Chance-constrained Optimization Problem Research Articles

Interval reservoir computing: theory and case studies

The time series data in many applications, for example, wind power and vehicle trajectory, show significant uncertainty. Using a single prediction value of wind power as feedback information for wind turbine control or unit commitment is not enough since the uncertainty of the prediction is not described. This paper addresses the uncertainty issue in time series data forecasting by proposing the novel interval reservoir computing method. The proposed interval reservoir computing can capture the underlying evolution of the stochastic dynamical system for time series data using the recurrent neural network (RNN). On the other hand, by formulating a chance-constrained optimization problem, interval reservoir computing outputs a set of parameters in the RNN, which maps to an interval of prediction values. The capacity of the interval is the smallest one satisfying the condition that the probability of having a prediction inside the interval is lower than the required level. The scenario approach solves the formulated chance-constrained optimization problem. We implemented an experimental data-based validation to evaluate the proposed method. The validation results show that the proposed interval reservoir computing can give a tight interval of time series data forecasting values for wind power and traffic trajectory. In addition, the confidence probability over the feasibility goes to 1 very quickly as the sample number increases.

Chance-Constrained Optimization Formulation for Ship Conceptual Design: A Comparison of Metaheuristic Algorithms

This paper presents a new chance-constrained optimization (CCO) formulation for the bulk carrier conceptual design. The CCO problem is modeled through the scenario design approach. We conducted extensive numerical experiments comparing the convergence of both canonical and state-of-the-art metaheuristic algorithms on the original and CCO formulations and showed that the CCO formulation is substantially more difficult to solve. The two best-performing methods were both found to be differential evolution-based algorithms. We then provide an analysis of the resulting solutions in terms of the dependence of the distribution functions of the unit transportation costs and annual cargo capacity of the ship design on the probability of violating the chance constraints.

Inner Moreau Envelope of Nonsmooth Conic Chance-Constrained Optimization Problems

Optimization problems with uncertainty in the constraints occur in many applications. Particularly, probability functions present a natural form to deal with this situation. Nevertheless, in some cases, the resulting probability functions are nonsmooth, which motivates us to propose a regularization employing the Moreau envelope of a scalar representation of the vector inequality. More precisely, we consider a probability function that covers most of the general classes of probabilistic constraints: [Formula: see text]where [Formula: see text] is a convex cone of a Banach space. The conic inclusion [Formula: see text] represents an abstract system of inequalities, and ξ is a random vector. We propose a regularization by applying the Moreau envelope to the scalarization of the function [Formula: see text]. In this paper, we demonstrate, under mild assumptions, the smoothness of such a regularization and that it satisfies a type of variational convergence to the original probability function. Consequently, when considering an appropriately structured problem involving probabilistic constraints, we can, thus, entail the convergence of the minimizers of the regularized approximate problems to the minimizers of the original problem. Finally, we illustrate our results with examples and applications in the field of (nonsmooth) joint, semidefinite, and probust chance-constrained optimization problems. Funding: P. Pérez-Aros was supported by Centro de Modelamiento Matemático [Grants ACE210010 and FB210005] and BASAL funds for center of excellence and ANID-Chile grant Fondecyt Regular [Grants 1200283 and 1190110] and Fondecyt Exploración [Grant 13220097]. C. Soto was supported by the National Agency for Research and Development (ANID)/Scholarship Program/Doctorado Nacional Chile [Grant 2017-21170428]. E. Vilches was supported by Centro de Modelamiento Matemático [Grants ACE210010 and FB210005] and BASAL funds for center of excellence and Fondecyt Regular [Grant 1200283] and Fondecyt Exploración [Grant 13220097] from ANID-Chile.

Efficient Scenario Generation for Heavy-Tailed Chance Constrained Optimization

We consider a generic class of chance-constrained optimization problems with heavy-tailed (i.e., power-law type) risk factors. As the most popular generic method for solving chance constrained optimization, the scenario approach generates sampled optimization problem as a precise approximation with provable reliability, but the computational complexity becomes intractable when the risk tolerance parameter is small. To reduce the complexity, we sample the risk factors from a conditional distribution given that the risk factors are in an analytically tractable event that encompasses all the plausible events of constraints violation. Our approximation is proven to have optimal value within a constant factor to the optimal value of the original chance constraint problem with high probability, uniformly in the risk tolerance parameter. To the best of our knowledge, our result is the first uniform performance guarantee of this type. We additionally demonstrate the efficiency of our algorithm in the context of solvency in portfolio optimization and insurance networks. Funding: The research of B. Zwart is supported by the NWO (Dutch Research Council) [Grant 639.033.413]. The research of J. Blanchet is supported by the Air Force Office of Scientific Research [Award FA9550-20-1-0397], the National Science Foundation [Grants 1820942, 1838576, 1915967, and 2118199], Defense Advanced Research Projects Agency [Award N660011824028], and China Merchants Bank.

Robust Power Allocation for Integrated Visible Light Positioning and Communication Networks

Integrated visible light positioning and communication (VLPC), capable of combining advantages of visible light communications (VLC) and visible light positioning (VLP), is a promising key technology for the future Internet of Things. In VLPC networks, positioning and communications are inherently coupled, which has not been sufficiently explored in the literature. We propose a robust power allocation scheme for integrated VLPC Networks by exploiting the intrinsic relationship between positioning and communications. Specifically, we derive explicit relationships between random positioning errors, following both a Gaussian distribution and an arbitrary distribution, and channel state information errors. Then, we minimize the Cramer-Rao lower bound (CRLB) of positioning errors, subject to the rate outage constraint and the power constraints, which is a chance-constrained optimization problem and generally computationally intractable. To circumvent the nonconvex challenge, we conservatively transform the chance constraints to deterministic forms by using the Bernstein-type inequality and the conditional value-at-risk for the Gaussian and arbitrary distributed positioning errors, respectively, and then approximate them as convex semidefinite programs. Finally, simulation results verify the robustness and effectiveness of our proposed integrated VLPC design schemes.

Proactive Scheduling for Zero-Energy Device Networks With Fast Uplink Grant

To address the inefficiency in terms of resource utilization of grant-free (GF) protocols, the fast uplink (FU) grant medium-access protocol has been proposed. In this work, we design a proactive wireless power transfer (WPT) framework for FU grant zero-energy massive machine-type communication. Specifically, zero-energy devices (ZEDs) first harvest energy during the WPT phase and then transmit data based on the FU grant protocol. Moreover, a multi-arm bandit traffic prediction scheme is adopted. Then, considering that due to the FU grant protocol ZEDs may harvest energy for several WPT rounds prior to data transmission, we extract the ZEDs’ outage probability during uplink. Based on the derived expressions, a chance-constrained optimization problem is formulated, which minimizes the energy consumption for WPT, while guaranteeing an acceptable outage probability for the ZEDs during data transmission. Simulations validate the theoretical analysis and illustrate that the proposed scheme outperforms GF access.

Unlock the Thermal Flexibility in Integrated Energy Systems: A Robust Nodal Pricing Approach for Thermal Loads

The thermal flexibility of buildings in integrated energy systems (IES) has great potential to improve the operational economy and wind power consumption. To incentivize the thermal flexibility of buildings under uncertainties, we propose a robust nodal pricing (RNP) model for thermal loads based on the Stackelberg game approach. The RNP model adopts a bilevel framework in which the IES operator plays the leader at the upper level while the thermal load aggregators (TLA) play the followers at the lower level. The IES operator problem optimizes the heat prices and dispatch plan, which is modeled as a two-stage robust optimization problem to address the uncertainties in the renewables and the loads. The TLA problem optimizes the thermal loads of buildings to minimize the energy cost and thermal comfort loss, which is modeled as a distributionally robust chance-constrained optimization problem to address the uncertainty in outdoor temperature. Then, we convert the TLA model into deterministic quadratic programming and prove that Slaters condition holds under a mild assumption. Using it, the TLA model is equivalently converted into Karush-Kuhn-Tucker conditions, leading to a reformulation of a classical two-stage robust optimization model for the RNP model. Case studies compare different pricing methods and verify the superiority of the proposed method.

Differential flatness based design of robust controllers using polynomial chaos for linear systems

The focus of this paper is the design of open-loop control profiles for linear differentially flat systems whose model parameters are uncertain and are represented probabilistically. To account for the time-invariant model parameter uncertainties, polynomial chaos is used to derive a surrogate model which permits easy evaluation of the mean and variance of the uncertain states. A chance constrained optimisation problem is posed to minimise the terminal error of the stochastic states for a prescribed risk of not satisfying the terminal state bounds. To permit posing a convex optimisation problem, the cost function which is the residual energy is approximated by a hypercube circumscribing the hypersphere which bounds the terminal residual error. This relaxation permits posing a convex optimisation problem to arrive at the robust control profile. The proposed technique is illustrated on two examples including the benchmark spring-mass-dashpot system undergoing a rest-to-rest maneuver and a UAV undergoing a translation in a prescribed time. Parametric studies are conducted where the impact of varying the order of the polynomial chaos expansion and the degree of the parameterised control profiles are used to conclude that a 2–3 degree polynomial chaos expansion is sufficient to capture the time evolution of the mean and variance of the states which are required for the chance constrained optimisation problems. As expected, the design results in improved performance as the number of free variables used to parameterise the control profiles increases.

A data-driven robust EVaR-PC with application to portfolio management.

We investigate the robust chance constrained optimization problem (RCCOP), which is a combination of the distributionally robust optimization (DRO) and the chance constraint (CC). The RCCOP plays an important role in modeling uncertain parameters within a decision-making framework. The chance constraint, which is equivalent to a constraint of Value-at-risk (VaR), is approximated by risk measures such as Entropic Value-at-risk (EVaR) or Conditional Value-at-risk (CVaR) due to its difficulty to be evaluated. An excellent approximation requires both tractability and non-conservativeness. In addition, the DRO assumes that we know partial information about the distribution of uncertain parameters instead of their known true underlying probability distribution. In this article, we develop a novel approximation EVaR- PC based on EVaR for CC. Then, we evaluate the proposed approximation EVaR- PC using a discrepancy-based ambiguity set with the wasserstein distance. From a theoretical perspective, the EVaR- PC is less conservative than EVaR and the wasserstein distance possesses many good theoretical properties; from a practical perspective, the discrepancy-based ambiguity set can make full use of the data to estimate the nominal distribution and reduce the sensitivity of decisions to priori knowledges. To show the advantages of our method, we show its application in portfolio management in detail and give the relevant experimental results.

Determination and Cost Allocation for Regulation Reserve With Renewables: A Data-Driven Assisted Approach

Proper setting of regulation reserve (RR) requirements is essential for the secure and economic operation of power grids. Currently, the RR requirement is normally determined based on ad-hoc experience or numerical methods that cannot comprehensively consider uncertainties and fluctuation characters. Regarding the increasing penetration of renewables, it becomes an obvious challenge for determining the proper RR requirement and reasonably allocating the RR costs to market participants. To address these issues, a data-driven assisted determination and cost allocation method for RR is proposed. A data-driven method considering fluctuation features is used to predict the RR requirement. To avoid the insufficiency of the RR resulting from prediction error, a compensation strategy based on error decomposition is proposed. Ensemble learning is used to handle the model error. The data error caused by the multiperiod forecast uncertainty of load and renewables is compensated by a computationally tractable chance-constrained optimization problem. To encourage grid-friendly behaviors, a hierarchical method for RR cost allocation is proposed to account for the comprehensive contributions of market participants. Case studies show that compared with existing methods, the proposed RR requirement determination method can achieve a more than 24% improvement in frequency performance without excessive costs. The reasonability of the proposed cost allocation method is also illustrated.

Model Predictive Control of Parabolic PDE Systems under Chance Constraints

Model predictive control (MPC) heavily relies on the accuracy of the system model. Nevertheless, process models naturally contain random parameters. To derive a reliable solution, it is necessary to design a stochastic MPC. This work studies the chance constrained MPC of systems described by parabolic partial differential equations (PDEs) with random parameters. Inequality constraints on time- and space-dependent state variables are defined in terms of chance constraints. Using a discretization scheme, the resulting high-dimensional chance constrained optimization problem is solved by our recently developed inner–outer approximation which renders the problem computationally amenable. The proposed MPC scheme automatically generates probability tubes significantly simplifying the derivation of feasible solutions. We demonstrate the viability and versatility of the approach through a case study of tumor hyperthermia cancer treatment control, where the randomness arises from the thermal conductivity coefficient characterizing heat flux in human tissue.

Data-Driven Certification of Neural Networks With Random Input Noise

Methods to certify the robustness of neural networks in the presence of input uncertainty are vital in safety-critical settings. Most certification methods in literature are designed for adversarial or worst-case inputs, but researchers have recently shown a need for methods that consider random input noise. In this article, we examine the setting where inputs are subject to random noise coming from an arbitrary probability distribution. We propose a robustness certification method that lower-bounds the probability that network outputs are safe. This bound is cast as a chance-constrained optimization problem, which is then reformulated using input–output samples to make the optimization constraints tractable. We develop sufficient conditions for the resulting optimization to be convex, as well as on the number of samples needed to make the robustness bound hold with overwhelming probability. Case studies on synthetic, MNIST, and CIFAR-10 networks experimentally demonstrate that this method is able to certify robustness against various input noise regimes over larger uncertainty regions than prior state-of-the-art techniques.

Love of Variety Based Latency Analysis for High Definition Map Updating: Age of Information and Distributional Robust Perspectives

High definition (HD) map is a key technology that enables autonomous driving, which has the characteristic of frequent updates and low delay requirements. In order to minimize the HD map updating latency, this paper jointly considers the generation and transmission processes of HD map. The generation process of HD map relies heavily on the data captured by different types of sensors, which is modeled by the federated analytics process. Also, in order to quantify the diversity of utilized sensors, the concept of love of variety is introduced in this paper. Then, considering HD map is composed of dynamic and static layers that have different latency requirements during the transmission process, this paper proposes a method to allocate edge server capacity to each HD map layer such that the overall information staleness can be minimized. Firstly, the deterministic edge capacity case is discussed and the solution is derived by obtaining the Karush—Kuhn—Tucker conditions. Then, considering the practice that an edge server provides services to multiple attached devices simultaneously, contentions among these devices make available capacity for the autonomous vehicle variational. Therefore, an uncertain edge capacity case is discussed as well, where the uncertainty is described by Wasserstein metrics and the problem is reformulated into distributional robust chance constrained optimization problem. And for the low latency purpose, we utilize an inner approximation to reduce the complexity of the original problem and find a suboptimal solution. Finally, simulation results demonstrate the effectiveness of our proposed methods that achieves the lowest latency for HD map updating.

Solving a multi-objective chance constrained hierarchical optimization problem under intuitionistic fuzzy environment with its application

Various optimization approaches have been developed and used for generating optimal solutions for different industry related optimization problems. The semantic representation of imprecise coefficients and various types of uncertainties arising in real life optimization problems are still a challenging task and require attention of academicians as well as professionals. Decision makers (DMs) often face problems in presenting the linguistically expressed constraints. The impreciseness of information also plays a key role in modelling and thus in decision making. The impreciseness of coefficients is usually handled by using various elements of fuzzy set theory. The uncertainty associated with constraints in optimization problem is handled by using stochastic programming approaches. These coefficients are usually revealed by a logical DM, who takes into account the scope of hesitation along with the imprecise nature of data. Such coefficients can be very well represented by using Intuitionistic fuzzy numbers. In this work, an approach for solving multi-objective bi-level chance constrained optimization problem in an intuitionistic fuzzy environment has been presented. The technological coefficients and the coefficients of objective functions are represented by Triangular Intuitionistic fuzzy numbers whereas normal random variables are used to represent the coefficient of resources. The concept of component wise optimization is used for the reduction of intuitionistic fuzzy objective into their equivalent deterministic form. The concept of alpha-beta cut, chance constrained programming and interval programming are then used to convert the problem into a deterministic multi-objective bi-level linear programming problem. The resultant deterministic optimization problem is then solved by using TOPSIS method. The production planning problem of a firm has been modelled and solved by using the proposed approach.

A steady-state optimal coordination strategy for DERs systems with guaranteed probabilistic performance

We consider the problem of coordinating multiple Distributed Energy Resources (DERs) so as to supply energy to the grid while minimizing its variability around a reference profile that must also be optimized. We focus on the case when each DER is equipped with solar panels and a battery storage device, and jointly design the disturbance compensation strategies for charging and discharging the batteries on a one-day time horizon. To this purpose, we linearly parameterize the strategies and search for a solution minimizing the fluctuations of the energy exchange with the grid in steady-state, with a bound on their extent that holds in probability given the stochastic nature of the solar energy. Interestingly, the probability measure of the resulting chance-constrained optimization problem depends on the parameters of the disturbance compensation strategies, which makes the application of the scenario approach not standard. The proposed scenario-based solution is feasible for the original steady-state chance-constrained optimization problem and proves effective in numerical simulations.

Hour-Ahead Optimization Strategy for Shared Energy Storage of Renewable Energy Power Stations to Provide Frequency Regulation Service

With the rapid growth of intermittent renewable energy sources, it is critical to ensure that renewable power generators have the capability to perform primary frequency response (PFR). This paper proposes a framework for using a shared battery energy storage system (BESS) to undertake the PFR obligations for multiple wind and photovoltaic (PV) power plants and provide commercial automatic generation control (AGC) service in the ancillary service market at the same time. Furthermore, an optimal bidding strategy in the hour-ahead energy and ancillary service markets is proposed to maximize the BESS' benefits, considering frequency regulation performance, battery degradation cost, and the state of charge (SoC) limits. The optimization model is first built as a chance-constrained optimization problem and then transformed into mixed-integer quadratically constrained programming using Wasserstein metric-based distributionally robust optimization technique. Simulation results show that the proposed optimal bidding strategy and solution method can effectively take advantage of a BESS to perform PFR and provide AGC services simultaneously and improve its economic benefits.

Probability maximization via Minkowski functionals: convex representations and tractable resolution

In this paper, we consider the maximizing of the probability $${\mathbb {P}}\left\{ \, \zeta \, \mid \, \zeta \, \in \, {\mathbf {K}}({\mathbf {x}}) \, \right\} $$ over a closed and convex set $${\mathcal {X}}$$ , a special case of the chance-constrained optimization problem. Suppose $${\mathbf {K}}({\mathbf {x}}) \, \triangleq \, \left\{ \, \zeta \, \in \, {\mathcal {K}}\, \mid \, c({\mathbf {x}},\zeta ) \, \ge \, 0 \right\} $$ , and $$\zeta $$ is uniformly distributed on a convex and compact set $${\mathcal {K}}$$ and $$c({\mathbf {x}},\zeta )$$ is defined as either $$c({\mathbf {x}},\zeta )\, \triangleq \, 1-\left| \zeta ^T{\mathbf {x}}\right| ^m$$ where $$m\ge 0$$ (Setting A) or $$c({\mathbf {x}},\zeta ) \, \triangleq \, T{\mathbf {x}}\, - \, \zeta $$ (Setting B). We show that in either setting, by leveraging recent findings in the context of non-Gaussian integrals of positively homogenous functions, $${\mathbb {P}}\left\{ \,\zeta \, \mid \, \zeta \, \in \, {\mathbf {K}}({\mathbf {x}}) \, \right\} $$ can be expressed as the expectation of a suitably defined continuous function $$F(\bullet ,\xi )$$ with respect to an appropriately defined Gaussian density (or its variant), i.e. $${\mathbb {E}}_{{{\tilde{p}}}} \left[ \, F({\mathbf {x}},\xi )\, \right] $$ . Aided by a recent observation in convex analysis, we then develop a convex representation of the original problem requiring the minimization of $$g\left( {\mathbb {E}}\left[ \, F(\bullet ,\xi )\, \right] \right) $$ over $${\mathcal {X}}$$ , where g is an appropriately defined smooth convex function. Traditional stochastic approximation schemes cannot contend with the minimization of $$g\left( {\mathbb {E}}\left[ F(\bullet ,\xi )\right] \right) $$ over $$\mathcal X$$ , since conditionally unbiased sampled gradients are unavailable. We then develop a regularized variance-reduced stochastic approximation (r-VRSA) scheme that obviates the need for such unbiasedness by combining iterative regularization with variance-reduction. Notably, (r-VRSA) is characterized by almost-sure convergence guarantees, a convergence rate of $$\mathcal {O}(1/k^{1/2-a})$$ in expected sub-optimality where $$a > 0$$ , and a sample complexity of $$\mathcal {O}(1/\epsilon ^{6+\delta })$$ where $$\delta > 0$$ . To the best of our knowledge, this may be the first such scheme for probability maximization problems with convergence and rate guarantees. Preliminary numerics on a portfolio selection problem (Setting A) and a set-covering problem (Setting B) suggest that the scheme competes well with naive mini-batch SA schemes as well as integer programming approximation methods.

Joint Chance Constrained Probabilistic Simple Temporal Networks via Column Generation (Extended Abstract)

Probabilistic Simple Temporal Networks (PSTN) are used to represent scheduling problems under uncertainty. In a temporal network that is Strongly Controllable (SC) there exists a concrete schedule that is robust to any uncertainty. We solve the problem of determining Chance Constrained PSTN SC as a Joint Chance Constrained optimisation problem via column generation, lifting the usual assumptions of independence and Boole's inequality typically leveraged in PSTN literature. Our approach offers on average a 10 times reduction in cost versus previous methods.

Air-breathing hypersonic vehicle trajectory optimization with uncertain no-fly zones

The air-breathing hypersonic planes are considered to be the future of commercial airlines, which can fly from NYC to London in under an hour. For air-breathing vehicles, 3D trajectory planning will become extremely important due to its significant impact on flight performance. Past research on this issue was not comprehensive enough. Thus, we proposed a hybrid method for generating the optimal trajectories efficiently. No-fly zones are specified for geopolitical restrictions in air-breathing hypersonic vehicle missions. However, previous studies focused on no-fly zone constraints with fixed locations and boundaries. For robust execution, we must take into account no-fly zones’ uncertainties, which arise due to uncertain localization, measurement errors, and no-fly zones’ movement. A chance-constrained approach is presented here to deal with such uncertainties. The key idea of this method is controlling risk when the flight path is close to the uncertain no-fly zones. First, the trajectory planning of air-breathing hypersonic vehicles is modeled as a chance-constrained optimization (CCOPT) problem. With the help of the convexification and linearization techniques, we can approximate the non 3.2.2 CCOPT problem as a convex second-order cone programing under the Gaussian distribution assumption. It can be solved to global optimality by the interior point method. Finally, we give numerical simulation results to prove the effectiveness of this method.

Online Learning for Chance-Constrained Observer of Leading Heavy-Duty Vehicle Power Capability

This paper proposes a stochastic observer for estimating power capability of a preceding heavy-duty vehicle, using its speed measurement and road slope information. A chance-constrained optimisation problem is formulated to take into consideration the uncertainties associated with measurement error in the speed and imperfect knowledge of the road slope. An online learning approach is proposed to solve the chance-constrained optimisation problem, which learns probability distribution of the measurements along the travelled distance. The effectiveness of the proposed observer is analysed in two case studies on real road topographies and compared with an existing deterministic leading vehicle observer. The results show that the proposed leading vehicle observer is robust against uncertainties.