Distributionally robust optimization for collaborative emergency response network design

Distributionally robust facility location with uncertain facility capacity and customer demand

Distributionally robust optimization (DRO) has been introduced for solving stochastic programs in which the distribution of the random variables is unknown and must be estimated by samples from that distribution. A key element of DRO is the construction of the ambiguity set, which is a set of distributions that contains the true distribution with a high probability. Assuming that the true distribution has a probability density function, we propose a class of ambiguity sets based on confidence bands of the true density function. As examples, we consider the shape-restricted confidence bands and the confidence bands constructed with a kernel density estimation technique. The former allows us to incorporate the prior knowledge of the shape of the underlying density function (e.g., unimodality and monotonicity), and the latter enables us to handle multidimensional cases. Furthermore, we establish the convergence of the optimal value of DRO to that of the underlying stochastic program as the sample size increases. The DRO with our ambiguity set involves functional decision variables and infinitely many constraints. To address this challenge, we apply duality theory to reformulate the DRO to a finite-dimensional stochastic program, which is amenable to a stochastic subgradient scheme as a solution method.

Decomposition methods have been well studied for solving two-stage and multi-stage stochastic programming problems, see Rockafellar and Wets (Math. Oper. Res. 16:119–147, 1991), Ruszczynski and Shapiro (Stochastic Programming, Handbook in OR & MS, North-Holland Publishing Company, Amsterdam, 2003) and Ruszczynski (Math. Program. 79:333–353, 1997). In this paper, we propose an algorithmic framework based on the fundamental ideas of the methods for solving two-stage minimax distributionally robust optimization (DRO) problems where the underlying random variables take a finite number of distinct values. This is achieved by introducing nonanticipativity constraints for the first stage decision variables, rearranging the minimax problem through Lagrange decomposition and applying the well-known primal-dual hybrid gradient (PDHG) method to the new minimax problem. The algorithmic framework does not depend on specific structure of the ambiguity set. To extend the algorithm to the case that the underlying random variables are continuously distributed, we propose a discretization scheme and quantify the error arising from the discretization in terms of the optimal value and the optimal solutions when the ambiguity set is constructed through generalized prior moment conditions, the Kantorovich ball and $$\phi$$ -divergence centred at an empirical probability distribution. Some preliminary numerical tests show the proposed decomposition algorithm featured with parallel computing performs well.

Distributionally robust optimization for dispatching the integrated operation of rural multi-energy system and waste disposal facilities under uncertainties

With the large-scale displacement of fossil fuel-tired units due to the increasing penetration of renewable generation, the generation expansion planning (GEP) problem has been facing a major challenge to deal with the renewable power uncertainty. To handle this issue, this paper introduces a distributionally robust GEP model which explicitly considering the large-scale displacement of fossil fuel-fired units. In the first stage, the long-period generation expansion planning takes into account the expectation of the worst case; While, in the second stage, the short-term operation is modeled based on the distributionally robust optimization (DRO). The wind power uncertainties are treated as ambiguity sets using the Wasserstein distance technique. Aiming to ease the computational complexity of the proposed model, Benders decomposition with a reformulation technique to deal with the worst-case constraints is introduced. Case studies on a modified IEEE RTS-79 system verify the effectiveness of the proposed approach.

In this article, a novel distributionally robust optimization (DRO) based multistage model predictive control (MPC) framework is proposed to hedge against the uncertainty in control problems. Without a priori knowledge of the exact uncertainty distribution, an ambiguity set, constructed based on principal component analysis, incorporates the first-order moment information instead. A linear performance measure is chosen so that the worst-case expected problem can be exactly dualized by adopting the affine decision rule. By considering input and state constraints robustly with respect to a support set, which specifies the domain of the uncertainty, the DRO-based MPC model is developed as a robust optimization problem. The proposed framework is illustrated on a two-mass-spring system and results show the improved control performance compared to traditional control strategies.

TimeGAN based distributionally robust optimization for biomass-photovoltaic-hydrogen scheduling under source-load-market uncertainties

Randomized decision making refers to the process of making decisions randomly according to the outcome of an independent randomization device, such as a dice roll or a coin flip. The concept is unconventional, and somehow counterintuitive, in the domain of mathematical programming, in which deterministic decisions are usually sought even when the problem parameters are uncertain. However, it has recently been shown that using a randomized, rather than a deterministic, strategy in nonconvex distributionally robust optimization (DRO) problems can lead to improvements in their objective values. It is still unknown, though, what is the magnitude of improvement that can be attained through randomization or how to numerically find the optimal randomized strategy. In this paper, we study the value of randomization in mixed-integer DRO problems and show that it is bounded by the improvement achievable through its continuous relaxation. Furthermore, we identify conditions under which the bound is tight. We then develop algorithmic procedures, based on column generation, for solving both single- and two-stage linear DRO problems with randomization that can be used with both moment-based and Wasserstein ambiguity sets. Finally, we apply the proposed algorithm to solve three classical discrete DRO problems: the assignment problem, the uncapacitated facility location problem, and the capacitated facility location problem and report numerical results that show the quality of our bounds, the computational efficiency of the proposed solution method, and the magnitude of performance improvement achieved by randomized decisions. Summary of Contribution: In this paper, we present both theoretical results and algorithmic tools to identify optimal randomized strategies for discrete distributionally robust optimization (DRO) problems and evaluate the performance improvements that can be achieved when using them rather than classical deterministic strategies. On the theory side, we provide improvement bounds based on continuous relaxation and identify the conditions under which these bound are tight. On the algorithmic side, we propose a finitely convergent, two-layer, column-generation algorithm that iterates between identifying feasible solutions and finding extreme realizations of the uncertain parameter. The proposed algorithm was implemented to solve distributionally robust stochastic versions of three classical optimization problems and extensive numerical results are reported. The paper extends a previous, purely theoretical work of the first author on the idea of randomized strategies in nonconvex DRO problems by providing useful bounds and algorithms to solve this kind of problems.

Coordinated operations of electricity and district heating networks offer a potential for mitigating inherent variability of renewable energy sources (RES) in the ongoing transition to smart grids. This paper proposes a two-stage distributionally robust optimization (DRO) approach to determine the optimal day-ahead unit commitment in coordinated electricity and district heating networks with variable RES power output. The proposed formulation is to minimize the worst-case expected total cost over an ambiguity set comprising a family of probability distributions with given support and moments of RES power output. As such, the proposed DRO approach can overcome the limitations of stochastic programming in its inherent dependence of exact probability distributions along with a huge computational burden, but also becomes less conservative than classical robust optimization. The pertinent DRO model is eventually reformulated as a tractable mixed-integer second-order cone (SOC) programming after employing linear decision rules and the SOC duality. Simplified affine policies are utilized to further improve computational tractability and performance. Finally, case studies are conducted based on Barry Island electricity and district heating networks. The numerical results demonstrate the decision-making superiority of the proposed method as compared with deterministic, stochastic programming, and robust optimization approaches. They also validate the computational improvement of the proposed approach by employing simplified affine policies.

Reliable Facility Location Under Event-Correlated Uncertainty

Natural disasters often disrupt communication networks and severely hamper emergency response and disaster management. Existing solutions, such as portable communication units and cloud-based network architectures, have improved disaster resilience but fall short if both the Radio Access Network (RAN) and backhaul infrastructure become inoperable. To address these challenges, we propose a demand-driven communication system supported by High Altitude Platform Stations (HAPS) to restore communication in an affected area and enable effective disaster relief. The proposed emergency response network is a promising solution as it provides a rapidly deployable, resilient communications infrastructure. The proposed HAPS-based communication can play a crucial role not only in ensuring connectivity for mobile users but also in restoring backhaul connections when terrestrial networks fail. As a bridge between the disaster management center and the affected areas, it can facilitate the exchange of information in real time, collect data from the affected regions, and relay crucial updates to emergency responders. Enhancing situational awareness, coordination between relief agencies, and ensuring efficient resource allocation can significantly strengthen disaster response capabilities. In this article, simulations show that HAPS with hybrid optical/THz links boosts backhaul capacity and resilience, even in harsh conditions. HAPS-enabled RAN in S- and Ka-bands ensures reliable communication for first responders and disaster-affected populations. This article also explores the integration of HAPS into emergency communication frameworks and standards, as it has the potential to improve network resilience and support effective disaster management.

<p style='text-indent:20px;'>In this paper, we survey the primary research on the theory and applications of distributionally robust optimization (DRO). We start with reviewing the modeling power and computational attractiveness of DRO approaches, induced by the ambiguity sets structure and tractable robust counterpart reformulations. Next, we summarize the efficient solution methods, out-of-sample performance guarantee, and convergence analysis. Then, we illustrate some applications of DRO in machine learning and operations research, and finally, we discuss the future research directions.</p>

Power-to-gas (P2G) can convert excessive renewable energy into hydrogen via electrolysis, which can then be transported by natural gas systems to bypass constrained electricity systems. However, the injection of hydrogen could impact gas security since gas composition fundamentally changes, adversely effecting the combustion, safety and lifespan of appliances. This paper develops a new gas security management scheme for hydrogen injection into natural gas systems produced from excessive wind power. It introduces four gas security indices for the integrated electricity and gas system (IEGS) measuring gas security, considering the coordinated operation of tightly coupled infrastructures. To maintain gas security under an acceptable range, the gas mixture of nitrogen and liquid petroleum gas with hydrogen is adopted to address the gas security violation caused by hydrogen injection. A distributionally robust optimization (DRO) modelled by Kullback-Leibler (KL) divergence-based ambiguity set is applied to flexibly control the robustness to capture wind uncertainty. The KL divergence-based ambiguity set defines uncertainties within a measured space which limits the shape of probability distributions. Case studies illustrate that wind power is maximally utilized and gas mixture is effectively managed, thus improving gas security and performance of IEGS. This work can bring many benefits: i) ensured gas security under hydrogen injection ii) low system operation cost and iii) high renewable energy penetration. It can be easily extended to manage injections of other green gases into IEGS.

Distributionally Robust Optimization (DRO), which aims to find an optimal decision that minimizes the worst case cost over the ambiguity set of probability distribution, has been widely applied in diverse applications, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e.g.</i>, network behavior analysis, risk management, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">etc.</i> Nevertheless, prevailing DRO techniques encounter three primary challenges in distributed environments: 1) addressing asynchronous updating efficiently; 2) leveraging the prior distribution effectively; 3) appropriately adjusting the degree of robustness based on varying scenarios. To this end, we propose an asynchronous distributed algorithm, named <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</b>synchronous <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</b>ingle-loo<bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</b> alternat<bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I</b>ve g<bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</b>adient proj<bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</b>ction (ASPIRE) algorithm with the it<bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</b>rative <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</b>ctive <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</b><bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</b>t method (EASE) to tackle the federated distributionally robust optimization (FDRO) problem. In addition, a new uncertainty set, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i.e.</i>, constrained <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$D$</tex-math></inline-formula>-norm uncertainty set, is developed to effectively leverage the prior distribution and flexibly control the degree of robustness. We further enhance the proposed framework by integrating various uncertainty sets and conducting a comprehensive theoretical analysis of the computational complexity associated with each uncertainty set. To expedite convergence speed, we also introduce ASPIRE-ADP, a method that can dynamically adjust the number of active workers. Finally, our theoretical analysis elucidates that the proposed algorithm is guaranteed to converge and the iteration complexity and communication complexity are also analyzed. Extensive empirical studies on real-world datasets validate that the proposed method excels not only in achieving fast convergence and robustness against data heterogeneity and malicious attacks but also in effectively managing the trade-off between robustness and performance.

This paper considers facility location, inventory pre‐positioning and vehicle routing as strategic and operational decisions corresponding to preparedness and response phases in disaster relief logistics planning. For balancing surpluses and shortages, an effective lateral transshipment strategy is proposed to evenly distribute the relief resources between warehouses after the disaster occurs. To handle ambiguity in the probability distribution of demand, we develop a risk‐averse two‐stage distributionally robust optimization (DRO) model for the disaster relief logistics planning problem, which specifies the worst‐case mean‐conditional value‐at‐risk (CVaR) as a risk measure. For computationally tractability, we transform the robust counterpart into its equivalent linear mixed‐integer programming model under the discrepancy‐based ambiguity set centered at the nominal (empirical) distributions on the observed demand from the historical data. We verify the effectiveness of the proposed DRO model and the value of lateral transshipment strategy by an illustrative small‐scale example. The numerical results show that the proposed DRO model has advantage on avoiding over‐conservative solutions compared to the classic robust optimization model. We also illustrate the applicability of the proposed DRO model by a real‐world case study of hurricanes in the southeastern United States. The computational results demonstrate that the proposed DRO model has superior out‐of‐sample performance and can mitigate the adverse effects of Optimizers' Curse compared with the traditional stochastic programming model.