Optimization Problems (MOOPs). One is the original MOOP, another one is that we take the Robustness Functions (RFs), robust counterparts of the original PFs, as optimization objectives. After solving these two MOOPs separately, two sets of solutions come out, namely the Pareto Performance Solutions (PP) and the Pareto Robustness Solutions (PR). Make a further development on these two sets, we can get two types of solutions, namely the Pareto Robustness Solutions among the Pareto Performance Solutions (PR(PP)), and the Pareto Performance Solutions among the Pareto Robustness Solutions (PP(PR)). Further more, the intersection of PR(PP) and PP(PR) can represent the intersection of PR and PP well. Then the designer can choose good solutions by comparing the results of PR(PP) and PP(PR). Thanks to this method, we can find out the optimal and robust solutions without setting the threshold of the variation of PFs nor losing the initial Pareto front. Finally, an illustrative example highlights the contributions of the paper.

scholarly journals Robust Optimization for Electricity Generation

2020 ◽  
Author(s):  
Haoxiang Yang ◽  
David P. Morton ◽  
Chaithanya Bandi ◽  
Krishnamurthy Dvijotham
Keyword(s):  
Renewable Energy ◽  
Robust Optimization ◽  
Electric Power ◽  
Power Flow ◽  
Optimal Power Flow ◽  
Optimization Problem ◽  
Convex Relaxation ◽  
Electric Power System ◽  
Robust Convex Optimization ◽  
Robust Optimization Problem

We consider a robust optimization problem in an electric power system under uncertain demand and availability of renewable energy resources. Solving the deterministic alternating current (AC) optimal power flow (ACOPF) problem has been considered challenging since the 1960s due to its nonconvexity. Linear approximation of the AC power flow system sees pervasive use, but does not guarantee a physically feasible system configuration. In recent years, various convex relaxation schemes for the ACOPF problem have been investigated, and under some assumptions, a physically feasible solution can be recovered. Based on these convex relaxations, we construct a robust convex optimization problem with recourse to solve for optimal controllable injections (fossil fuel, nuclear, etc.) in electric power systems under uncertainty (renewable energy generation, demand fluctuation, etc.). We propose a cutting-plane method to solve this robust optimization problem, and we establish convergence and other desirable properties. Experimental results indicate that our robust convex relaxation of the ACOPF problem can provide a tight lower bound.

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

Unsupervised Learning and Robust Optimization in the Portfolio Problem

2020 ◽  
pp. 4-9
Author(s):  
Grigorii I. Beliavsky ◽  
Natalia V. Danilova ◽  
Alexey D. Logunov
Keyword(s):  
Machine Learning ◽  
Decision Making ◽  
Robust Optimization ◽  
Unsupervised Learning ◽  
Optimization Problem ◽  
Clustering Algorithm ◽  
Learning Problems ◽  
Portfolio Problem ◽  
Robust Optimization Problem ◽  
Decision Making Model

Robust optimization in various problems of science and technology arises from the uncertainty of the parameters that determine the decision-making model. The parameters can be judged on the basis of a large number of observed examples. This challenge is to customize a solution based on a large number of examples that works well for examples that were not involved in customizing the solution. In this sense, the robust optimization problem belongs to machine learning problems. The paper uses a dichotomous clustering algorithm to determine the range of parameters for the optimal portfolio problem.

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