Conditional Value-at-risk Constraints Research Articles

Optimal Investment Strategy for DC Pension Plan with Stochastic Salary and Value at Risk Constraint in Stochastic Volatility Model

This paper studies the optimal asset allocation problem of a defined contribution (DC) pension plan with a stochastic salary and value under a constraint within a stochastic volatility model. It is assumed that the financial market contains a risk-free asset and a risky asset whose price process satisfies the Stein–Stein stochastic volatility model. To comply with regulatory standards and offer a risk management tool, we integrate the dynamic versions of Value-at-Risk (VaR), Conditional Value-at-Risk (CVaR), and worst-case CVaR (wcCVaR) constraints into the DC pension fund management model. The salary is assumed to be stochastic and characterized by geometric Brownian motion. In the dynamic setting, a CVaR/wcCVaR constraint is equivalent to a VaR constraint under a higher confidence level. By using the Lagrange multiplier method and the dynamic programming method to maximize the constant absolute risk aversion (CARA) utility of terminal wealth, we obtain closed-form expressions of optimal investment strategies with and without a VaR constraint. Several numerical examples are provided to illustrate the impact of a dynamic VaR/CVaR/wcCVaR constraint and other parameters on the optimal strategy.

Joint Maritime Bunker Hedging and Operational Consumption Based on CVaR Optimization

Maritime shipping is an important driver of global economic growth. Efficient green maritime technologies are critical for both the profitability and sustainability of shipping carriers due to the fact that fuel consumption has already made up 45–55% of the total operational cost of a ship. Moreover, the application of green maritime technologies also challenges the input/output of the maritime industry. Currently, there is a lack of coordination between the two strategies of maritime bunker management: one is the bunker procurement, which is faced with the fierce volatility of bunker fuel prices, and the other is the bunker consumption of vessel operation scheduling with applicable maritime technologies. To address the challenge posed by the new sulfur emission regulations, the two isolated strategies are inefficient. This study presents an integrated model that takes both the financial technology (bunker hedging) and the operational bunker cost efficiency (sailing speed and routing optimization under emission regulations) into account. The objective is to maximize the total rate of portfolio return considering the revenue and the cost simultaneously. By analyzing the Conditional Value at Risk (CVaR) risk measure, we examined the effects of the bunker spot, contract, and hedging in futures markets on the optimal joint solution. Numerical results from a real-world case study show that the optimized integrating financial and operational strategies yield the lowest expected total costs as well as the highest revenue with CVaR constraints. The findings provide a prospect for maritime shipping as an effective decision tool for bunker management under environmental regulations. The management insights of our study will benefit the corporate participants, policy makers, and researchers in liner shipping revenue and risk management.

Minimum capital requirement and portfolio allocation for non-life insurance: a semiparametric model with Conditional Value-at-Risk (CVaR) constraint

We present an optimization problem to determine the minimum capital requirement for a non-life insurance company. The optimization problem imposes a non-positive Conditional Value-at-Risk (CVaR) of the insurer’s net loss and a portfolio performance constraint. When expressing the optimization problem in a semiparametric form, we demonstrate its convexity for any integrable random variable representing the insurer’s liability. Furthermore, we prove that the function defining the CVaR constraint in the semiparametric formulation is continuously differentiable when the insurer’s liability has a continuous distribution. We use the Kelley-Cheney-Goldstein algorithm to solve the optimization problem in the semiparametric form and show its convergence. An empirical analysis is carried out by assuming three different liability distributions: a lognormal distribution, a gamma distribution, and a mixture of Erlang distributions with a common scale parameter. The numerical experiments show that the choice of the liability distribution plays a crucial role since marked differences emerge when comparing the mixture distribution with the other two distributions. In particular, the mixture distribution describes better the right tail of the empirical distribution of liabilities with respect to the other two distributions and implies higher capital requirements and different assets in the optimal portfolios.

Efficient Off-Policy Safe Reinforcement Learning Using Trust Region Conditional Value At Risk

This letter aims to solve a safe reinforcement learning (RL) problem with risk measure-based constraints. As risk measures, such as conditional value at risk (CVaR), focus on the tail distribution of cost signals, constraining risk measures can effectively prevent a failure in the worst case. An on-policy safe RL method, called TRC, deals with a CVaR-constrained RL problem using a trust region method and can generate policies with almost zero constraint violations with high returns. However, to achieve outstanding performance in complex environments and satisfy safety constraints quickly, RL methods are required to be sample efficient. To this end, we propose an off-policy safe RL method with CVaR constraints, called off-policy TRC. If off-policy data from replay buffers is directly used to train TRC, the estimation error caused by the distributional shift results in performance degradation. To resolve this issue, we propose novel surrogate functions, in which the effect of the distributional shift can be reduced, and introduce an adaptive trust-region constraint to ensure a policy not to deviate far from replay buffers. The proposed method has been evaluated in simulation and real-world environments and satisfied safety constraints within a few steps while achieving high returns even in complex robotic tasks.

Optimal trade-off of integrated river basin water resources allocation considering water market: A bi-level multi-objective model with conditional value-at-risk constraints

The predictable water shortage crisis is intensifying due to frequent extreme weather caused by climate change and human activities. Motivated by urgent practical needs of more scientific and balanced river basin water resources allocation (RBWRA) schemes, a comprehensive bi-level multi-objective programming (BLMOP) model for RBWRA considering uncertain risks and water market is constructed in this paper, which pursues the objectives of environmental protection, equality, and economic development. River basin authority, as the leader of RBWRA, is committed to reducing pollutants while introducing Gini coefficient to maintain the equality of economic development between sub-areas. Meanwhile, the sub-areas (i.e., followers) are dedicated to improving the economic benefit. In addition, the water market is introduced as water redistribution mechanism and its effect is probed into in detail. Moreover, conditional value-at-risk (CVaR) constraints are integrated into the model to control the economic loss caused by the risks of uncertain water availability. To solve the BLMOP model, a novel bi-level interaction based adaptive global equilibrium optimization (BI-aGEO) approach is proposed, which can obtain the best equilibrium solution that satisfies all decision-making participants. Furthermore, the proposed model and approach are applied to the Hanjiang River Basin in central China to elaborate on their feasibility and effectiveness. Results show that the proposed model is effective in ensuring the sustainability, equality and efficiency of RBWRA. The introduction of water market is shown to be able to improve water efficiency and promote water conservation and CVaR constraints play a significant role in controlling economic risks. Finally, some managerial suggestions are provided for different RBWRA participants for more scientific and effective integrated water resources management (IWRM).

TRC: Trust Region Conditional Value at Risk for Safe Reinforcement Learning

As safety is of paramount importance in robotics, reinforcement learning that reflects safety, called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">safe RL</i> , has been studied extensively. In safe RL, we aim to find a policy which maximizes the desired return while satisfying the defined safety constraints. There are various types of constraints, among which constraints on conditional value at risk (CVaR) effectively lower the probability of failures caused by high costs since CVaR is a conditional expectation obtained above a certain percentile. In this paper, we propose a trust region-based safe RL method with CVaR constraints, called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">TRC</i> . We first derive the upper bound on CVaR and then approximate the upper bound in a differentiable form in a trust region. Using this approximation, a subproblem to get policy gradients is formulated, and policies are trained by iteratively solving the subproblem. TRC is evaluated through safe navigation tasks in simulations with various robots and a sim-to-real environment with a Jackal robot from Clearpath. Compared to other safe RL methods, the performance is improved by 1.93 times while the constraints are satisfied in all experiments.

Distributionally robust and risk-averse optimisation for the stochastic multi-product disassembly line balancing problem with workforce assignment

Existing works usually focus on the single-product disassembly line balancing problem (DLBP). In practice, end-of-life (EOL) products to be disassembled may be heterogeneous, and the actual processing time of each task may vary with its assigned worker. This work studies a stochastic multi-product DLBP with workforce assignment, to minimise the system cost. Due to historical data scarcity, we assume that only partial distributional information of uncertain task processing times is known. Exceeding the preset cycle time may lead to a disassembly performance reduction, thus we control the cycle time violation via conditional Value-at-Risk (CVaR) constraints, i.e. in a risk-averse fashion. For the problem, we first propose a novel formulation with distributionally robust CVaR constraints. Then some valid inequalities are proposed, leading to an improved model. Two solution approaches, i.e. an exact cutting-plane method and an approximation method, are further proposed and compared, via numerical experiments. Some managerial insights are also drawn.

Online learning based risk-averse stochastic MPC of constrained linear uncertain systems

This paper investigates the problem of designing data-driven stochastic Model Predictive Control (MPC) for linear time-invariant systems under additive stochastic disturbance, whose probability distribution is unknown but can be partially inferred from data. We propose a novel online learning based risk-averse stochastic MPC framework in which Conditional Value-at-Risk (CVaR) constraints on system states are required to hold for a family of distributions called an ambiguity set. The ambiguity set is constructed from disturbance data by leveraging a Dirichlet process mixture model that is self-adaptive to the underlying data structure and complexity. Specifically, the structural property of multimodality is exploited, so that the first- and second-order moment information of each mixture component is incorporated into the ambiguity set. A novel constraint tightening strategy is then developed based on an equivalent reformulation of distributionally robust CVaR constraints over the proposed ambiguity set. As more data are gathered during the runtime of the controller, the ambiguity set is updated online using real-time disturbance data, which enables the risk-averse stochastic MPC to cope with time-varying disturbance distributions. The online variational inference algorithm employed does not require all collected data be learned from scratch, and therefore the proposed MPC is endowed with the guaranteed computational complexity of online learning. The guarantees on recursive feasibility and closed-loop stability of the proposed MPC are established via a safe update scheme. Numerical examples are used to illustrate the effectiveness and advantages of the proposed MPC.

A vehicle routing problem with distribution uncertainty in deadlines

This article considers a stochastic vehicle routing problem with probability constraints. The probability that customers are served before their (uncertain) deadlines must be higher than a pre-specified target. It is unrealistic to expect that the perfect knowledge on the probability distributions of deadlines is always available. To this end, we propose a distributionally robust optimisation framework to study worst bounds of the problem, which exploits the moment information of the historical observations. This framework includes two steps. We first use Conditional Value-at-Risk (CVaR) as a risk approximation to the probability of missing customer deadlines. The resulting nonlinear model is then transformed into a semi-infinite mixed integer program, using the dual form of the CVaR approximation. A sample approximation approach is then used to address the computational challenges. As the standard CVaR approximation to probability constraints is rather conservative, we suggest a relaxation to the approximation and develop an iterative algorithm to find the right value of the parameter that is introduced to the relaxed CVaR constraints. The extensive numerical experiments show that the routing policies developed by the proposed solution framework are robust and able to achieve the required target, regardless of deadline distributions.

Segregated Linear Decision Rules for Distributionally Robust Control With Linear Dynamics and Quadratic Cost

In this article, we investigate the multistage stochastic constrained control problem with linear dynamics and quadratic costs when only partial information of the disturbance distribution (i.e., the first two moments) is known. We adopt a distributionally robust chance constraint (DRCC)-based approach, where the DRCC holds (with high probability) as long as the true distribution of uncertainty belongs to a distribution family or the ambiguity set (constructed based on the known information). We approximate the DRCC with the worst-case conditional value-at-risk (CVaR) constraint, which bounds the expected constraint violation with respect to all distributions in the ambiguity set. The worst-case CVaR problem is known to be computationally tractable under linear decision rules (LDRs). To improve the performance of LDRs, we propose to apply the segregated linear decision rules (SLDRs) on dynamical control systems with the worst-case CVaR approximation. To deal with the tractability issue of the worst-case CVaR constraint, we construct a special group of segregation (of the random disturbance) that is shown to be without loss of optimality. The proposed segregation method enables us to establish the equivalence between the stochastic control problem (with worst-case CVaR constraints and under SLDRs) and a tractable semidefinite program.

Distributionally Robust CVaR Constraints for Power Flow Optimization

Recent research on optimal power flow (OPF) in networks with renewable power involves optimizing both first and second stage variables that adjust the decision once the uncertainty is revealed. In general, only partial information on the underlying probability distribution of renewable power production is available. This paper considers a distributionally robust framework for solving the OPF problem. The formulation stipulates a probability mass function of wind power production, whose probabilities and scenario locations vary in a box of ambiguity with bounds that can be tuned based on historical data. Distributionally robust optimization (DRO) is used to derive new conditional value-at-risk (CVaR) constraints that limit the frequency and severity of branch flow limit violations whenever the renewable power generation deviates from its forecast. Numerical results are reported on networks with up to 2736 nodes and contrasted with classical robust optimization (RO) and stochastic optimization (SO) solutions. The results show an advantage to adopting the proposed DRO for load flow control. In particular, the solution benefits from the observed correlation amongst the uncertain parameters to mitigate branch flow limit violations, and yet maintain an acceptable worst-case expected operational cost as compared to RO and SO solutions.

Risk-Aware Motion Planning and Control Using CVaR-Constrained Optimization

We propose a risk-aware motion planning and decision-making method that systematically adjusts the safety and conservativeness in an environment with randomly moving obstacles. The key component of this method is the conditional value-at-risk (CVaR) used to measure the safety risk that a robot faces. Unlike chance constraints, CVaR constraints are coherent, convex, and distinguish between tail events. We propose a two-stage method for safe motion planning and control: A reference trajectory is generated by using RRT* in the first stage, and then a receding horizon controller is employed to limit the safety risk by using CVaR constraints in the second stage. However, the second stage problem is nontrivial to solve, as it is a triple-level stochastic program. We develop a computationally tractable approach through 1) a reformulation of the CVaR constraints; 2) a sample average approximation; and 3) a linearly constrained mixed integer convex program formulation. The performance and utility of this risk-aware method are demonstrated via simulation using a 12-dimensional model of quadrotors.

Varying confidence levels for CVaR risk measures and minimax limits

Conditional value at risk (CVaR) has been widely studied as a risk measure. In this paper we add to this work by focusing on the choice of confidence level and its impact on optimization problems with CVaR appearing in the objective and also the constraints. We start by considering a problem in which CVaR is minimized and investigate the way in which it approximates the minimax robust optimization problem as the confidence level is driven to one. We make use of a consistent tail condition which ensures that the CVaR of a random function will converge uniformly to its supremum as the confidence level increases, and establish an error bound for the CVaR optimal solution under second order growth conditions. The results are extended to a minimization problem with a constraint on the CVaR value which in the limit as the confidence level approaches one coincides with a problem having semi-infinite constraints. We study the sample average approximation scheme for the CVaR constraints and establish an exponential rate of convergence for the sample averaged optimal solution. We propose a procedure to explore the possibility of varying the confidence level to a lower value which can give an advantage when there is a need to find good solutions to CVaR-constrained problems out of sample. Our numerical results demonstrate that using the optimal solution to an adjusted problem with lower confidence level can lead to better overall performance.

Fitting heavy-tailed mixture models with CVaR constraints

Abstract Standard methods of fitting finite mixture models take into account the majority of observations in the center of the distribution. This paper considers the case where the decision maker wants to make sure that the tail of the fitted distribution is at least as heavy as the tail of the empirical distribution. For instance, in nuclear engineering, where probability of exceedance (POE) needs to be estimated, it is important to fit correctly tails of the distributions. The goal of this paper is to supplement the standard methodology and to assure an appropriate heaviness of the fitted tails. We consider a new Conditional Value-at-Risk (CVaR) distance between distributions, that is a convex function with respect to weights of the mixture. We have conducted a case study demonstrating e˚ciency of the approach. Weights of mixture are found by minimizing CVaR distance between the mixture and the empirical distribution. We have suggested convex constraints on weights, assuring that the tail of the mixture is as heavy as the tail of empirical distribution.

Extended cross decomposition for mixed-integer linear programs with strong and weak linking constraints

Large-scale mixed-integer linear programming (MILP) problems, such as those from two-stage stochastic programming, usually have a decomposable structure that can be exploited to design efficient optimization methods. Classical Benders decomposition can solve MILPs with weak linking constraints (which are decomposable when linking variables are fixed) but not strong linking constraints (which are not decomposable even when linking variables are fixed). In this paper, we first propose a new rigorous bilevel decomposition strategy for solving MILPs with strong and weak linking constraints, then extend a recently developed cross decomposition method based on this strategy. We also show how to apply the extended cross decomposition method to two-stage stochastic programming problems with conditional-value-at-risk (CVaR) constraints. In the case studies, we demonstrate the significant computational advantage of the proposed extended cross decomposition method as well as the benefit of including CVaR constraints in stochastic programming.

EPO Dosage Optimization for Anemia Management: Stochastic Control under Uncertainty Using Conditional Value at Risk

Due to insufficient endogenous production of erythropoietin, chronic kidney disease patients with anemia are often treated by the administration of recombinant human erythropoietin (EPO). The target of the treatment is to keep the patient’s hemoglobin level within a normal range. While conventional methods for guiding EPO dosing used by clinicians normally rely on a set of rules based on past experiences or retrospective studies, model predictive control (MPC) based dosage optimization is receiving attention recently. The objective of this paper is to incorporate the hemoglobin response model uncertainty into the dosage optimization decision making. Two methods utilizing Conditional Value at Risk (CVaR) are proposed for hemoglobin control in chronic kidney disease under model uncertainty. The first method includes a set-point tracking controller with the addition of CVaR constraints. The second method involves the use of CVaR directly in the cost function of the optimal control problem. The methods are compared to set-point tracking MPC and Zone-tracking MPC through computer simulations. Simulation results demonstrate the benefits of utilizing CVaR in stochastic predictive control for EPO dosage optimization.

Index tracking model, downside risk and non-parametric kernel estimation

In this paper, we propose an index tracking model with the conditional value-at-risk (CVaR) constraint based on a non-parametric kernel (NPK) estimation framework. In theory, we demonstrate that the index tracking model with the CVaR constraint is a convex optimization problem. We then derive NPK estimators for tracking errors and CVaR, and thereby construct the NPK index tracking model. Monte Carlo simulations show that the NPK method outperforms the linear programming (LP) method in terms of estimation accuracy. In addition, the NPK method can enhance computational efficiency when the sample size is large. Empirical tests show that the NPK method can effectively control downside risk and obtain higher excess returns, in both bearish and bullish market environments.

Optimal Control of Conditional Value-at-Risk in Continuous Time

We consider continuous-time stochastic optimal control problems featuring conditional value-at-risk (CVaR) in the objective. The major difficulty in these problems arises from time inconsistency, which prevents us from directly using dynamic programming. To resolve this challenge, we convert to an equivalent bilevel optimization problem in which the inner optimization problem is standard stochastic control. Furthermore, we provide conditions under which the outer objective function is convex and differentiable. We compute the outer objective's value via a Hamilton--Jacobi--Bellman equation and its gradient via the viscosity solution of a linear parabolic equation, which allows us to perform gradient descent. The significance of this result is that we provide an efficient dynamic-programming-based algorithm for optimal control of CVaR without lifting the state space. To broaden the applicability of the proposed algorithm, we propose convergent approximation schemes in cases where our key assumptions do not hold and characterize relevant suboptimality bounds. In addition, we extend our method to a more general class of risk metrics, which includes mean variance and median deviation. We also demonstrate a concrete application to portfolio optimization under CVaR constraints. Our results contribute an efficient framework for solving time-inconsistent CVaR-based sequential optimization.

Optimal scheduling of hydro-thermal power systems considering the flood risk of cascade reservoirs

ABSTRACTThe increased flood risk linked to global warming affects the safety of cascade reservoirs, with direct effects on the stable operation of power systems. In this article, an optimal scheduling framework for hydro-thermal power systems considering the flood risk of cascade reservoirs is presented. First, the extreme value theory-based peak over threshold model is adopted to build a generalized Pareto distribution of extreme flood inflow for a single reservoir. Then, the Copula function is used to build a joint probability distribution function of extreme inflow for cascade reservoirs during a flood period. Based on the superior performance of the conditional value at risk (CVaR) in characterizing the tail risk of the cascade reservoir spillway safety margin, a CVaR constraint for cascade reservoir flood prevention is proposed, and a scheduling model for hydro-thermal power systems considering the flood prevention risk of head-dependent cascade reservoirs is presented. Secondly, Rockafeller and Uryasey reformulation and sample average approximation are employed to transform the proposed model with a CVaR constraint into a convex solvable optimization model. Finally, a modified IEEE 14-node system is used to verify the better performance of the proposed model than that of the models with the chance constraint and with the independent normal distribution of extreme flood inflow. The impacts of flood prevention confidence level and Monte Carlo sample size on the optimal scheduling results are analysed quantitatively.

Optimal reinsurance under dynamic VaR constraint

This paper deals with the optimal reinsurance strategy from an insurer’s point of view. Our objective is to find the optimal policy that maximises the insurer’s survival probability. To meet the requirement of regulators and provide a tool to risk management, we introduce the dynamic version of Value-at-Risk (VaR), Conditional Value-at-Risk (CVaR) and worst-case CVaR (wcCVaR) constraints in diffusion model and the risk measure limit is proportional to company’s surplus in hand. In the dynamic setting, a CVaR/wcCVaR constraint is equivalent to a VaR constraint under a higher confidence level. Applying dynamic programming technique, we obtain closed form expressions of the optimal reinsurance strategies and corresponding survival probabilities under both proportional and excess-of-loss reinsurance. Several numerical examples are provided to illustrate the impact caused by dynamic VaR/CVaR/wcCVaR limit in both types of reinsurance policy.