Stochastic Constraints Research Articles

Distributionally robust portfolio optimization with second-order stochastic dominance based on wasserstein metric

In portfolio optimization, we may be dealing with misspecification of a known distribution, that stock returns follow it. The unknown true distribution is considered in terms of a Wasserstein-neighborhood of P to examine the tractable formulations of the portfolio selection problem. This study considers a distributionally robust portfolio optimization problem with an ambiguous stochastic dominance constraint by assuming the unknown distribution of asset returns. The objective is to maximize the worst-case expected return and subject to an ambiguous second-order stochastic dominance constraint. The expected return robustly stochastically dominates the benchmark in the second order over all possible distributions within an ambiguity set. It is also shown that the Wasserstein-moment ambiguity set-based distributionally robust portfolio optimization can be reduced to a semidefinite program and second-order conic programming. We use a cutting plane to solve our second-order stochastic dominance constraint portfolio optimization problem with ambiguity sets based on the Wasserstein metric. Then we decompose this class of distributionally robust portfolio optimization into semi-infinite programming and apply the cutting surface method to solve it. The captured optimization programs are applied to real-life data for more efficient comparison. The problems are examined in depth using the optimal solutions of the optimization programs based on the different setups.

Optimal control of nonlinear systems with integer‐valued control inputs and stochastic constraints

AbstractPractical industrial process is usually a dynamic process including uncertainty. Stochastic constraints can be used for industrial process modeling, when system sate and/or control input constraints cannot be strictly satisfied. Thus, optimal control of nonlinear systems with stochastic constraints can be available to address practical industrial process problems with integer‐valued control inputs. In general, obtaining an analytical solution of this optimal control problem is very difficult due to the discrete nature of the control inputs and the complexity of stochastic constraints. To obtain a numerical solution, this problem is formulated as a finite dimensional static constrained optimization problem. First, the integer‐valued control input is relaxed into a continuous‐valued control input by imposing a penalty term on the objective function. Convergence results show that the relaxed solution obtained is an integer solution as long as the penalty parameter is sufficiently large. In addition, it should be noted that no any constraint is introduced in the novel relaxation technique, which can effectively avoid introducing any extra extreme point for the original optimal control problem. Next, a novel smooth approximation function is used to construct a subset of feasible region for this optimal control problem. It is proved that the smooth approximation can converge uniformly to the stochastic constraints as the adjusting parameter reduces. Following that, a numerical computation method is proposed for solving the original optimal control problem. Finally, in order to illustrate the effectiveness of the proposed method, an electric vehicle energy management problem is extended by considering some stochastic constraints. The numerical results show that the proposed method is less conservative compared with the existing typical approaches and can obtain a stable and robust performance when considering the initial condition small perturbations.

Short Communication: Minimal Quantile Functions Subject to Stochastic Dominance Constraints

Short Communication: Minimal Quantile Functions Subject to Stochastic Dominance Constraints

Optimal Energy-Efficient Antenna Selection and Power Adaptation for Interference-Outage Constrained Underlay Spectrum Sharing

Underlay spectrum sharing addresses spectrum scarcity but imposes constraints on the interference the secondary system causes to the primary receiver. For a secondary transmitter that is subject to the stochastic and general interference-outage constraint and the peak transmit power constraint, we present a novel joint antenna selection and power adaptation rule. It addresses the twin goals of improving the spectral efficiency and reducing the power consumed of an underlay secondary system with low hardware complexity and cost. We prove that the rule maximizes the energy-efficiency (EE) of the secondary system. Its form differs from all other rules considered in the literature. We then present an insightful geometrical characterization of the optimal power of the selected antenna in terms of the channel gains within the secondary system and between the secondary and primary systems. Our approach leads to several special cases, which are themselves novel, and novel analytical insights about the performance and structure of the optimal rule. We also present an iterative subgradient-based algorithm and a simpler non-iterative bound-based algorithm to compute the rule’s parameters. The rule achieves a markedly higher EE compared to conventional approaches, and serves as a new fundamental benchmark for antenna selection in underlay spectrum sharing.

Optimal Resource Allocation of 5G Machine-Type Communications for Situation Awareness in Active Distribution Networks

Advanced metering infrastructure targets the real-time collection of massive multisource data with smart meters (SMs), forming a key component of active distribution networks (ADNs). To achieve comprehensive situation awareness for ADNs, the use of high-speed and reliable two-way communications, such as 5G communications, is the basis. However, it is challenging to optimally allocate wireless resources for massive data, especially when emergency events occur randomly. This study discusses an energy-efficient and optimal wireless resource allocation method employing 5G uplink-based machine-type communications for achieving ADN situation awareness. We propose a 5G-based framework that can reasonably reserve resource blocks for emergency data and model the resource allocation scheme as a problem of maximizing energy efficiency (EE). Then, by using the sample average approximation theory, we transform the stochastic constraint and summarize the scheme as a mixed-integer linear programming problem. Finally, we construct an iterative algorithm based on the Lagrange dual algorithm for searching the global optimal allocation scheme. The results of simulations and experiments illustrate that even if emergency events occur randomly, our algorithm can optimally support the cooperative transmission of all sampled data while providing the best EE for large-scale SM communications.

Finding Feasible Systems for Subjective Constraints Using Recycled Observations

We consider the problem of finding a set of feasible or near-feasible systems among a finite number of simulated systems in the presence of stochastic constraints. When the constraints are subjective, a decision maker may want to test multiple threshold values for the constraints. Or the decision maker may simply want to determine how a set of feasible systems changes as constraints become more strict with the objective of pruning systems or finding the system with the best performance. When only the constraint thresholds change for the same set of underlying systems, it is natural to reuse observations collected from the feasibility check with a different threshold value. We present an indifference-zone procedure that recycles observations and provide an overall probability of correct decision for all threshold values. Our numerical experiments show that the proposed procedure performs well in reducing the required number of observations while providing a statistical guarantee on the probability of correct decision.Summary of Contribution: We consider the problem of determining the feasibility of a finite number of systems in the presence of subjective constraints on performance measures that can be estimated only by stochastic simulation. Specifically, our work focuses on the situation where the decision maker is willing to relax some constraint thresholds if necessary to achieve feasibility. Also, we discuss how our proposed procedures can help select the best system in the presence of multiple objectives. This is the first work that considers subjective constraints in the field of ranking and selection in simulation and it provides a practically useful decision-making tool with more flexibility on feasibility determination.History: Accepted by Bruno Tuffin, Area Editor for Simulation.Supplemental Material: The online supplement is available at https://doi.org/10.1287/ijoc.2022.1227 .

Constrained tracking control of stochastic multivariable nonlinear systems via Gaussian process predictions

To overcome the difficulty of propagating the stochastic uncertainties through a nonlinear model for successful online implementation of the stochastic nonlinear model predictive control (SNMPC) framework, this paper proposes the utilisation of Gaussian processes (GPs) in the context of SNMPC for the stochastic multivariable nonlinear systems to track a given trajectory in the presence of stochastic uncertainties and system constraints. Taking advantage of the GP regression which not only provides predictions, but also uncertainty quantification in the function estimation, the proposed GP-SNMPC architecture exploits the probability distribution of stochastic uncertainties and utilises the predictions provided by the GPs to formulate the model cost and constraint functions, which yields a tractable framework for handling nonlinear constrained control problems with Gaussian parametric uncertainties. By employing a cancellation strategy, the control law consists of two components, that is, tracking control law and disturbance rejection control law. Theoretical results regarding the performance bounds of the closed-loop system are derived. Numerical examples are provided to verify the effectiveness of the proposed GP-SNMPC scheme.

Moment based model predictive control for linear systems: Additive perturbations case

AbstractIn this article, we present a novel linear model predictive control (MPC) strategy for controlling uncertain dynamical systems. The moment based MPC strategy is based on the (centralized) moments of the stochastic cost and constraint functions. We show that moment based MPC formulations yield computationally tractable predictive control problems, due to the explicit analytic expressions of the moments and linearity in the dynamics. This article considers (i) the output regulation MPC configuration, where the output is steered towards a reference and state vector is not correctly accessible to the controller at the decision instants; (ii) the moment based MPC formulation with uncertainties that are non‐Gaussian random variables, and (iii) bound and polytopic constraints in the uncertain MPC problem subject to either Gaussian disturbances or uncertainties with symmetric distributions. We demonstrate the effectiveness of the proposed MPC technique on simulation examples such as a double integrator system and a quadruple tank system.

A Lyapunov-Based Methodology for Constrained Optimization with Bandit Feedback

In a wide variety of applications including online advertising, contractual hiring, and wireless scheduling, the controller is constrained by a stringent budget constraint on the available resources, which are consumed in a random amount by each action, and a stochastic feasibility constraint that may impose important operational limitations on decision-making. In this work, we consider a general model to address such problems, where each action returns a random reward, cost, and penalty from an unknown joint distribution, and the decision-maker aims to maximize the total reward under a budget constraint B on the total cost and a stochastic constraint on the time-average penalty. We propose a novel low-complexity algorithm based on Lyapunov optimization methodology, named LyOn, and prove that for K arms it achieves square root of KBlog(B) regret and zero constraint-violation when B is sufficiently large. The low computational cost and sharp performance bounds of LyOn suggest that Lyapunov-based algorithm design methodology can be effective in solving constrained bandit optimization problems.

A penalty function-based random search algorithm for optimal control of switched systems with stochastic constraints and its application in automobile test-driving with gear shifts

Practical industrial process is usually a dynamic process including uncertainty. Stochastic constraints can be used for industrial process modeling, when system sate and/or control input constraints cannot be strictly satisfied. Thus, optimal control of switched systems with stochastic constraints can be available to address practical industrial process problems with different modes. In general, obtaining an analytical solution of the optimal control problem is usually very difficult due to the discrete nature of the switching law and the complexity of stochastic constraints. To obtain a numerical solution, this problem is formulated as a constrained nonlinear parameter selection problem (CNPSP) based on a relaxation transformation (RT) technique, an adaptive sample approximation (ASA) method, a smooth approximation (SA) technique, and a control parameterization (CP) method. Following that, a penalty function-based random search (PFRS) algorithm is designed for solving the CNPSP based on a novel search rule-based penalty function (NSRPF) method and a novel random search (NRS) algorithm. The convergence results show that the proposed method is globally convergent. Finally, an optimal control problem in automobile test-driving with gear shifts (ATGS) is further extended to illustrate the effectiveness of the proposed method by taking into account some stochastic constraints. Numerical results show that compared with other typical methods, the proposed method is less conservative and can obtain a stable and robust performance when considering the small perturbations in initial system state. In addition, to balance the computation amount and the numerical solution accuracy, a tolerance setting method is also provided by the numerical analysis technique.

Semiparametric modelling of two-component mixtures with stochastic dominance.

In this work, we studied a two-component mixture model with stochastic dominance constraint, a model arising naturally from many genetic studies. To model the stochastic dominance, we proposed a semiparametric modelling of the log of density ratio. More specifically, when the log of the ratio of two component densities is in a linear regression form, the stochastic dominance is immediately satisfied. For the resulting semiparametric mixture model, we proposed two estimators, maximum empirical likelihood estimator (MELE) and minimum Hellinger distance estimator (MHDE), and investigated their asymptotic properties such as consistency and normality. In addition, to test the validity of the proposed semiparametric model, we developed Kolmogorov–Smirnov type tests based on the two estimators. The finite-sample performance, in terms of both efficiency and robustness, of the two estimators and the tests were examined and compared via both thorough Monte Carlo simulation studies and real data analysis.Supplementary InformationThe online version contains supplementary material available at 10.1007/s10463-022-00835-5.

Using stochastic hierarchical aggregation constraints to nowcast regional economic aggregates

Recent decades have seen advances in using econometric methods to produce more timely and higher frequency estimates of economic activity at the national level, enabling better tracking of the economy in real-time. These advances have not generally been replicated at the sub-national level, likely because of the empirical challenges that nowcasting at a regional level presents, notably, the short time series of available data, changes in data frequency over time, and the hierarchical structure of the data. This paper develops a mixed-frequency Bayesian VAR model to address common features of the regional nowcasting context, using an application to regional productivity in the UK. We evaluate the contribution that different features of our model provide to the accuracy of point and density nowcasts, in particular, the role of hierarchical aggregation constraints. We show that these aggregation constraints, imposed in stochastic form, play a crucial role in delivering improved regional nowcasts; they prove more important than adding region-specific predictors when the equivalent national data are known, but not when this aggregate is unknown.

Impact of implementing a price-based demand response program on the system reliability in security-constrained unit commitment problem coupled with wind farms in the presence of contingencies

Large scale penetration of highly intermittent wind energy sources has caused new challenges to the reliable operation of power systems. To address these challenges, demand response program is considered as an efficient solution to manage shift-able loads. In this paper, the impact of demand response on power system reliability has been evaluated by developing a two-stage stochastic security constraint unit commitment that considers both the wind volatility and line outages at the same time. The allocation of ramping products and spinning reserves to cover these uncertainties has also been modeled. wind curtailment and load shedding are considered as alternative strategies, and an optimal exchange between them and the use of spinning reserves and flexible ramping products has been modeled in the objective function. To assess the system reliability, the expected load not served (ELNS) index was evaluated in different cases. The proposed scheme as a mixed-integer linear programming problem was solved by a scenario-based two-stage stochastic programming method. The IEEE test systems 6-bus, modified 24-bus, and 118-bus were examined numerically. The results show that the simultaneous consideration of demand response and flexible ramping product coordinated with wind energy has adequate flexibility to reduce the reliability index and system operational cost.

Neural learning fixed-time adaptive tracking control of complex stochastic constraint nonlinear systems

This paper investigates the adaptive fixed-time tracking control problem for a class of stochastic nonlinear systems with constrained states and input saturation. By utilizing a barrier Lyapunov function (BLF), the issue caused by constrained states is settled. Simultaneously, a smooth function is constructed to approximate input saturation and the unknown nonlinear function is approximated by neural networks (NNs). In view of the fixed-time tracking control problem of stochastic nonlinear systems, a fixed-time stability lemma for stochastic nonlinear systems has been established and proved. Under the designed controller, the closed-loop system is semi-globally fixed-time stable in probability (SGFSP), and all the states are constrained within a compact set. Moreover, the controlled system in this paper achieves equilibrium within a fixed-time internal. Lastly, the presented method is verified by a simulation example.

Constrained tracking control of stochastic multivariable nonlinear systems using unscented Kalman filter

To overcome the challenges, including propagation of the stochastic uncertainties through a nonlinear model and full determination of the states over the prediction horizon for successful online implementation of the stochastic nonlinear model predictive control (SNMPC) framework, this paper proposes the utilisation of unscented Kalman filter (UKF) in the context of SNMPC for the stochastic multivariable nonlinear systems to track a given trajectory in the presence of stochastic uncertainties and system constraints. Taking advantage of the UKF which estimates and propagates the entire conditional distribution of the states along the time horizon by taking into account errors due to external disturbances and measurement noises as well as errors introduced due to the imperfect knowledge of the states through the state estimation, the proposed UKF-SNMPC framework utilises the conditional mean and conditional covariance of the states to reformulate the model cost and constraint functions, which yields a tractable framework for handling nonlinear chance-constrained tracking control problems. By employing a cancellation strategy, the control law consists of two components, that is, a tracking control law is designed to follow the desired trajectory and an uncertainty compensation control law designed to accommodate the nonlinearities in the plant. The concept of ‘robust horizon’ is introduced as the terminal constraint to prevent the covariance from the explosion and guarantee the convergence of the UKF-SNMPC framework. Performance bounds of the closed-loop system are theoretically analysed. Comparative studies on numerical examples are carried out to verify the effectiveness of the proposed UKF-SNMPC scheme.

Sample average approximation with heavier tails I: non-asymptotic bounds with weak assumptions and stochastic constraints

We derive new and improved non-asymptotic deviation inequalities for the sample average approximation (SAA) of an optimization problem. Our results give strong error probability bounds that are “sub-Gaussian” even when the randomness of the problem is fairly heavy tailed. Additionally, we obtain good (often optimal) dependence on the sample size and geometrical parameters of the problem. Finally, we allow for random constraints on the SAA and unbounded feasible sets, which also do not seem to have been considered before in the non-asymptotic literature. Our proofs combine different ideas of potential independent interest: an adaptation of Talagrand’s “generic chaining” bound for sub-Gaussian processes; “localization” ideas from the Statistical Learning literature; and the use of standard conditions in Optimization (metric regularity, Slater-type conditions) to control fluctuations of the feasible set.

Stochastic gravitational-wave background searches and constraints on neutron-star ellipticity

ABSTRACTRotating neutron stars (NSs) are promising sources of gravitational waves (GWs) in the frequency band of ground-based detectors. They are expected to emit quasi-monochromatic, long-duration GW signals, called continuous waves (CWs), due to their deviations from spherical symmetry. The degree of such deformations, and hence the information about the internal structure of an NS, is encoded in a dimension-less parameter ε called ellipticity. Searches for CW signals from isolated Galactic NSs have shown to be sensitive to ellipticities as low as $\varepsilon \sim \mathcal {O}(10^{-9})$. These searches are optimal for detecting and characterizing GWs from individual NSs, but they are not designed to measure the properties of NSs as population, such as the average ellipticity εav. These ensemble properties can be determined by the measurement of the stochastic gravitational-wave background (SGWB) arising from the superposition of GW signals from individually undetectable NSs. In this work, we perform a cross-correlation search for such a SGWB using the data from the first three observation runs of Advanced LIGO and Virgo. Finding no evidence for an SGWB signal, we set upper limits on the dimension-less energy density parameter Ωgw(f). Using these results, we also constrain the average ellipticity of Galactic NSs and five NS ‘hotspots’, as a function of the number of NSs emitting GWs within the frequency band of the search Nband. We find $\varepsilon _{\mathrm{av}} \lesssim 1.8 \times 10^{-8}$, with Nband = 1.6 × 107, for Galactic NSs, and $\varepsilon _{\mathrm{av}} \lesssim [3.5-11.8]\times 10^{-7}$, with Nband = 1.6 × 1010, for NS hotspots.

Stochastic Generation Scheduling of Insular Grids with High Penetration of Photovoltaic and Battery Energy Storage Systems: South Andaman Island Case Study

Insular grids are fragile owing to lower inertia and the absence of interconnection with other grids. With the increasing penetration of non-dispatchable renewable energy sources, the vulnerability of such insular grids increases further. The government of India has proposed several projects to improve the photovoltaic systems (PV) penetration in the Andaman and Nicobar Islands’ grid. This paper investigates joint stochastic scheduling of energy and reserve generation for insular grids fed from diesel and gas-based generators, PV, and battery energy storage systems (BESS). The proposed stochastic scheduling model considers a wide range of probabilistic forecast scenarios instead of a deterministic model that assumes a single-point forecast. Hence, it provides an optimal solution that is technically feasible for a wide range of PV power forecast scenarios. The striking feature of the model developed in this work is the inclusion of stochastic constraints that represent (i) the coordination between PV and BESS, (ii) reserve constraints, (iii) battery charging/discharging limit constraints, and (iv) non-anticipatory constraints that ensure technical viability of scheduling decisions. The proposed model is validated on the dataset for South Andaman Island. Results reveal the applicability and feasibility of the proposed stochastic dispatch model for different generation mix scenarios.

Robust job-shop scheduling under deterministic and stochastic unavailability constraints due to preventive and corrective maintenance

This paper presents a modeling and solution approach for a robust job-shop scheduling problem (JSP) under deterministic and stochastic machine unavailability caused, respectively, by planned preventive maintenance (PM) and unplanned corrective maintenance (CM) following random breakdowns. The goal is to optimize the sequence of jobs and run-based planned preventive maintenance tasks in a robust manner while considering the degradation of machines over time. The time to failure for each machine is assumed to follow a Weibull distribution. The robust objective function to be minimized is a weighted sum of the expected values of the makespan and the gross positive deviation between the actual and planned start times of the jobs, as proxies for quality robustness and solution robustness, respectively. Two metaheuristic algorithms that aim at providing a good balance between performance quality and solution robustness are developed. In both algorithms, the “true” makespan objective function (with both PM and CM) is approximated using three surrogate measures. Genetic algorithm (GA) is first used to optimize the surrogate functions, then the fittest solutions from the three oracles are simulated with random breakdown scenarios and the best among them is introduced as an elite member in the next GA iteration. The first algorithm terminates as soon as the solution obtained is worse than the best know solution or when the maximum number of iterations is reached, whereas the second algorithm applies a rule inspired by Simulated Annealing for termination. Numerical experimentation on benchmark instances from the literature showed excellent performance of the proposed algorithms in terms of average marginal improvement and runtime. These results show that the proposed framework can generate high-quality robust job shop schedules under deterministic and stochastic machine unavailability constraints. Moreover, the sensitivity analysis results recommended some key insights and directions for future research.

Distributionally robust optimization with multivariate second-order stochastic dominance constraints with applications in portfolio optimization

In this paper, we study the stochastic optimization problem with multivariate second-order stochastic dominance (MSSD) constraints where the distribution of uncertain parameters is unknown. Instead, only some historical data are available. Using the Wasserstein metric, we construct an ambiguity set and develop a data-driven distributionally robust optimization model with multivariate second-order stochastic dominance constraints (DROMSSD). By utilizing the linear scalarization function, we transform MSSD constraints into univariate constraints. We present a stability analysis focusing on the impact of the variation of the ambiguity set on the optimal value and optimal solutions. Moreover, we carry out quantitative stability analysis for the DROMSSD problems as the sample size increases. Specially, in the context of the portfolio, we propose a convex lower reformulation of the corresponding DROMSSD models under some mild conditions. Finally, some preliminary numerical test results are reported. We compare the DROMSSD model with the sample average approximation model through out-of-sample performance, certificate and reliability. We also use real stock data to verify the effectiveness of the DROSSM model.