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In this article, a risk-aware motion control scheme is considered for mobile robots to avoid randomly moving obstacles when the true probability distribution of uncertainty is unknown. We propose a novel model-predictive control (MPC) method for limiting the risk of unsafety even when the true distribution of the obstacles’ movements deviates, within an ambiguity set, from the empirical distribution obtained using a limited amount of sample data. By choosing the ambiguity set as a statistical ball with its radius measured by the Wasserstein metric, we achieve a probabilistic guarantee of the out-of-sample risk, evaluated using new sample data generated independently of the training data. To resolve the infinite-dimensionality issue inherent in the distributionally robust MPC problem, we reformulate it as a finite-dimensional nonlinear program using modern distributionally robust optimization techniques based on the Kantorovich duality principle. To find a globally optimal solution in the case of affine dynamics and output equations, a spatial branch-and-bound algorithm is designed using McCormick relaxation. The performance of the proposed method is demonstrated and analyzed through simulation studies using nonlinear dynamic and kinematic vehicle models and a linearized quadrotor model. The simulation results indicate that, even when the sample size is small, the proposed method can successfully avoid randomly moving obstacles with a guarantee of out-of-sample risk, while its sample average approximation counterpart fails to do so. [ABSTRACT FROM AUTHOR]

We propose a novel computational framework for the robust optimization of highly nonlinear, non‐convex models that possess uncertainty in their parameter data. The proposed method is a generalization of the robust cutting‐set algorithm that can handle models containing irremovable equality constraints, as is often the case with models in the process systems engineering domain. Additionally, we accommodate general forms of decision rules to facilitate recourse in second‐stage (control) variables. In particular, we compare and contrast the use of various types of decision rules, including quadratic ones, which we show in certain examples to be able to decrease the overall price of robustness. Our proposed approach is demonstrated on three process flow sheet models, including a relatively complex model for amine‐based CO2 capture. We thus verify that the generalization of the robust cutting‐set algorithm allows for the facile identification of robust feasible designs for process systems of practical relevance. [ABSTRACT FROM AUTHOR]

We propose an optimization for space-efficient implementations of explicit model-predictive controllers (MPC) for robust control of linear time-invariant (LTI) systems on embedded platforms. We obtain an explicitform robust model-predictive controller as a solution to a multi-parametric linear programming problem. The structure of the controller is a polyhedral decomposition of the control domain, with an affine map for each domain. While explicit MPC is suited for embedded devices with low computational power, the memory requirements for such controllers can be high. We provide an optimization algorithm for a mixed-precision implementation of the controller, where the deviation of the implemented controller from the original one is within the robustness margin of the robust control problem. The core of the mixed-precision optimization is an iterative static analysis that co-designs a robust controller and a low-bitwidth approximation that is statically guaranteed to always be within the robustness margin of the original controller. We have implemented our algorithm and show on a set of benchmarks that our optimization can reduce space requirements by up to 20.9% and on average by 12.6% compared to a minimal uniform precision implementation of the original controller. [ABSTRACT FROM AUTHOR]

• A novel concept of power-to-gas (P2G)-based virtual power plant (GVPP) is designed. • Power-gas-power effect of GVPP brought by P2G and gas storage tank is discussed. • A multi-objective flexible risk aversion model for GVPP operation is proposed. • Payoff table of optimization model is applied for multiple iterative weighting method. • Different operation cases for verifying the proposed operation strategy are defined. Power-to-gas (P2G) provides a new means for accommodating abandoned new energy that will support the optimal operation of virtual power plants (VPPs) in the future. In this study, a novel structure of a P2G-based virtual power plant (GVPP) is designed. A flexible risk aversion model for GVPP operation is proposed with two objectives—maximum operation profit and minimum operation risk. In the model, the conditional value at risk method and robust optimization theory are utilized to reflect uncertainty risks. To solve the multi-objective model, a solution algorithm was constructed for iteratively obtaining the optimal weight coefficient for transforming the multi-objective model into a single-objective model based on a payoff table and a rough set method. Four simulation cases were set for comparative analysis based on a nine-node energy hub system. The results show the following outcomes: (1) GVPP realizes the complementary utilization of distributed energy and forms a power-gas-power recycling mode, (2) the proposed model can provide an effective decision-making tool for different risk-attitude decision makers by setting a reasonable confidence level and robust coefficient, (3) P2G and price-based demand response (PBDR) improve the grid-connected space of clean energy, in particular, PBDR improves the flexibility of system operation and reduces operation risk, and (4) if the maximum emission trade allowance (META) is considered, P2G preferentially converts carbon dioxide into methane when the META is lower, while clean energy is preferentially used to satisfy load demand and the methane produced by P2G can be sold to the natural gas network when the META is higher. Overall, the proposed optimal decision model achieves the maximum utilization of clean energy to obtain higher economic benefits while rationally controlling operation risks; hence, providing reliable support for decision makers. [ABSTRACT FROM AUTHOR]

• Studying service network design considering multiple types of services. • Proposing both deterministic and two-stage robust optimization formulations. • Implementing a column-and-constraint generation approach to solve the robust model. • The proposed algorithm outperforms the Benders decomposition method. • The diversification of service types reduces total cost and the number of vehicles. Integrated transportation systems providing multiple types of services are attracting increasing attention due to the diversification of demands. In this paper, we propose both deterministic and two-stage robust optimization formulations for the service network design problem considering multiple types of services. A probability-free uncertainty set is employed to deal with demand uncertainty in the proposed robust model. We implement a column-and-constraint generation approach to solve the robust model, which outperforms the Benders decomposition method. Comparative results indicate that it helps reduce the total cost and the number of vehicles to consider multiple service types in the network. [ABSTRACT FROM AUTHOR]

In the present paper, a robust approach is used to locate hub facilities considering network risks. An additional objective function, minimax regret, is added to the classical objective function in the hub location problem. In the proposed model, risk factors such as availability, security, delay time, environmental guidelines and regional air pollution are considered using triangular fuzzy-stochastic numbers. Then an equivalent crisp single objective model is proposed and solved by the Benders decomposition method. Finally, the results of both Benders decomposition and commercial optimization software are compared for different instances. Numerical instances were developed based on the well-known Civil Aeronautics Board (CAB) data set, considering different levels of uncertainty in parameters. The results show that the proposed model is capable of selecting nodes as sustainable hubs. Also, the results confirm that using Benders decomposition is more efficient than using classical solution methods for large-scale problems. [ABSTRACT FROM AUTHOR]

Independent operation of single microgrids (MGs) faces problems such as low self-consumption of local renewable energy, high operation cost and frequent power exchange with the grid. Interconnecting multiple MGs as a multi-microgrid (MMG) is an effective way to improve operational and economic performance. However, ensuring the optimal collaborative operation of a MMG is a challenging problem, especially under disturbances of intermittent renewable energy. In this paper, the economic and collaborative operation of MMGs is formulated as a unit commitment problem to describe the discrete characteristics of energy transaction combinations among MGs. A two-stage adaptive robust optimization based collaborative operation approach for a residential MMG is constructed to derive the scheduling scheme which minimizes the MMG operating cost under the worst realization of uncertain PV output. Transformed by its KKT optimality conditions, the reformulated model is efficiently solved by a column-and-constraint generation (C&CG) method. Case studies verify the effectiveness of the proposed model and evaluate the benefits of energy transactions in MMGs. The results show that the developed MMG operation approach is able to minimize the daily MMG operating cost while mitigating the disturbances of uncertainty in renewable energy sources. Compared to the non-interactive model, the proposed model can not only reduce the MMG operating cost but also mitigate the frequent energy interaction between the MMG and the grid. [ABSTRACT FROM AUTHOR]

In this paper we consider ambiguous stochastic constraints under partial information consisting of means and dispersion measures of the underlying random parameters. Whereas the past literature used the variance as the dispersion measure, here we use the mean absolute deviation from the mean (MAD). This makes it possible to use the 1972 result of Ben-Tal and Hochman (BH) in which tight upper and lower bounds on the expectation of a convex function of a random variable are given. First, we use these results to treat ambiguous expected feasibility constraints to obtain exact reformulations for both functions that are convex and concave in the components of the random variable. This approach requires, however, the independence of the random variables and, moreover, may lead to an exponential number of terms in the resulting robust counterparts.We then show how upper bounds can be constructed that alleviate the independence restriction, and require only a linear number of terms, by exploiting models in which random variables are linearly aggregated. Moreover, using the BH bounds we derive three new safe tractable approximations of chance constraints of increasing computational complexity and quality. In a numerical study, we demonstrate the efficiency of our methods in solving stochastic optimization problems under mean-MAD ambiguity. [ABSTRACT FROM AUTHOR]

We treat uncertain linear programming problems by utilizing the notion of weighted analytic centers and notions from the area of multi-criteria decision making. After introducing our approach, we develop interactive cutting-plane algorithms for robust optimization, based on concave and quasi-concave utility functions. In addition to practical advantages, due to the flexibility of our approach, we prove that under a theoretical framework proposed by Bertsimas and Sim (Oper Res 52:35-53, 2004), which establishes the existence of certain convex formulation of robust optimization problems, the robust optimal solutions generated by our algorithms are at least as desirable to the decision maker as any solution generated by many other robust optimization algorithms in the theoretical framework. We present some probabilistic bounds for feasibility of robust solutions and evaluate our approach by means of computational experiments. [ABSTRACT FROM AUTHOR]

Robust optimization is a powerful means to handle optimization problems where there is a set of parameters that are uncertain. The effectiveness of the method is especially noticeable when these parameters are only known to lie inside some uncertainty region. Unfortunately, there are important computational considerations that have prevented the methodology from being fully adopted in fields of practice where the cost function that needs to be robustified is nonlinear with respect to such parameters. In this paper, we propose a new robust optimization formulation that circumvent the computational burden in problems where the cost decomposes as the sum of costs that each involve a single decision variable. This is done by exploiting the fact that in this formulation the worst-case cost function can be expressed as a convex combination between a nominal and an upper-bounding cost function. One can still control the conservatism of the robust solution by adjusting how many terms of this total cost can simultaneously reach their respective most pessimistic value. To demonstrate the potential of this "practicable robust counterpart" formulation, we present how it can be employed for the robust optimization of packet routing on a telecommunication network with congestion. In such problems, an important source of uncertainty stems from the queueing delay, which is usually approximated by a nonlinear convex function using the theory of M/M/1 queues. Computational results on a large number of problem instances of realistic size confirmthat it is possible to identify robust solutions that significantly outperform a deterministic approach in terms of both the amount of congestion and the risks of excessive congestion. Moreover, our proposed method also improves significantly the quality of solutions that are obtained using two other natural approximation schemes. [ABSTRACT FROM AUTHOR]

In this paper, we consider the robust mean variance optimization problem where the probability distribution of assets’ returns is multivariate normal and the uncertain mean and covariance are controlled by a constraint involving Rényi divergence. We present the closed-form solutions for the robust mean variance optimization problem and find that the choice of order parameter which is related to the Rényi divergence measure will not impact optimal portfolio strategy under the cases that the mean vector and the covariance matrix are uncertain, respectively. Moreover, we obtain the closed-form solution for the robust mean variance optimization problem under the case that the mean vector and the covariance matrix are both uncertain. We illustrate the efficiency of our results with an example. [ABSTRACT FROM PUBLISHER]

Motivated by weakly convex optimization and quadratic optimization problems, we first show that there is no duality gap between a difference of convex (DC) program over DC constraints and its associated dual problem. We then provide certificates of global optimality for a class of nonconvex optimization problems. As an application, we derive characterizations of robust solutions for uncertain general nonconvex quadratic optimization problems over nonconvex quadratic constraints. [ABSTRACT FROM PUBLISHER]

In this paper, we study risk-averse models for multicriteria optimization problems under uncertainty. We use a weighted sum-based scalarization and take a robust approach by considering a set of scalarization vectors to address the ambiguity and inconsistency in the relative weights of each criterion. We model the risk aversion of the decision makers via the concept of multivariate conditional value-at-risk ( CVaR). First, we introduce a model that optimizes the worst-case multivariate CVaR and show that its optimal solution lies on a particular type of stochastic efficient frontier. To solve this model, we develop a finitely convergent delayed cut generation algorithm for finite probability spaces. We also show that the proposed model can be reformulated as a compact linear program under certain assumptions. In addition, for the cut generation problem, which is in general a mixed-integer program, we give a stronger formulation than the existing ones for the equiprobable case. Next, we observe that similar polyhedral enhancements are also useful for a related class of multivariate CVaR-constrained optimization problems that has attracted attention recently. In our computational study, we use a budget allocation application to benchmark our proposed maximin type risk-averse model against its risk-neutral counterpart and a related multivariate CVaR-constrained model. Finally, we illustrate the effectiveness of the proposed solution methods for both classes of models. [ABSTRACT FROM AUTHOR]

In our study, we integrate the data uncertainty of real-world models into our regulatory systems and robustify them. We newly introduce and analyse robust time-discrete target–environment regulatory systems under polyhedral uncertainty through robust optimization. Robust optimization has reached a great importance as a modelling framework for immunizing against parametric uncertainties and the integration of uncertain data is of considerable importance for the model’s reliability of a highly interconnected system. Then, we present a numerical example to demonstrate the efficiency of our new robust regression method for regulatory networks. The results indicate that our approach can successfully approximate the target–environment interaction, based on the expression values of all targets and environmental factors. [ABSTRACT FROM AUTHOR]

Probabilistic robustness evaluation is a promising approach to evolutionary robust optimization; however, high computational time arises. In this paper, we apply this approach to the Bayesian optimization algorithm (BOA) with a view to improving its computational time. To this end, we analyze the Bayesian networks constructed in BOA in order to extract the patterns of non-robust solutions. In each generation, the solutions that match the extracted patterns are detected and then discarded from the process of evaluation; therefore, the computational time in discovering the robust solutions decreases. The experimental results demonstrate that our proposed method reduces computational time, while increasing the robustness of solutions. [ABSTRACT FROM AUTHOR]

This paper proposes a novel approach for the offering strategy of a virtual power plant that participates in the day-ahead and the real-time energy markets. The virtual power plant comprises a conventional power plant, a wind-power unit, a storage facility, and flexible demands, which participate in the day-ahead and the real-time markets as a single entity in order to optimize their energy resources. We model the uncertainty in the wind-power production and in the market prices using confidence bounds and scenarios, respectively, which allows us to formul-ate the strategic offering problem as a stochastic adaptive robust optimization model. Results of a case study are provided to show the applicability of the proposed approach. [ABSTRACT FROM PUBLISHER]

Advanced control systems are tuned using dynamic models and optimization techniques. This approach frequently involves satisfying multiple conflicting objectives. Tuning robust controllers requires considering a framework that represents the system uncertainties, and its definition is not a trivial task. When dealing with a nonlinear model with many parameters, a high-quality representation requires a massive sampling of variations. In many cases, this represents an inaccessible computational cost for the optimization process. This work presents a new methodology for parameter uncertainty modeling that is oriented to tuning robust controllers based on multiobjective optimization techniques. The uncertainty modeling formulated represents a feasible computational cost and leads to robust solutions without attributing excessive conservatism. The novelty of this process consists in using the multiobjective space to define a set of scenarios with highly representative properties of the global uncertainty framework that formulate the control problem under a predefined minimization strategy. To demonstrate the effectiveness of this methodology, we present a temperature control design in a micro-CHP system under worst-case minimization. Based on the results, particular interest is given to verifying the appropriate formulation of the uncertainty modeling, which represents a 92.8% reduction of the computational cost involved in solving the robust optimization problem under a global uncertainty framework. • Tuning robust controllers requires considering system uncertainty. • Consider feasible computational cost and conservatism. • A new methodology is proposed to define highly relevant models. • The novelty consists of using the multi-objective space to set a characterization. • This process formulates the robust control under a realistic worst case. [ABSTRACT FROM AUTHOR]

This work proposes model based approaches for on-line or off-line economic optimization of batch reactors in the presence of model error (uncertainty). Polynomial Chaos Expansions are used as an effective and computationally efficient tool to propagate the error in model parameters into the optimizations’ cost functions. The computational efficiency of the proposed uncertainty propagation approach is essential for the implementation of the on-line approach that takes into account feedback corrections. The role of feedback, as applied in the on-line formulation, is proved to be instrumental for reducing conservatism of the optimization results. The proposed approaches can serve to design recipes for maximizing productivity in batch, fed-batch or perfusion operation of bioreactors. [ABSTRACT FROM AUTHOR]

This paper provides characterizations of the weakly minimal elements of vector optimization problems and the global minima of scalar optimization problems posed on locally convex spaces whose objective functions are deterministic while the uncertain constraints are treated under the robust (or risk-averse) approach, i.e. requiring the feasibility of the decisions to be taken for any possible scenario. To get these optimality conditions we provide Farkas-type results characterizing the inclusion of the robust feasible set into the solution set of some system involving the objective function and possibly uncertain parameters. In the particular case of scalar convex optimization problems, we characterize the optimality conditions in terms of the convexity and closedness of an associated set regarding a suitable point. [ABSTRACT FROM AUTHOR]

In this paper, we develop a unified framework for studying constrained robust optimal control problems with adjustable uncertainty sets . In contrast to standard constrained robust optimal control problems with known uncertainty sets, we treat the uncertainty sets in our problems as additional decision variables. In particular, given a finite prediction horizon and a metric for adjusting the uncertainty sets, we address the question of determining the optimal size and shape of the uncertainty sets, while simultaneously ensuring the existence of a control policy that will keep the system within its constraints for all possible disturbance realizations inside the adjusted uncertainty set. Since our problem subsumes the classical constrained robust optimal control design problem, it is computationally intractable in general. Nevertheless, we demonstrate that by restricting the families of admissible uncertainty sets and control policies, the problem can be formulated as a tractable convex optimization problem. We show that our framework captures several families of (convex) uncertainty sets of practical interest, and illustrate our approach on a demand response problem of providing control reserves for a power system. [ABSTRACT FROM AUTHOR]

The design of decentralized energy supply systems is a complex task and thus best addressed by mathematical optimization. However, design problems typically rely on uncertain input data, such as future energy demands or prices. Still, conventional optimization models are usually deterministic and thus neglect uncertainties. For this reason, the deterministic optimal solution is in general suboptimal or even infeasible. Robust design methods are available to guarantee security of energy supply, however, they usually lead to significant additional costs. In this work, we show that energy supply systems with guaranteed secure energy supply are not expensive per se. For this purpose, we propose the T wo-stage R ob u stnes s T rade-off (TRusT) approach. The TRusT approach considers the trade-off between expected costs in the nominal scenario and costs in the worst case while guaranteeing security of energy supply. Thereby, the TRusT approach identifies balanced robust energy supply systems which are cost-efficient in both the daily business and the worst case. The TRusT approach can be applied and solved efficiently. In a case study, we identify robust design options which ensure security of energy supply at low additional costs. Hence, the TRusT approach is a suitable tool to design cost-efficient and secure energy systems. [ABSTRACT FROM AUTHOR]

The separation of water and acetic acid mixtures is a challenging task in the terephthalic acid production process. This study addresses the techno-economical optimization of an heteroextractive distillation column that uses p -xylene as entrainer. The optimization of this column is performed by means of robust optimizers belonging to the BzzMath library, coupled with the detailed process simulation of the downstream section performed with the commercial software SimSci PRO/II. A comparison with respect to the commercial simulator optimizer and a quantification of benefits are given. Results show that the BzzMath robust optimizer reaches a lower Total Annual Cost (−3.5%) with respect to the commercial software optimization. Finally, the comparison with other entrainers proposed in literature for the advanced distillation of the water/acetic acid mixtures demonstrates the economical and operational appeal of this process configuration. [ABSTRACT FROM AUTHOR]

One of the major challenges of today's world is the energy crisis, which has significant impact on other issues, such as world economy or the environment. Due to its vital role in manufacturing, energy saving should be considered as an important element of production planning, especially for energy intensive manufacturing plants. In this paper, an aggregate planning model is developed to consider energy planning, demand and production capacity simultaneously. The model happens to be a linear program with three objective functions, which minimize the operational cost, the energy cost and carbon emission. The proposed model is solved as a single-objective model applying a goal attainment technique. To overcome the effects of uncertain input data, we apply the robust optimization approach to this model. The practicality of the proposed approach is illustrated by applying it to a smelting manufacturer, which produces a wide range of products such as aluminum, copper and lead. Considering uncertainty in objective function parameters, we provide some insights into the relationship between the budgets of uncertainty and optimal values. Moreover, regarding uncertainty in product demands, we generate the robust solutions which have a better performance than the deterministic solutions; also we test “ Aggregate constraints approach ” to the maximum production capacity parameters successfully. The sensitivity of different uncertain parameters is also analyzed. [ABSTRACT FROM AUTHOR]

Task-space inverse dynamics (TSID) is a well-known optimization-based technique for the control of highly redundant mechanical systems, such as humanoid robots. One of its main flaws is that it does not take into account any of the uncertainties affecting these systems: poor torque tracking, sensor noises, delays, and model uncertainties. As a consequence, the resulting control-state trajectories may be feasible for the ideal system, but not for the real one. We propose to improve the robustness of TSID by modeling uncertainties in the joint torques, either as Gaussian random variables or as bounded deterministic variables. Then we try to immunize the constraints of the system to any—or at least most—of the realizations of these uncertainties. When the resulting optimization problem is computationally too expensive for online control, we propose ways to approximate it that lead to computation times below 1 ms. Extensive simulations in a realistic environment show that the proposed robust controllers greatly outperform the classic one, even when other unmodeled uncertainties affect the system (e.g., errors in the inertial parameters, delays in the velocity estimates). [ABSTRACT FROM AUTHOR]

Multiple Tuned Mass Dampers (MTMD) is one of the simplest and most reliable solution to control the vibrations of a structure, e.g. allowing to deal with a wide distribution of structural natural frequencies and usually showing inherent stability. Nevertheless, uncertainties in the behavior of structures under random dynamic input may induce implications on the MTMD effectiveness, e.g. detuning or amplifications. Typically, uncertainties in the input are exclusively considered, while uncertainties affect as much also the mechanical parameters. This paper proposes a robust optimum design of the MTMD, considering uncertainties both in the structural parameters and in the earthquake input. At this aim, a random vibration analysis of the response is adopted, and a direct linear perturbation method is applied on the uncertain parameters. Results show that a deterministic optimization is inappropriate for sensible systems as TMDs, while a robust optimization aims to better control the response with low computational efforts. Furthermore, the selection of the objective functions significantly affects the optimal design parameters, as the number of TMDs influences the robust designs. [ABSTRACT FROM AUTHOR]

A novel infill sampling criterion is proposed for efficient estimation of the global robust optimum of expensive computer simulation based problems. The algorithm is especially geared towards addressing problems that are affected by uncertainties in design variablesandproblem parameters. The method is based on constructing metamodels using Kriging and adaptively sampling the response surface via a principle of expected improvement adapted for robust optimization. Several numerical examples and an engineering case study are used to demonstrate the ability of the algorithm to estimate the global robust optimum using a limited number of expensive function evaluations. [ABSTRACT FROM AUTHOR]

This paper considers minimum cost flow problem in dynamic networks with uncertain costs. First, we present a short introduction of dynamic minimum cost flow. Then, we survey discrete and continuous dynamic minimum cost flow problems, their properties and relationships between them. After that, the minimum cost flow problem in discrete dynamic network with uncertainty in the cost vector is considered such that the arc cost can be changed within an interval. Finally, we propose an algorithm to find the optimal solution of the proposed model. [ABSTRACT FROM AUTHOR]

This study considers a basic inventory management problem with nonzero fixed order costs under interval demand uncertainty. The existing robust formulations obtained by applying well-known robust optimization methodologies become computationally intractable for large problem instances due to the presence of binary variables. This study resolves this intractability issue by proposing a new robust formulation that is shown to be solvable in polynomial time when the initial inventory is zero or negative. Because of the computational efficiency of the new robust formulation, it is implemented on a folding-horizon basis, leading to a new heuristic for the problem. The computational results reveal that the new heuristic is not only superior to the other formulations regarding the computing time needed, but also outperforms the existing robust formulations in terms of the actual cost savings on the larger instances. They also show that the actual cost savings yielded by the new heuristic are close to a lower bound on the optimal expected cost. [ABSTRACT FROM AUTHOR]

We consider optimal decision-making problems in an uncertain environment. In particular, we consider the case in which the distribution of the input is unknown, yet there is some historical data drawn from the distribution. In this paper, we propose a new type of distributionally robust optimization model called the likelihood robust optimization (LRO) model for this class of problems. In contrast to previous work on distributionally robust optimization that focuses on certain parameters (e.g., mean, variance, etc.) of the input distribution, we exploit the historical data and define the accessible distribution set to contain only those distributions that make the observed data achieve a certain level of likelihood. Then we formulate the targeting problem as one of optimizing the expected value of the objective function under the worst-case distribution in that set. Our model avoids the over-conservativeness of some prior robust approaches by ruling out unrealistic distributions while maintaining robustness of the solution for any statistically likely outcomes. We present statistical analyses of our model using Bayesian statistics and empirical likelihood theory. Specifically, we prove the asymptotic behavior of our distribution set and establish the relationship between our model and other distributionally robust models. To test the performance of our model, we apply it to the newsvendor problem and the portfolio selection problem. The test results show that the solutions of our model indeed have desirable performance. [ABSTRACT FROM AUTHOR]

This article presents a robust optimization reformulation of the dual-response problem developed in response surface methodology. The dual-response approach fits separate models for the mean and the variance and analyzes these two models in a mathematical optimization setting. We use metamodels estimated from experiments with both controllable and environmental inputs. These experiments may be performed with either real or simulated systems; we focus on simulation experiments. For the environmental inputs, classic approaches assume known means, variances, or covariances and sometimes even a known distribution. We, however, develop a method that uses only experimental data, so it does not need a known probability distribution. Moreover, our approach yields a solution that is robust against the ambiguity in the probability distribution. We also propose an adjustable robust optimization method that enables adjusting the values of the controllable factors after observing the values of the environmental factors. We illustrate our novel methods through several numerical examples, which demonstrate their effectiveness. [ABSTRACT FROM AUTHOR]

To consider the interval uncertainty in a practical optimal design problem, a new methodology for efficient robust optimizations is proposed. The proposed methodology uses a constrained formulation for robust performances not only in alleviating the inefficiency of the existing approaches in modeling interval uncertainties but also in avoiding the deficiency in the biasing force selection. The gradient information is used as both a trigger to activate the uncertain quantification procedure and the steepest increment direction to develop a fast searching phase. The stochastic approximation method is employed to minimize the computational burdens in computing the gradients. The numerical results on a case study are reported to validate the proposed methodology. [ABSTRACT FROM AUTHOR]

Environmentalism has become an important global issue in the present century. During the recent years, the closed-loop green supply chain management has been increasingly a center of attention as a result of governmental laws and legislations (regarding the environmental effects of any activity) and the soaring consumers' expectations. In this study, effort has been made to investigate a facility location/allocation model for a multi-product closed-loop green supply chain network consisting of manufacturing/remanufacturing and collection/inspection centers as well as disposal center and markets. To design the network, we have proposed a mixed-integer linear programming model capable of reducing the network total costs. The model has been so developed as to consider such environmental objectives as reducing the rate of carbon dioxide emission in the environment throughout the network in question. Moreover, the model has been developed using a robust fuzzy programming approach to investigate the effects of uncertainties of the variable costs, as well as the demand rate, on the network design. To solve the bi-objective programming model, use has been made of the ε-constraint approach and a numerical illustration of Copiers Industry is used to show the applicability of the proposed optimization model. Results have revealed that the model is capable of controlling the network uncertainties as a result of which a robustness price will be imposed on the system. [ABSTRACT FROM AUTHOR]

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In this paper, a cost minimization problem is formulated to intelligently schedule energy generations for microgrids equipped with unstable renewable sources and combined heat and power (CHP) generators. In such systems, the fluctuant net demands (i.e., the electricity demands not balanced by renewable energies) and heat demands impose unprecedented challenges. To cope with the uncertainty nature of net demand and heat demand, a new flexible uncertainty model is developed. Specifically, we introduce reference distributions according to predictions and field measurements and then define uncertainty sets to confine net and heat demands. The model allows the net demand and heat demand distributions to fluctuate around their reference distributions. Another difficulty existing in this problem is the indeterminate electricity market prices. We develop chance constraint approximations and robust optimization approaches to firstly transform and then solve the prime problem. Numerical results based on real-world data evaluate the impacts of different parameters. It is shown that our energy generation scheduling strategy performs well and the integration of combined heat and power (CHP) generators effectively reduces the system expenditure. Our research also helps shed some illuminations on the investment policy making for microgrids. [ABSTRACT FROM AUTHOR]

With the development of microgrid technology, for an energy intensive corporate such as an iron and steel plant energy consumptions and costs can be saved significantly by achieving the balance between self-generation and consumption. In this paper, we present a self-balancing and robust scheduling model with flexible batch loads for an energy intensive corporate. The model is a multi-level optimization model with the objective to minimize the unbalance cost in the worst case. Load following properties are given to determine whether the uncertain loads can be followed or not, and the self-balancing capability of an energy intensive corporate is analyzed. The problem is equivalently converted from the multi-level model to a single-level optimization model with a set of constraints. In this way, the iteration between the outer and inner level can be avoided. Case study based on an energy intensive corporate microgrid is tested and the results show that the unbalance cost can be significantly reduced by using the robust self-balancing model. In addition, compared the approach method with iterative solving method, computational efficiency can be improved and local optimum can be avoided. [ABSTRACT FROM AUTHOR]

We propose an approach to the timing of markdowns in a continuous-time finite-horizon setting that does not require the precise knowledge of the underlying probabilities, instead relying on range forecasts for the arrival rates of the demand processes, and that captures the degree of the manager's risk aversion through intuitive budget of uncertainty functions. These budget functions bound the cumulative deviation of the arrival rates from their nominal values over the lengths of time for which a product is offered at a given price. A key issue is that using lengths of time as decision variables introduces non-convexities when budget functions are concave. In the single-product case, we describe a tractable and intuitive framework to incorporate uncertainty on customers' arrival rates, formulate the resulting robust optimization model, describe an efficient procedure to compute the optimal sale times, and provide theoretical insights. We then describe how to use the solution of the static robust optimization model to implement a dynamic markdown policy. We also extend the robust optimization approach to multiple products and suggest the idea of constraint aggregation to preserve performance for this type of problem structure. [ABSTRACT FROM AUTHOR]

In this paper, a robust optimization approach is used to solve the redundancy allocation problem (RAP) in series-parallel systems with component mixing where uncertainty exists in components’ reliabilities. In real world, the reliabilities of components are imprecisely estimated or the reliability of some components may vary due to some realistic factors. Therefore, we may deal with a system where there are many components with uncertain values of reliabilities. To deal with this problem, for the first time a robust optimization approach is applied to RAP with component mixing to produce a robust solution, which is relatively insensitive with respect to uncertainty in reliability of components. In addition, the advantages of the proposed robust technique are illustrated by considering a series-parallel system and finding the suitable redundancy levels and then Monte Carlo simulation is implemented to examine the quality of the robust solutions. The results indicate that applying the proposed robust RAP can be more reliable to determine system reliability in the designing phase of systems. [ABSTRACT FROM AUTHOR]

Two new approximate formulations to joint chance-constrained optimization problems are proposed in this paper. The relationships of CVaR (conditional-value-at-risk), chance constrains and robust optimization are reviewed. Firstly, two new upper bounds on E((·) + ) are proposed, where E stands for the expectation and x + = max(0, x ), based on which two approximate formulations for individual chance-constrained problems are derived. The approximations are proved to be the robust optimization with the corresponding uncertain sets. Then the approximations are extrapolated to joint chance-constrained problem. Finally numerical studies are performed to compare the solutions of individual and joint chance constraints approximations and the results demonstrate the validity of our method. [ABSTRACT FROM AUTHOR]

Solving optimization problems under hybrid uncertainty bears a heavy computational burden. In this study, we propose a unified structured optimization approach, termed robust granular optimization (RGO), to tackle the optimization problems under hybrid manifold uncertainties in a computationally tractable manner. Essentially, the RGO can be regarded as a complementary fusion of granular computing and robust optimization techniques. The paradigm of RGO consists of three core phases: 1) uncertainty identification, 2) information granulation in which basic granular units (BGUs) are formed, and 3) robust optimization realized over the BGUs. Following the proposed paradigm, we develop two classes of RGO models for general single-stage and two-stage optimization problems with separable and higher order hybrid uncertainties, respectively. It is shown that both types RGO models can be equivalently transformed into linear programs or mixed integer linear programs that can be handled efficiently by off-the-shelf solvers. Furthermore, a target-based tradeoff model is developed to enhance the flexibility of the RGO models in balancing the granularity level (or robustness level) and the solution conservativeness. The tradeoff model can also be efficiently solved by a binary search algorithm. Finally, sufficient computational studies are presented, and comparisons with the existing approaches show that the RGO models can bring much higher computational efficiency and scalability without losing much optimality, and the RGO solutions exhibit a stronger resistance to the uncertainty. [ABSTRACT FROM PUBLISHER]

The optimum design of base isolation system to control seismic vibration considering uncertain system parameters are usually performed by minimizing the unconditional expected value of mean square response of a structure without any consideration to the variance of such responses due to system parameter uncertainty. However, the unconditional mean square response based designed may have larger variance of responses due to uncertainty in system parameters and the overall system performance may be sensitive. But, it is desirable that the optimum design should reduce both the mean and variance of dynamic performance measure under system parameter uncertainty. The present study deals with robust design optimization (RDO) of base isolation system considering random system parameters characterizing the structure, isolator and ground motion model. The RDO is performed by minimizing the weighted sum of the expected value of the maximum root mean square acceleration of the structure as well its standard deviation. A numerical study elucidates the importance of the RDO procedure for design of base isolation system by comparing the proposed RDO results with the results obtained by the conventional stochastic structural optimization procedure and the unconditional response based optimization. [ABSTRACT FROM AUTHOR]

Urban bulk water systems supply water with high reliability and, in the event of extreme drought, must avoid catastrophic economic and social collapse. In view of the deep uncertainty about future climate change, it is vital that robust solutions be found that secure urban bulk water systems against extreme drought. To tackle this challenge an approach was developed integrating: 1) a stochastic model of multi-site streamflow conditioned on future climate change scenarios; 2) Monte Carlo simulation of the urban bulk water system incorporated into a robust optimization framework and solved using a multi-objective evolutionary algorithm; and 3) a comprehensive decision space including operating rules, investment in new sources and source substitution and a drought contingency plan with multiple actions with increasingly severe economic and social impact. A case study demonstrated the feasibility of this approach for a complex urban bulk water supply system. The primary objective was to minimize the expected present worth cost arising from infrastructure investment, system operation and the social cost of “normal” and emergency restrictions. By introducing a second objective which minimizes either the difference in present worth cost between the driest and wettest future climate change scenarios or the present worth cost for driest climate scenario, the trade-off between efficiency and robustness was identified. The results show that a significant change in investment and operating strategy can occur when the decision maker expresses a stronger preference for robustness and that this depends on the adopted robustness measure. Moreover, solutions are not only impacted by the degree of uncertainty about future climate change but also by the stress imposed on the system and the range of available options. [ABSTRACT FROM AUTHOR]

Uncertainty can be defined as the probability of difference between the forecasted value and the real value. As this probability is small, the operation cost of the power system will be less. This purpose necessitates modeling of system random variables (such as the output power of renewable resources and the load demand) with appropriate and practicable methods. In this paper, an adequate procedure is proposed in order to do an optimal energy management on a typical micro-grid with regard to the relevant uncertainties. The point estimate method is applied for modeling the wind power and solar power uncertainties, and robust optimization technique is utilized to model load demand uncertainty. Finally, a comparison is done between deterministic and probabilistic management in different scenarios and their results are analyzed and evaluated. [ABSTRACT FROM AUTHOR]