CVaR-constrained stochastic programming reformulation for stochastic nonlinear complementarity problems

We reformulate a stochastic nonlinear complementarity problem as a stochastic programming problem which minimizes an expected residual defined by a restricted NCP function with nonnegative constraints and CVaR constraints which guarantee the stochastic nonlinear function being nonnegative with a high probability. By applying smoothing technique and penalty method, we propose a penalized smoothing sample average approximation algorithm to solve the CVaR-constrained stochastic programming. We show that the optimal solution of the penalized smoothing sample average approximation problem converges to the solution of the corresponding nonsmooth CVaR-constrained stochastic programming problem almost surely. Finally, we report some preliminary numerical test results.     

A shortage probability metric for battery depletion risk

In this paper, we propose a battery depletion risk (BDR) metric that evaluates the probability of depleting an energy storage system in the context of a centralized power system with stochastic demand and supply for electricity across multiple consecutive time periods. This newly proposed reliability metric informs battery sizing decisions to mitigate outage risks. We demonstrate the applicability of this metric by embedding it in an economic dispatch model with thermal generation units and a centralized energy storage system.     

A smoothing sample average approximation method for stochastic optimization problems with CVaR risk measure

This paper is concerned with solving single CVaR and mixed CVaR minimization problems. A CHKS-type smoothing sample average approximation (SAA) method is proposed for solving these two problems, which retains the convexity and smoothness of the original problem and is easy to implement. For any fixed smoothing constant ε, this method produces a sequence whose cluster points are weak stationary points of the CVaR optimization problems with probability one. This framework of combining smoothing technique and SAA scheme can be extended to other smoothing functions as well. Practical numerical examples arising from logistics management are presented to show the usefulness of this method.     

基于CVaR投资组合优化问题的光滑化方法

对选定的风险资产进行组合投资,以条件风险价值(CVaR)作为度量风险的工具,建立单期投资组合优化问题的CVaR模型。目标函数中含有多重积分与plus函数,产生情景矩阵将多重积分计算转化成求和运算,提出plus函数的一个新的一致光滑逼近函数并给出求解CVaR模型的光滑化方法,最后的实证研究表明了本文算法的优越性。     

TTRISK: Tensor train decomposition algorithm for risk averse optimization

This article develops a new algorithm named TTRISK to solve high-dimensional risk-averse optimization problems governed by differential equations (ODEs and/or partial differential equations [PDEs]) under uncertainty. As an example, we focus on the so-called Conditional Value at Risk (CVaR), but the approach is equally applicable to other coherent risk measures. Both the full and reduced space formulations are considered. The algorithm is based on low rank tensor approximations of random fields discretized using stochastic collocation. To avoid nonsmoothness of the objective function underpinning the CVaR, we propose an adaptive strategy to select the width parameter of the smoothed CVaR to balance the smoothing and tensor approximation errors. Moreover, unbiased Monte Carlo CVaR estimate can be computed by using the smoothed CVaR as a control variate. To accelerate the computations, we introduce an efficient preconditioner for the Karush–Kuhn–Tucker (KKT) system in the full space formulation.The numerical experiments demonstrate that the proposed method enables accurate CVaR optimization constrained by large-scale discretized systems. In particular, the first example consists of an elliptic PDE with random coefficients as constraints. The second example is motivated by a realistic application to devise a lockdown plan for United Kingdom under COVID-19. The results indicate that the risk-averse framework is feasible with the tensor approximations under tens of random variables.     

Sample average approximation of expected value constrained stochastic programs

We propose a sample average approximation (SAA) method for stochastic programming problems with expected value constraints. Such problems arise, for example, in portfolio selection with constraints on conditional value-at-risk (CVaR). We provide a convergence analysis and a statistical validation scheme for the proposed method.     

规避风险的多阶段最优库存研究

基于市场需求是随机的,并且在进行市场销售前,就要确定每个阶段的生产数量的背景下,建立了具有规避风险的多阶段库存凸随机规划模型.该模型以最小化损失函数的期望值为目标函数,以规避风险为约束条件,以价值风险(VaR)和条件价值风险(CVaR)为风险度量;采用样本平均近似方法(SAA)求解该模型,并分析样本平均近似方法的收敛性;最后,给出数值结果.     

A Smoothing SAA Method for a Stochastic Linear Complementarity Problem

Utilizing the well-known aggregation technique, we propose a smoothing sample average approximation (SAA) method for a stochastic linear complementarity problem, where the underlying functions are represented by expectations of stochastic functions. The method is proved to be convergent and the preliminary numerical results are reported.     

Stochastic Optimization Problems with CVaR Risk Measure and Their Sample Average Approximation

We provide a refined convergence analysis for the SAA (sample average approximation) method applied to stochastic optimization problems with either single or mixed CVaR (conditional value-at-risk) measures. Under certain regularity conditions, it is shown that any accumulation point of the weak GKKT (generalized Karush-Kuhn-Tucker) points produced by the SAA method is almost surely a weak stationary point of the original CVaR or mixed CVaR optimization problems. In addition, it is shown that, as the sample size increases, the difference of the optimal values between the SAA problems and the original problem tends to zero with probability approaching one exponentially fast.     

On reformulations for the one-warehouse multi-retailer problem

Computing optimal stochastic portfolio execution strategies under an appropriate risk consideration presents many computational challenges. Using Monte Carlo simulations, we investigate an approach based on smoothing and parametric rules to minimize mean and Conditional Value-at-Risk (CVaR) of the execution cost. The proposed approach reduces computational complexity by smoothing the nondifferentiability arising from the simulation discretization and by employing a parametric representation of a stochastic strategy. We further handle constraints using a smoothed exact penalty function. Using the downside risk as an example, we show that the proposed approach can be generalized to other risk measures. In addition, we computationally illustrate the effect of including risk on the stochastic optimal execution strategy.     

An exact method for constrained maximization of the conditional value-at-risk of a class of stochastic submodular functions

We consider a class of risk-averse submodular maximization problems (RASM) where the objective is the conditional value-at-risk (CVaR) of a random nondecreasing submodular function at a given risk level. We propose valid inequalities and an exact general method for solving RASM under the assumption that we have an efficient oracle that computes the CVaR of the random function. We demonstrate the proposed method on a stochastic set covering problem that admits an efficient CVaR oracle for the random coverage function.     

Stochastic Multiobjective Optimization: Sample Average Approximation and Applications

We investigate one stage stochastic multiobjective optimization problems where the objectives are the expected values of random functions. Assuming that the closed form of the expected values is difficult to obtain, we apply the well known Sample Average Approximation (SAA) method to solve it. We propose a smoothing infinity norm scalarization approach to solve the SAA problem and analyse the convergence of efficient solution of the SAA problem to the original problem as sample sizes increase. Under some moderate conditions, we show that, with probability approaching one exponentially fast with the increase of sample size, an ϵ-optimal solution to the SAA problem becomes an ϵ-optimal solution to its true counterpart. Moreover, under second order growth conditions, we show that an efficient point of the smoothed problem approximates an efficient solution of the true problem at a linear rate. Finally, we describe some numerical experiments on some stochastic multiobjective optimization problems and report preliminary results.     

Asymptotic Analysis of Sample Average Approximation for Stochastic Optimization Problems with Joint Chance Constraints via Conditional Value at Risk and Difference of Convex Functions

Conditional Value at Risk (CVaR) has been recently used to approximate a chance constraint. In this paper, we study the convergence of stationary points, when sample average approximation (SAA) method is applied to a CVaR approximated joint chance constrained stochastic minimization problem. Specifically, we prove under some moderate conditions that optimal solutions and stationary points, obtained from solving sample average approximated problems, converge with probability one to their true counterparts. Moreover, by exploiting the recent results on large deviation of random functions and sensitivity results for generalized equations, we derive exponential rate of convergence of stationary points. The discussion is also extended to the case, when CVaR approximation is replaced by a difference of two convex functions (DC-approximation). Some preliminary numerical test results are reported.     

Monte Carlo methods for mean-risk optimization and portfolio selection

Stochastic programming is a well-known instrument to model many risk management problems in finance. In this paper we consider a stochastic programming model where the objective function is the variance of a random function and the constraint function is the expected value of the random function. Instead of using popular scenario tree methods, we apply the well-known sample average approximation (SAA) method to solve it. An advantage of SAA is that it can be implemented without knowing the distribution of the random data. We investigate the asymptotic properties of statistical estimators obtained from the SAA problem including examining the rate of convergence of optimal solutions of the SAA problem as sample size increases. By using the classical penalty function technique and recent results on uniform exponential convergence of sample average random functions, we show that under some mild conditions the statistical estimator of the optimal solution converges to its true counterpart at an exponential rate. We apply the proposed model and the numerical method to a portfolio management problem and present some numerical results.     

基于CVaR的债券投资组合模型

针对债券投资组合中的风险度量难题,用CVaR作为风险度量方法,构建了基于CVaR的债券投资组合优化模型.采用历史模拟算法处理模型中的随机收益率向量,将随机优化模型转化为确定性优化模型,并且证明了算法的收敛性.通过线性化技术处理CVaR中的非光滑函数,将该模型转化为一般的线性规划模型.结合10只债券的组合投资实例,验证了模型与算法的有效性.     

求解带有交易费的CVaR投资组合模型的L-S算法

本文假设投资者是风险厌恶型,用CVaR作为测量投资组合风险的方法.在预算约束的条件下,以最小化CVaR为目标函数,建立了带有交易费用的投资组合模型.将模型转化为两阶段补偿随机优化模型,构造了求解模型的随机L-S算法.为了验证算法的有效性,用中国证券市场中的股票进行数值试验,得到了最优投资组合、VaR和CVaR的值.而且对比分析了有交易费和没有交易费的最优投资组合的不同,给出了相应的有效前沿.     

Stochastic and Risk Management Models and Solution Algorithm for Natural Gas Transmission Network Expansion and LNG Terminal Location Planning

Due to the increasing demands for natural gas, it is playing a more important role in the energy system, and its system expansion planning is drawing more attentions. In this paper, we propose expansion planning models which include both natural gas transmission network expansion and LNG (Liquified Natural Gas) terminals location planning. These models take into account the uncertainties of demands and supplies in the future, which make the models stochastic mixed integer programs with discrete subproblems. Also we consider risk control in our models by including probabilistic constraints, such as a limit on CVaR (Conditional Value at Risk). In order to solve large-scale problems, especially with a large number of scenarios, we propose the embedded Benders decomposition algorithm, which applies Benders cuts in both first and second stages, to tackle the discrete subproblems. Numerical results show that our algorithm is efficient for large scale stochastic natural gas transportation system expansion planning problems.     

A smoothing method for solving portfolio optimization with CVaR and applications in allocation of generation asset

This paper focuses on the computation issue of portfolio optimization with scenario-based CVaR. According to the semismoothness of the studied models, a smoothing technology is considered, and a smoothing SQP algorithm then is presented. The global convergence of the algorithm is established. Numerical examples arising from the allocation of generation assets in power markets are done. The computation efficiency between the proposed method and the linear programming (LP) method is compared. Numerical results show that the performance of the new approach is very good. The remarkable characteristic of the new method is threefold. First, the dimension of smoothing models for portfolio optimization with scenario-based CVaR is low and is independent of the number of samples. Second, the smoothing models retain the convexity of original portfolio optimization problems. Third, the complicated smoothing model that maximizes the profit under the CVaR constraint can be reduced to an ordinary optimization model equivalently. All of these show the advantage of the new method to improve the computation efficiency for solving portfolio optimization problems with CVaR measure.     

A smoothing SAA method for a stochastic mathematical program with complementarity constraints

A smoothing sample average approximation (SAA) method based on the log-exponential function is proposed for solving a stochastic mathematical program with complementarity constraints (SMPCC) considered by Birbil et al. (S. I. Birbil, G. Gürkan, O. Listes: Solving stochastic mathematical programs with complementarity constraints using simulation, Math. Oper. Res. 31 (2006), 739–760). It is demonstrated that, under suitable conditions, the optimal solution of the smoothed SAA problem converges almost surely to that of the true problem as the sample size tends to infinity. Moreover, under a strong second-order sufficient condition for SMPCC, the almost sure convergence of Karash-Kuhn-Tucker points of the smoothed SAA problem is established by Robinson’s stability theory. Some preliminary numerical results are reported to show the efficiency of our method.     

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In this paper, we introduced a transaction costs function and established a portfolio model of risk management with second stochastic dominance constraints. This model does not need to make any assumptions about the utility function of the investors and the distribution of the risk assets income, and it can ensure that the choices of the risk-averse investor can be randomly better than a reference value, so it can avoid the high risk investment. We provide a smoothing penalty sample average approximation method for solving this optimization problem. We prove that the smoothing penalty problem is equivalent to the original problem. Numerical results prove that the model and the method are efficient.