Robust Optimization Problem Research Articles

A Robust Networking Model With Quantum Evolution for Internet of Things

The Internet of Things (IoT), which includes massive energy-limited sensor nodes, has become the foundation of smart city. Improving the ability of network topology to resist node cascading failures, namely robustness, is the key for IoT to provide stable data-aware services for upper-layer applications. However, the robustness optimization problem of complex topology is an NP-hard problem and cannot be optimally solved in polynomial time. The existing researches try to find the approximate solution by heuristic algorithms, but there are problems of slow convergence and easy to fall into local optimum. Quantum computing has more diverse search spaces due to the existence of quantum superposition states, which can jump out of the local optimum. Therefore, this paper firstly combines quantum computing with topology robustness evolution, and proposes a robust networking model based on quantum evolution. By using the quantum encoding, we design a novel quantum measurement method to collapse quantum states towards a more robust network topology. The experimental results show that our model can jump out of the local optimum with fewer population individuals and achieve higher robustness.

A unified approach to inverse robust optimization problems

A variety of approaches has been developed to deal with uncertain optimization problems. Often, they start with a given set of uncertainties and then try to minimize the influence of these uncertainties. The reverse view is to first set a budget for the price one is willing to pay and then find the most robust solution. In this article, we aim to unify these inverse approaches to robustness. We provide a general problem definition and a proof of the existence of its solution. We study properties of this solution such as closedness, convexity, and boundedness. We also provide a comparison with existing robustness concepts such as the stability radius, the resilience radius, and the robust feasibility radius. We show that the general definition unifies these approaches. We conclude with an example that demonstrates the flexibility of the introduced concept.

Data‐driven robust look‐ahead power dispatch with preidentification of the worst‐case scenario using combined multilayer perceptron

AbstractTo address the uncertainty caused by renewable energy generation, two‐stage robust optimization (RO) has been proposed for intraday look‐ahead power dispatch (LAPD) at multiple time scales. However, the intensive computational burden of solving RO problems limits its online application. To address this issue, we developed a novel data‐driven robust LAPD method that exploits information transfer in the worst‐case scenario (WCS) involving uncertain parameters between the two stages of RO. By employing the multilayer perceptron (MLP), the WCSs of uncertain parameters are tentatively preidentified as preliminary inputs of the column‐and‐constraint generation (C&CG) algorithm to reduce the stress of the WCS search during the subsequent iterative procedure. In addition, a combined multilayer perceptron is proposed to address correlations between uncertain parameters in the WCS. The MLP model is trained in distinct time segments to reduce the model training complexity and improve the precision of WCS preidentification. Case studies with various system scales and rolling window sizes indicate that the proposed method is effective at accelerating the C&CG process.

Robust design optimization of a multi-body system with aleatory and epistemic uncertainty

There are several issues in robust design optimization of the multi-body system with aleatory and epistemic uncertainty, such as the difficulty in the multi-source uncertainty characterization and low efficiency in dynamic modeling, analysis and the corresponding solving strategy for its robust design optimization. To address these problems, this paper firstly constructs a unified analysis framework for aleatory and epistemic uncertainty for the multi-body system based on probability theory and evidence theory. Taking a multi-body system of a shipborne artillery model as an example, the transfer matrix method is next employed to rapidly calculate the dynamic response of the muzzle. The results are compared with those by the finite element method to verify the accuracy and efficiency of the transfer matrix method. A decision model of the robust design optimization with both aleatory and epistemic uncertainty is proposed. Based on the polynomial chaos expansion-Kriging model, subset simulation optimization method and a weighted single-objective solving strategy, the multi-objective robust design optimization problem involving aleatory and epistemic uncertainty is then transformed into a single-objective optimization one. Finally, results of both the multi-objective and weighted single-objective robust design optimization methods for the shipborne artillery model are compared to demonstrate the performance of the proposed method.

Reliable Frequency Regulation Through Vehicle-to-Grid: Encoding Legislation with Robust Constraints

Problem definition: Vehicle-to-grid increases the low utilization rate of privately owned electric vehicles by making their batteries available to electricity grids. We formulate a robust optimization problem that maximizes a vehicle owner’s expected profit from selling primary frequency regulation to the grid and guarantees that market commitments are met at all times for all frequency deviation trajectories in a functional uncertainty set that encodes applicable legislation. Faithfully modeling the energy conversion losses during battery charging and discharging renders this optimization problem nonconvex. Methodology/results: By exploiting a total unimodularity property of the uncertainty set and an exact linear decision rule reformulation, we prove that this nonconvex robust optimization problem with functional uncertainties is equivalent to a tractable linear program. Through extensive numerical experiments using real-world data, we quantify the economic value of vehicle-to-grid and elucidate the financial incentives of vehicle owners, aggregators, equipment manufacturers, and regulators. Managerial implications: We find that the prevailing penalties for nondelivery of promised regulation power are too low to incentivize vehicle owners to honor the delivery guarantees given to grid operators. Funding: This work was supported by the Institut Vedecom. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0154 .

Designing tractable piecewise affine policies for multi-stage adjustable robust optimization

AbstractWe study piecewise affine policies for multi-stage adjustable robust optimization (ARO) problems with non-negative right-hand side uncertainty. First, we construct new dominating uncertainty sets and show how a multi-stage ARO problem can be solved efficiently with a linear program when uncertainty is replaced by these new sets. We then demonstrate how solutions for this alternative problem can be transformed into solutions for the original problem. By carefully choosing the dominating sets, we prove strong approximation bounds for our policies and extend many previously best-known bounds for the two-staged problem variant to its multi-stage counterpart. Moreover, the new bounds are—to the best of our knowledge—the first bounds shown for the general multi-stage ARO problem considered. We extensively compare our policies to other policies from the literature and prove relative performance guarantees. In two numerical experiments, we identify beneficial and disadvantageous properties for different policies and present effective adjustments to tackle the most critical disadvantages of our policies. Overall, the experiments show that our piecewise affine policies can be computed by orders of magnitude faster than affine policies, while often yielding comparable or even better results.

A Direct Yaw Moment Control Framework Through Robust T-S Fuzzy Approach Considering Vehicle Stability Margin

The direct yaw moment control system of distributed drive electric vehicles provides a flexible control scheme to enhance the vehicle stability through four independently driven in-wheel motors. However, the vehicle nonlinearities may lead to the invalidation of control. To this end, a Takagi–Sugeno (T-S) fuzzy-based robust <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H∞</i> control method is proposed to ensure the vehicle performance while addressing the nonlinear challenge. First, thanks to the T-S fuzzy modeling technology, the tire nonlinear characteristics are described by fuzzy rules, based on which the vehicle lateral dynamics model is established. Next, the safety region represented by the tire slip angles phase plane is presented to evaluate the vehicle stability performance. Considering the control priority of vehicle handling performance and stability control with different stability margins, a multiobjective optimization function is transformed into a standard robust performance optimization problem with dynamic weight coefficients. A T-S fuzzy-based robust <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H∞</i> state feedback controller is then designed to ensure the system stability and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H∞</i> performance. Finally, the hardware-in-the-loop tests are conducted to validate the proposed controller. Comparative results show the effectiveness to improve the vehicle handling performance while ensuring the stability.

Sufficient optimality, duality and saddle point analysis in terms of convexificators for nonsmooth robust fractional interval-valued optimization problems

In this study, sufficient optimality criteria for a robust fractional interval-valued optimization problem (RP) is highlighted based on the idea of convexificators. The use of generalized invexity tools to support sufficient optimality conditions is demonstrated by an example. Also, saddle point condition of a Lagrange function is examined for RP. Furthermore, Mond–Weir-type robust dual model is constructed and by employing the concept of generalized invexity, usual duality results are discussed.

Adjustability in robust linear optimization

AbstractWe investigate the concept of adjustability—the difference in objective values between two types of dynamic robust optimization formulations: one where (static) decisions are made before uncertainty realization, and one where uncertainty is resolved before (adjustable) decisions. This difference reflects the value of information and decision timing in optimization under uncertainty, and is related to several other concepts such as the optimality of decision rules in robust optimization. We develop a theoretical framework to quantify adjustability based on the input data of a robust optimization problem with a linear objective, linear constraints, and fixed recourse. We make very few additional assumptions. In particular, we do not assume constraint-wise separability or parameter nonnegativity that are commonly imposed in the literature for the study of adjustability. This allows us to study important but previously under-investigated problems, such as formulations with equality constraints and problems with both upper and lower bound constraints. Based on the discovery of an interesting connection between the reformulations of the static and fully adjustable problems, our analysis gives a necessary and sufficient condition—in the form of a theorem-of-the-alternatives—for adjustability to be zero when the uncertainty set is polyhedral. Based on this sharp characterization, we provide two efficient mixed-integer optimization formulations to verify zero adjustability. Then, we develop a constructive approach to quantify adjustability when the uncertainty set is general, which results in an efficient and tight poly-time algorithm to bound adjustability. We demonstrate the efficiency and tightness via both theoretical and numerical analyses.

Active-learning Kriging-assisted robust design optimization of tuned mass dampers: Vibration mitigation of a steel-arch footbridge

Many typical footbridges are designed as slender structures to highlight their aesthetic characteristic. As these bridges can be highly susceptible to pedestrian-induced excitations, tuned mass dampers have been adopted as energy absorbers in many applications to mitigate this issue. The design of tuned mass dampers can be formulated as a robust design optimization problem considering various sources of uncertainties related to the bridge state and environment. To alleviate the computational burden of analyzing the statistical moments, surrogate model-based approaches have been introduced to build cheaper-to-evaluate substitutes, and the analytical expression of the robustness has been discussed to facilitate the optimization procedure. Motivated by this, the present paper derives an analytical expression for the robustness index, considering both deterministic and uncertain variables as inputs of the ordinary Kriging surrogate model. An active-learning framework is proposed to streamline the design optimization of tuned mass dampers for the vibration mitigation of slender footbridges. Two numerical examples are firstly investigated to validate the performance of the proposed method; and then, a practical application of a steel-arch footbridge is carefully studied to demonstrate the engineering feasibility and superiority of the proposed method. The results exhibit that the proposed method can efficiently and accurately facilitate the robust design of vibration mitigation devices for practical engineering projects, and showcases the potential to be used in other engineering problems.

Secure state estimation via robust optimization for nonlinear cyber‐physical systems

AbstractThis article proposes secure state estimation for cyber‐physical systems against sensor attacks. The attack and defense strategies are established via additional historical data, and the defender aims to reduce the estimation error maximally while the attacker aims to degrade the system performance maximally. The algorithm is implemented in the Nash equilibrium framework where the defender first designs the defense strategy and then the attacker designs corresponding attack parameters to launch attacks. Then, a robust optimization problem is formulated using Wasserstein ambiguity sets, which turn out to be equivalent to a convex program. A novel secure observer is proposed, where the attack estimation is used to mitigate attacks. Moreover, the detector is to monitor system behavior and detects the existence of sensor attacks. Finally, simulation results and comparative results illustrate the effectiveness of the defense strategy.

On semidefinite programming relaxations for a class of robust SOS-convex polynomial optimization problems

On semidefinite programming relaxations for a class of robust SOS-convex polynomial optimization problems

A Two-Step Approach to Wasserstein Distributionally Robust Chance- and Security-Constrained Dispatch

This paper considers a security constrained dispatch problem involving generation and line contingencies in the presence of the renewable generation. The uncertainty due to renewables is modeled using joint chance-constraint and the mismatch caused by contingencies and renewables are handled using reserves. We consider a distributionally robust approach to solve the chance-constrained program. We assume that samples of the uncertainty are available. Using them, we construct a set of distributions, termed ambiguity set, containing all distributions that are close to the empirical distribution under the Wasserstein metric. The chance constraint is imposed for all distributions in the ambiguity set to form the distributionally robust optimization problem. This problem is nonconvex and computationally heavy to solve exactly. We adopt a two-step approach to find an approximate solution. In the first step, we construct a polyhedral set in the space of uncertainty that contains enough mass under all distributions in the ambiguity set. This set is constructed by solving several two-dimensional distributionally robust problems. In the second step, we solve a linear robust optimization problem where the uncertain constraint is imposed for all uncertainty values lying in the polyhedral set. We demonstrate the scalability and robustness of our method using numerical experiments.

Value of Resilience for Self-Sustained Highway Transportation Systems

Nowadays, self-sustained highway transportation systems (HTSs) are calling for the inter-operation of road networks, distribution networks (DNs), and electric vehicles (EVs) towards the unexpected impacts inducted by extreme weather events (EWEs). A novel joint resilience assessment scheme is proposed to quantify the resilience of self-sustained HTSs under tropical cyclone (TC) events. The impacts of TCs on transmission lines and highways are formulated using a two-step ambiguity set. In the first step, a multitask learning (MTL) based model with a novel training strategy is proposed to simulate the future TC tracks; the impacts of TCs are then calculated in the second step based on the simulation results. Using the ambiguity set, the self-sustained HTSs resilience assessment problem is formulated based on the dynamic optimal traffic assignment problem, where EVs are integrated as third-party emergency resources before the advent of TCs. This scheme is then formulated as a two-stage distributionally robust optimization (DRO) problem and reformulated as a mixed-integer linear programming (MILP) problem. Case studies have been conducted on a self-sustained HTS consisting of a modified IEEE-14 bus test system with 2 wind farms (WFs) and a 14-node transportation network with 4 charging stations (CSs). Numerical results indicate that the proposed scheme can quantify the resilience of self-sustained highway transportation systems under the joint operation of EVs regarding the unmet travel demand and load-shedding costs. In comparison to the statistical TC forecasting models, the proposed MTL-based model is more accurate and can reduce the system cost in most cases.

Relaxed Total Fenchel-Lagrange Duality and Optimality Conditions for the Robust Composite Convex Optimization Problem

Relaxed Total Fenchel-Lagrange Duality and Optimality Conditions for the Robust Composite Convex Optimization Problem

Valuation of distributed predictive information in robust economic dispatch

Robust economic dispatch is an essential way to deal with the uncertainties of renewable generations, but its performance highly depends on the pre-built uncertainty set. This paper proposes a novel robust economic dispatch model in which the operator can buy predictions from distributed forecasters to build a better uncertainty set and enhance its dispatch efficiency. The value of distributed predictive information can be quantified by the operator’s payment for buying it. The proposed model renders a two-stage robust optimisation problem with decision-dependent uncertainty (DDU). By analysing the structure of the uncertainty set, the proposed model is equivalently transformed into a two-stage robust optimisation problem with decision-independent uncertainty (DIU) so that it can be effectively solved by applying the column-and-constraint generation (C&amp;CG) algorithm. Case studies show that the proposed method is effective.

Probabilistic Forecasting-Based Reserve Determination Considering Multi-Temporal Uncertainty of Renewable Energy Generation

Operating reserve, among the most important ancillary services, is a powerful prescription for mitigating increasing uncertainty of renewable generation. Traditional reserve determination neglects uncertainty of renewable generation variations within the dispatch interval, which cannot gurantee sufficient reserve deliverability in real-time operation. This paper proposes a probabilistic forecasting-based reserve determination method to efficiently deal with multi-temporal uncertainty of renewable energy in power systems. A novel probabilistic forecasting approach is proposed by integrating Dirichlet process Gaussian mixture model into bootstrap-based extreme learning machine. A unified uncertainty model is constructed for depicting both interval-averaged uncertainty and intra-interval uncertainty by combing the probabilistic forecasts and linearized Itô-process model. In current business practices of many independent system operators, the coordinated determination of two well-designed reserve products, including regulating reserve and ramp capability reserve, is formulated in a two-stage robust optimization framework. A Bernstein polynomial-based model reformulation approach is then employed to handle the existing heterogeneity in the sub-models of each stage. Consequently, the integrated reserve determination is embedded in a generic two-stage robust optimization problem, which can be efficiently solved by the adopted modified column-and-constraint generation method. Finally, numerical simulations are implemented to validate the effectiveness and profitability of the proposed approach.

Using Gaussian Processes for Metamodeling in Robust Optimization Problems

&#x0D; This article proposes an approach based on Gaussian Processes for building metamodels for robust optimization problems that seek to reduce the computational effort required to quantify uncertainties. The approach is applied to two cases: a low-dimensional benchmark problem and a high-dimensional structural design, which consists of minimizing the mass of a structure formed by bars of different materials and diameters, subjected to point loads in different locations. The cases are modeled as robust optimization problems, where the objective function is estimated by a Gaussian Process and the optimization procedure uses a population meta-heuristic. The results indicate that the proposed approach is effective in reducing the number of objective function evaluations required to obtain a robust solution, with no significant statistical differences in the quality of solutions achieved.&#x0D;

On constraint qualifications and optimality conditions for robust optimization problems through pseudo-differential

On constraint qualifications and optimality conditions for robust optimization problems through pseudo-differential

Code and Data Repository for A Decision Rule Approach for Two-Stage Data-Driven Distributionally Robust Optimization Problems with Random Recourse

Code and Data Repository for A Decision Rule Approach for Two-Stage Data-Driven Distributionally Robust Optimization Problems with Random Recourse