A transformation-proximal bundle algorithm for multistage adaptive robust optimization and application to constrained robust optimal control

This paper proposes a novel transformation-proximal bundle algorithmic framework to solve multistage adaptive robust optimization (ARO) problems. Different from existing solution methods, the proposed algorithmic framework partitions recourse decisions into state decisions and local decisions. It applies affine decision rule only to state decision variables and allows local decision variables to be fully adjustable. In this way, the original multistage ARO problem is proved to be transformed into a two-stage ARO problem. The proximal bundle algorithm with the Moreau- Yosida regularization is further developed for the exact solution of the resulting two-stage ARO problem. The transformation-proximal bundle algorithmic framework could generate less conservative solutions compared with the decision rule based approach, while enjoying a high computational efficiency. An application on multiperiod inventory control problem under demand uncertainty is presented to demonstrate the effectiveness and superiority of the proposed algorithm.

We develop a modular and tractable framework for solving an adaptive distributionally robust linear optimization problem, where we minimize the worst-case expected cost over an ambiguity set of probability distributions. The adaptive distributionally robust optimization framework caters for dynamic decision making, where decisions adapt to the uncertain outcomes as they unfold in stages. For tractability considerations, we focus on a class of second-order conic (SOC) representable ambiguity set, though our results can easily be extended to more general conic representations. We show that the adaptive distributionally robust linear optimization problem can be formulated as a classical robust optimization problem. To obtain a tractable formulation, we approximate the adaptive distributionally robust optimization problem using linear decision rule (LDR) techniques. More interestingly, by incorporating the primary and auxiliary random variables of the lifted ambiguity set in the LDR approximation, we can significantly improve the solutions, and for a class of adaptive distributionally robust optimization problems, exact solutions can also be obtained. Using the new LDR approximation, we can transform the distributionally adaptive robust optimization problem to a classical robust optimization problem with an SOC representable uncertainty set. Finally, to demonstrate the potential for solving management decision problems, we develop an algebraic modeling package and illustrate how it can be used to facilitate modeling and obtain high-quality solutions for medical appointment scheduling and inventory management problems. The electronic companion is available at https://doi.org/10.1287/mnsc.2017.2952 . This paper was accepted by Noah Gans, optimization.

Data-driven adaptive robust optimization for energy systems in ethylene plant under demand uncertainty

A machine learning approach to two-stage adaptive robust optimization

A novel two‐stage adaptive robust optimization (ARO) approach to production scheduling of batch processes under uncertainty is proposed. We first reformulate the deterministic mixed‐integer linear programming model of batch scheduling into a two‐stage optimization problem. Symmetric uncertainty sets are then introduced to confine the uncertain parameters, and budgets of uncertainty are used to adjust the degree of conservatism. We then apply both the Benders decomposition algorithm and the column‐and‐constraint generation (C&CG) algorithm to efficiently solve the resulting two‐stage ARO problem, which cannot be tackled directly by any existing optimization solvers. Two case studies are considered to demonstrate the applicability of the proposed modeling framework and solution algorithms. The results show that the C&CG algorithm is more computationally efficient than the Benders decomposition algorithm, and the proposed two‐stage ARO approach returns 9% higher profits than the conventional robust optimization approach for batch scheduling. © 2015 American Institute of Chemical Engineers AIChE J, 62: 687–703, 2016

Acceleration techniques for adaptive robust optimization transmission network expansion planning problems

Despite the modeling power for problems under uncertainty, robust optimization (RO) and adaptive RO (ARO) can exhibit too conservative solutions in terms of objective value degradation compared with the nominal case. One of the main reasons behind this conservatism is that, in many practical applications, uncertain constraints are directly designed as constraint-wise without taking into account couplings over multiple constraints. In this paper, we define a coupled uncertainty set as the intersection between a constraint-wise uncertainty set and a coupling set. We study the benefit of coupling in alleviating conservatism in RO and ARO. We provide theoretical tight and computable upper and lower bounds on the objective value improvement of RO and ARO problems under coupled uncertainty over constraint-wise uncertainty. In addition, we relate the power of adaptability over static solutions with the coupling of uncertainty set. Computational results demonstrate the benefit of coupling in applications. Funding: I. Wang was supported by the NSF CAREER Award [ECCS 2239771] and Wallace Memorial Honorific Fellowship from Princeton University. B. Stellato was supported by the NSF CAREER Award [ECCS 2239771].

This paper compares risk-averse optimization methods to address the self-scheduling and market involvement of a virtual power plant (VPP). The decision-making problem of the VPP involves uncertainty in the wind speed and electricity price forecast. We focus on two methods: risk-averse two-stage stochastic programming (SP) and two-stage adaptive robust optimization (ARO). We investigate both methods concerning formulations, uncertainty and risk, decomposition algorithms, and their computational performance. To quantify the risk in SP, we use the conditional value at risk (CVaR) because it can resemble a worst-case measure, which naturally links to ARO. We use two efficient implementations of the decomposition algorithms for SP and ARO; we assess (1) the operational results regarding first-stage decision variables, estimate of expected profit, and estimate of the CVaR of the profit and (2) their performance taking into consideration different sample sizes and risk management parameters. The results show that similar first-stage solutions are obtained depending on the risk parameterizations used in each formulation. Computationally, we identified three cases: (1) SP with a sample of 500 elements is competitive with ARO; (2) SP performance degrades comparing to the first case and ARO fails to converge in four out of five risk parameters; (3) SP fails to converge, whereas ARO converges in three out of five risk parameters. Overall, these performance cases depend on the combined effect of deterministic and uncertain data and risk parameters. Summary of Contribution: The work presented in this manuscript is at the intersection of operations research and computer science, which are intrinsically related with the scope and mission of IJOC. From the operations research perspective, two methodologies for optimization under uncertainty are studied: risk-averse stochastic programming and adaptive robust optimization. These methodologies are illustrated using an energy scheduling problem. The study includes a comparison from the point of view of uncertainty modeling, formulations, decomposition methods, and analysis of solutions. From the computer science perspective, a careful implementation of decomposition methods using parallelization techniques and a sample average approximation methodology was done . A detailed comparison of the computational performance of both methods is performed. Finally, the conclusions allow establishing links between two alternative methodologies in operations research: stochastic programming and robust optimization.

In this paper, we consider a variant of adaptive robust combinatorial optimization problems where the decision maker can prepare K solutions and choose the best among them upon knowledge of the true data realizations. We suppose that the uncertainty may affect the objective and the constraints through functions that are not necessarily linear. We propose a new exact algorithm for solving these problems when the feasible set of the nominal optimization problem does not contain too many good solutions. Our algorithm enumerates these good solutions, generates dynamically a set of scenarios from the uncertainty set, and assigns the solutions to the generated scenarios using a vertex p-center formulation, solved by a binary search algorithm. Our numerical results on adaptive shortest path and knapsack with conflicts problems show that our algorithm compares favorably with the methods proposed in the literature. We additionally propose a heuristic extension of our method to handle problems where it is prohibitive to enumerate all good solutions. This heuristic is shown to provide good solutions within a reasonable solution time limit on the adaptive knapsack with conflicts problem. Finally, we illustrate how our approach handles nonlinear functions on an all-or-nothing subset problem taken from the literature. Summary of Contribution: Our paper describes a new exact algorithm for solving adaptive robust combinatorial optimization problems when the feasible set of the nominal optimization problems does not contain too many good solutions. Its development relies on a progressive relaxation of the problem augmented with a row-and-column generation technique. Its efficient execution requires a reformulation of this progressive relaxation, coupled with dominance rules and a binary search algorithm. The proposed algorithm is amenable to exploiting the special structures of the problems considered as illustrated with various applications throughout the paper. A practical view is provided by the proposition of a heuristic variant. Our computational experiments show that our proposed exact solution method outperforms the existing methodologies and therefore pushes the computational envelope for the class of problems considered.

An efficient adaptive algorithm for robust control optimization using StoSAG

Optimal sizing of a reliability-constrained, stand-alone hybrid renewable energy system using robust satisficing

There are different uncertainties in microgrids’ (MG) operation such as output power of renewable energy sources (RESs), electricity price and load demand. Ignoring these existing uncertainties in the optimization problem imposes high cost to the system and the lack of reliability. This paper presents a general linear framework for microgrid optimization problem using robust optimization method. Adaptive robust optimization (ARO) model is a min-max-min problem in which the first level targets to determine the on/off status of dispatchable units, the second one aims find the worst case of uncertain parameters and eventually in third level the operational costs are minimized. This model is converted to a min-max one by using Karush-Kuhn-Tucker (KKT) conditions and then the ensuing model is linearized. A control parameter named budget of uncertainty is considered to determine the level of robustness and being conservative. The more budget of uncertainty we consider, the more robust model we obtain. An optimum point in which the expected cost is minimal and a compromise between the level of robustness and operational cost is reached. A modified IEEE-33 bus system is considered to evaluate the adequacy of proposed linear ARO model. Simulation results prove that the proposed ARO model is appropriately able to deal with the existing uncertainties and results in lower expected cost compared to deterministic model.

The distribution networks are experiencing important changes driven by the massive integration of renewable energy conversion systems. However, the lack of direct controllability of the Distributed Generations (DGs) supplying Active Distribution Networks (ADNs) represents a major obstacle to the increase of the penetration of renewable energy resources characterized by a non-negligible volatility. The successful development of ADNs depends on the combination of i) specific control tools and ii) availability of new technologies and controllable resources. Within this context, this thesis focuses on developing practical and scalable methodologies for the ADN planning and operation with particular reference to the integration of Energy Storage Systems (ESSs) owned, and directly controlled, by the Distribution Network Operators (DNOs). In this respect, an exact convex formulation of Optimal Power Flow (OPF), called AR-OPF, is first proposed for the case of radial power networks. The proposed formulation takes into account the correct model of the lines and the security constraints related to the nodal voltage magnitudes, as well as, the lines ampacity limits. Sufficient conditions are provided to guarantee that the solution of the AR-OPF is feasible and optimal (i.e., the relaxation used is exact). Moreover, by analyzing the exactness conditions, it is revealed that they are mild and hold for real distribution networks. The AR-OPF is further augmented by suitably incorporating radiality constraints in order to develop an optimization model for optimal reconfiguration of ADNs. Then, a two-stage optimization problem for day-ahead resource scheduling in ADNs, accounting for the uncertainties of nodal injections, is proposed. The Adaptive Robust Optimization (ARO) and stochastic optimization techniques are successfully adapted to solve this optimization problem. The solutions of ARO and stochastic optimization reveal that the ARO provides a feasible solution for any realization of the uncertain parameters even if its solution is optimal only for the worst case realization. On the other hand, the stochastic optimization provides a solution taking into account the probability of the considered scenarios. Finally, the problem of optimal resource planning in ADNs is investigated with particular reference to the ESSs. In this respect, the AR-OPF and the proposed ADN reconfiguration model, are employed to develop optimization models for the optimal siting and sizing of ESSs in ADNs. The objective function aims at finding the optimal trade-off between technical and economical goals. In particular, the proposed procedures accounts for (i) network voltage deviations, (ii) feeders/lines congestions, (iii) network losses, (iv) cost of supplying loads (from external grid or local producers) together with the cost of ESS investment/maintenance, (v) load curtailment and (vi) stochasticity of loads and renewables production. The use of decomposition methods for solving the targeted optimization problems with discrete variables and probable large size is investigated. More specifically, Benders decomposition and Alternative Direction Method of Multipliers (ADMM) techniques are successfully applied to the targeted problems. Using real and standard networks, it is shown that the ESSs could possibly prevent load and generation curtailment, reduce the voltage deviations and lines congestions, and do the peak shaving.

A robust optimization approach to energy and reserve dispatch in electricity markets

Purpose:Accuracy of dose calculation models and robustness under various uncertainties are key factors influencing the quality of intensity modulated proton therapy (IMPT) plans. In this work, a robust IMPT optimization based on accurate Monte Carlo (MC) dose calculation is developed.Methods:We used an in‐house developed and graphics processing unit (GPU) accelerated MC for dose calculation. For robust optimization, dose volume histograms (DVHs) were computed for each uncertainty scenario at each optimization iteration. A gradient based adaptive method was used to improve the DVHs with adjustable scenario weights. GPUs were employed to accelerate the optimization process. Uncertainties in patient setup and proton range were considered in all cases studied. Additionally, the uncertainty of intra‐fraction relative shift between fields was considered for craniospinal irradiation cases. The adaptive robust optimization method was compared with for clinical cases at several different disease sites.Results:Comparing with the traditional optimization target volume (OTV) based method, the adaptive robust optimization spared critical structures better while maintain the target coverage in clinical cases. For example, the right parotid hot spot dose was reduced from 78.5Gy to 74.5Gy as shown in Fig. 1. For craniospinal irradiation, the adaptive method found the robust solution at field junctions without manual feathering of the match lines. Even for relatively large head‐and‐neck cases and craniospinal cases, the whole process of MC dose calculation and robust optimization can be done within 30 minutes on a system of 100 Nvidia GeForce GTX Titan cards.Conclusion:A robust IMPT treatment planning system is developed utilizing an adaptive method. The treatment planning optimization is based on MC dose calculation and is accelerated by GPU to be clinically viable.This work is supported in part by Varian Medical Systems.