High-dimensional Stochastic Problems Research Articles

Fair Incentives for Repeated Engagement

We study a decision-maker’s problem of finding optimal monetary incentive schemes for retention when faced with agents whose participation decisions (stochastically) depend on the incentive they receive. Our focus is on policies constrained to fulfill two fairness properties that preclude outcomes wherein different groups of agents experience different treatment on average. We formulate the problem as a high-dimensional stochastic optimization problem and study it through the use of a closely related deterministic variant. We show that the optimal static solution to this deterministic variant is asymptotically optimal for the dynamic problem under fairness constraints. Though solving for the optimal static solution gives rise to a nonconvex optimization problem, we uncover a structural property that allows us to design a tractable, fast-converging heuristic policy. Traditional schemes for retention ignore fairness constraints; indeed, the goal in these is to use differentiation to incentivize repeated engagement with the system. Our work (i) shows that even in the absence of explicit discrimination, dynamic policies may unintentionally discriminate between agents of different types by varying the type composition of the system, and (ii) presents an asymptotically optimal policy to avoid such discriminatory outcomes.

StocIPNet: A novel probabilistic interpretable network with affine-embedded reparameterization layer for high-dimensional stochastic inverse problems

The stochastic inverse problem (StocIP), which aims to align push-forward and observed output distributions by estimating probability distributions of unknown system inputs, often faces optimization challenges and the curse of dimensionality. A novel deep network called StocIPNet which comprises an affine-embedded reparameterization subnetwork (ReparNet) and a complex system metamodeling subnetwork (MetaNet) is proposed to alleviate these issues. The ReparNet subnetwork embeds the affine transformation to convert the statistical parameters of the physical random vector into learnable weights and biases, effectively implementing the reparameterization trick by separating random and deterministic elements in the stochastic sampling operation to preserve differentiability. In parallel, the MetaNet subnetwork offers a computationally efficient alternative to time-consuming forward solvers, facilitating the generation of push-forward distributions. The entire StocIPNet utilizes the kernel maximum mean discrepancy (MMD) as a distribution-free loss function, quantifying the discrepancy between push-forward and observed output distributions. By leveraging the ReparNet’s advantage of reformulating the sampling process as a differentiable transformation and combining two subnetworks seamlessly, the StocIP is reconfigured into a pure network training paradigm preserving differentiability perfectly, which allows for direct modeling and efficient inference of uncertainty within the network using automatic differentiation, backpropagation and gradient-based optimization methods, enabling ease of scaling to high-dimensional problems. The proposed framework has been theoretically demonstrated to be equivalent to the maximum likelihood method, ensuring its solid probabilistic interpretable foundation. The proposed framework is applied to perform stochastic model updating on a numerical and an experimental structure, which effectively demonstrates the framework’s remarkable effectiveness and high efficiency in treating high-dimensional StocIPs.

Consumer service level-oriented resilience optimisation for e-commerce supply chain considering hybrid strategies with blockchain adoption

ABSTRACT The booming development of customised e-commerce makes e-commerce supply chains experience a severe test of consumer service level (CSL). An efficient e-commerce supply chain resilience optimisation method addressing economic performance and service performance is proposed to improve customised services under disruption risks. The proposed method includes a hybrid strategy considering both a resistance strategy with blockchain (BCT) adoption and time-dependent recovery strategies simultaneously. A two-stage multi-period multi-product stochastic programming with BCT adoption is then proposed, which (1) implements BCT as the resistance strategy in the pre-disruption stage; (2) collaborates four recovery strategies in the post-disruption stage; (3) supports the e-tailer in making decisions and optimises his profit and order fulfillment time while taking product priorities into account. Using the actual data of Chinese e-commerce during the public health emergency in 2020, it is demonstrated that (1) the applicability of the model with BCT adoption in both supply chain resilience and CSL improvement; (2) the performance of the hybrid strategy in long-term disruption management; (3) the efficiency and robustness of the proposed solution approach for multi-objective high-dimensional stochastic programming problems. E-tailers can optimise decision-making under long-term disruptions by the proposed methodology in practices.

Optimal scheduling of active distribution network based on RBF-stochastic response surface method

As the high proportion of distributed generation (DG) and diversification of loads increase, the distribution network faces increasingly greater impact. To address the high-dimensional stochastic problems arising from the multi-dimensional uncertainty of power source and load, this paper proposes a two-stage optimization scheduling scheme for AC distribution networks based on RBF-stochastic response surface method (RBF-SRSM). This scheme takes into account economy, environmental protection and safety, and consists of day-ahead and intra-day stages. The day-ahead decision scheduling is based on an optimal low-carbon economy operation plan, while the intra-day reactive power optimization is achieved by converting nonlinear integral indicators into discrete linear indicators through probabilistic power flow calculation and the first moment derivation principle inspired by SRSM. The mixed integer second-order cone programming (MISOCP) method is used to solve the dispatching plan, and the results suggest that the proposed method is more efficient than the SRSM.

A decentralized algorithm for a mean field control problem of piecewise deterministic Markov processes

This paper provides a decentralized approach for the control of a population of N agents to minimize an aggregate cost. Each agent evolves independently according to a Piecewise Deterministic Markov dynamics controlled via unbounded jumps intensities. The N-agent high dimensional stochastic control problem is approximated by the limiting mean field control problem. A Lagrangian approach is proposed. Although the mean field control problem is not convex, it is proved to achieve zero duality gap. A stochastic version of the Uzawa algorithm is shown to converge to the primal solution. At each dual iteration of the algorithm, each agent solves its own small dimensional sub problem by means of the Dynamic Programming Principal, while the dual multiplier is updated according to the aggregate response of the agents. Finally, this algorithm is used in a numerical simulation to coordinate the charging of a large fleet of electric vehicles in order to track a target consumption profile.

Mean Field Approximations via Log-Concavity

Abstract We propose a new approach to deriving quantitative mean field approximations for any probability measure $P$ on $\mathbb {R}^{n}$ with density proportional to $e^{f(x)}$, for $f$ strongly concave. We bound the mean field approximation for the log partition function $\log \int e^{f(x)}dx$ in terms of $\sum _{i \neq j}\mathbb {E}_{Q^{*}}|\partial _{ij}f|^{2}$, for a semi-explicit probability measure $Q^{*}$ characterized as the unique mean field optimizer, or equivalently as the minimizer of the relative entropy $H(\cdot \,|\,P)$ over product measures. This notably does not involve metric-entropy or gradient-complexity concepts which are common in prior work on nonlinear large deviations. Three implications are discussed, in the contexts of continuous Gibbs measures on large graphs, high-dimensional Bayesian linear regression, and the construction of decentralized near-optimizers in high-dimensional stochastic control problems. Our arguments are based primarily on functional inequalities and the notion of displacement convexity from optimal transport.

Empirical Assessment of Non-Intrusive Polynomial Chaos Expansions for High-Dimensional Stochastic CFD Problems

Empirical Assessment of Non-Intrusive Polynomial Chaos Expansions for High-Dimensional Stochastic CFD Problems

A stochastic LATIN method for stochastic and parameterized elastoplastic analysis

The LATIN method has been developed and successfully applied to a variety of deterministic problems, but few work has been developed for nonlinear stochastic problems. This paper presents a stochastic LATIN method to solve stochastic and/or parameterized elastoplastic problems. To this end, the stochastic solution is decoupled into spatial, temporal and stochastic spaces, and approximated by the sum of a set of products of triplets of spatial functions, temporal functions and random variables. Each triplet is then calculated in a greedy way using a stochastic LATIN iteration. The high efficiency of the proposed method relies on two aspects: The nonlinearity is efficiently handled by inheriting advantages of the classical LATIN method, and the randomness and/or parameters are effectively treated by a sample-based approximation of stochastic spaces. Further, the proposed method is not sensitive to the stochastic and/or parametric dimensions of inputs due to the sample description of stochastic spaces. It can thus be applied to high-dimensional stochastic and parameterized problems. Five numerical examples demonstrate the promising performance of the proposed stochastic LATIN method.

Inverse reliability analysis and design for tunnel face stability considering soil spatial variability

The traditional deterministic analysis for tunnel face stability neglects the uncertainties of geotechnical parameters, while the simplified reliability analysis which models the potential uncertainties by means of random variables usually fails to account for soil spatial variability. To overcome these limitations, this study proposes an efficient framework for conducting reliability analysis and reliability-based design (RBD) of tunnel face stability in spatially variable soil strata. The three-dimensional (3D) rotational failure mechanism of the tunnel face is extended to account for the soil spatial variability, and a probabilistic framework is established by coupling the extended mechanism with the improved Hasofer-Lind-Rackwits-Fiessler recursive algorithm (iHLRF) as well as its inverse analysis formulation. The proposed framework allows for rapid and precise reliability analysis and RBD of tunnel face stability. To demonstrate the feasibility and efficacy of the proposed framework, an illustrative case of tunnelling in frictional soils is presented, where the soil's cohesion and friction angle are modelled as two anisotropic cross-correlated lognormal random fields. The results show that the proposed method can accurately estimate the failure probability (or reliability index) regarding the tunnel face stability and can efficiently determine the required supporting pressure for a target reliability index with soil spatial variability being taken into account. Furthermore, this study reveals the impact of various factors on the support pressure, including coefficient of variation, cross-correlation between cohesion and friction angle, as well as autocorrelation distance of spatially variable soil strata. The results also demonstrate the feasibility of using the forward and/or inverse first-order reliability method (FORM) in high-dimensional stochastic problems. It is hoped that this study may provide a practical and reliable framework for determining the stability of tunnels in complex soil strata.

Stochastic virtual element methods for uncertainty propagation of stochastic linear elasticity

This paper presents stochastic virtual element methods for propagating uncertainty in linear elastic stochastic problems. We first derive stochastic virtual element equations for 2D and 3D linear elastic problems that may involve uncertainties in material properties, external forces, boundary conditions, etc. A stochastic virtual element space that couples the deterministic virtual element space and the stochastic space is constructed for this purpose and used to approximate the unknown stochastic solution. Two numerical frameworks are then developed to solve the derived stochastic virtual element equations, including a Polynomial Chaos approximation based approach and a weakly intrusive approximation based approach. In the Polynomial Chaos based framework, the stochastic solution is approximated using the Polynomial Chaos basis and solved via an augmented deterministic virtual element equation that is generated by applying the stochastic Galerkin procedure to the original stochastic virtual element equation. In the weakly intrusive approximation based framework, the stochastic solution is approximated by a summation of a set of products of random variables and deterministic vectors, where the deterministic vectors are solved via converting the original stochastic problem to deterministic virtual element equations by the stochastic Galerkin approach, and the random variables are solved via converting the original stochastic problem to one-dimensional stochastic algebraic equations by the classical Galerkin procedure. This method avoids the curse of dimensionality in high-dimensional stochastic problems successfully since all random inputs are embedded into one-dimensional stochastic algebraic equations whose computational effort weakly depends on the stochastic dimension. Numerical results on 2D and 3D problems with low- and high-dimensional random inputs demonstrate the good performance of the proposed methods.

Reliability analysis of pile-reinforced slopes in width-limited failure mode considering three-dimensional spatial variation of soil strength

Stabilization pile is a common measurement for landslide mitigation. In previous literatures, the deterministic safety factor Fs method has been well established for pile-reinforced slope stability analysis, for example by limit analysis. However, the Fs method by itself cannot reflect the inherent uncertainty and spatial variability of mechanical properties of soil. This paper presents a probabilistic analysis procedure for width-constrained pile-reinforced slope where the failure pattern and soil spatial variability are both accounted for under three-dimensional (3D) condition. A modified discretization kinematic analysis-based mechanism is employed as the deterministic model. The spatial variability of cohesion c and friction angle φ is modeled as lognormal random fields by using the Karhunen–Loève expansion. The sparse polynomial chaos expansion (SPCE) is used to construct the meta-model to reduce the computational cost in high dimensional stochastic problems. The failure probability, probability density function (PDF), and other useful reliability results can be provided by performing Monte Carlo simulations of the proposed SPCE meta-model. Finally, the influences of soil spatial variability, slope geometry, and pile parameters on the slope reliability are analyzed. The combination of the modified discretization mechanism and SPCE provides a useful tool for the probabilistic analysis of pile-reinforced earth slopes under 3D condition. The probabilistic simulation results are also helpful for reliability design of pile-reinforced slope in practice.

Decomposition of Convex High Dimensional Aggregative Stochastic Control Problems

We consider the framework of convex high dimensional stochastic control problems, in which the controls are aggregated in the cost function. As first contribution, we introduce a modified problem, whose optimal control is under some reasonable assumptions an $\varepsilon$-optimal solution of the original problem. As second contribution, we present a decentralized algorithm whose convergence to the solution of the modified problem is established. Finally, we study the application of the developed tools in an engineering context, studying a coordination problem for large populations of domestic thermostatically controlled loads (TCLs)

Deep treasury management for banks.

Retail banks use Asset Liability Management (ALM) to hedge interest rate risk associated with differences in maturity and predictability of their loan and deposit portfolios. The opposing goals of profiting from maturity transformation and hedging interest rate risk while adhering to numerous regulatory constraints make ALM a challenging problem. We formulate ALM as a high-dimensional stochastic control problem in which monthly investment and financing decisions drive the evolution of the bank's balance sheet. To find strategies that maximize long-term utility in the presence of constraints and stochastic interest rates, we train neural networks that parametrize the decision process. Our experiments provide practical insights and demonstrate that the approach of Deep ALM deduces dynamic strategies that outperform static benchmarks.

On the Feasibility Guarantees of Deep Reinforcement Learning Solutions for Distribution System Operation

Deep reinforcement learning (DRL) has scored unprecedented success in finding near-optimal solutions in high-dimensional stochastic problems, leading to its extensive use in operational research, including the operation of power systems. However, in practice, it has been adopted with extreme caution because the standard DRL does not guarantee the satisfaction of operational constraints. In this paper, the feasibility of solutions given by a DRL agent trained to operate a distribution system is guaranteed by modifying the exploration process and optimality criterion of standard DRL. To that end, first, a convex feasibility set in the form of a multi-dimensional polyhedron, called feasibility diamond, is formed inside the region defined by the power flow constraints, using which the feasibility of solutions given by DRL is checked in real-time. Solutions outside the feasibility diamond are projected on the diamond’s surface, and the new modified action is used for DRL training. Further, the distance of infeasible solutions to their feasible projection is penalized in the DRL reward function. The impact of the proposed method on the feasibility and optimality of DRL solutions are tested on three test distribution systems, indicating that modifying the exploration process and a soft penalization of infeasibilities works best in achieving near-optimal and reliable DRL-trained operators.

Efficient structural reliability analysis via a weak-intrusive stochastic finite element method

This paper presents a novel methodology for structural reliability analysis by means of the stochastic finite element method (SFEM). The key issue of structural reliability analysis is to determine the limit state function and corresponding multidimensional integral that are usually related to the structural stochastic displacement and/or its derivative, e.g., the stress and strain. In this paper, a novel weak-intrusive SFEM is first used to calculate structural stochastic displacements of all spatial positions. In this method, the stochastic displacement is decoupled into a combination of a series of deterministic displacements with random variable coefficients. An iterative algorithm is then given to solve the deterministic displacements and the corresponding random variables. Based on the stochastic displacement obtained by the SFEM, the limit state function described by the stochastic displacement (and/or its derivative) and the corresponding multidimensional integral encountered in reliability analysis can be calculated in a straightforward way. Failure probabilities of all spatial positions can be obtained at once since the stochastic displacements of all spatial points have been known by using the proposed SFEM. Furthermore, the proposed method can be applied to high-dimensional stochastic problems without any modification. One of the most challenging problems encountered in high-dimensional reliability analysis, known as the curse of dimensionality, can be circumvented with great success. Three numerical examples, including low- and high-dimensional reliability analysis, are given to demonstrate the good accuracy and the high efficiency of the proposed method.

A stochastic finite element scheme for solving partial differential equations defined on random domains

This paper proposes a novel stochastic finite element scheme to solve partial differential equations defined on random domains. A geometric mapping algorithm first transforms the random domain into a reference domain. By combining the mesh topology (i.e. the node numbering and the element numbering) of the reference domain and random nodal coordinates of the random domain, random meshes of the original problem are obtained by only one mesh of the reference domain. In this way, the original problem is still discretized and solved on the random domain instead of the reference domain. A random isoparametric mapping of random meshes is then proposed to generate the stochastic finite element equation of the original problem. We adopt a weak-intrusive method to solve the obtained stochastic finite element equation. In this method, the unknown stochastic solution is decoupled into a sum of the products of random variables and deterministic vectors. Deterministic vectors are computed by solving deterministic finite element equations, and corresponding random variables are solved by a proposed sampling method. The computational effort of the proposed method does not increase dramatically as the stochastic dimension increases and it can solve high-dimensional stochastic problems with low computational effort, thus the proposed method avoids the curse of dimensionality successfully. Four numerical examples are given to demonstrate the good performance of the proposed method.

Solving Stochastic Optimal Control Problem via Stochastic Maximum Principle with Deep Learning Method

In this paper, we aim to solve the high dimensional stochastic optimal control problem from the view of the stochastic maximum principle via deep learning. By introducing the extended Hamiltonian system which is essentially a Forward Backward Stochastic Differential Equation (FBSDE) with a maximum condition, we reformulate the original control problem as a new one. According to whether the optimal control has an explicit representation, three algorithms are proposed to solve the new control problem. Numerical results for different examples demonstrate the effectiveness of our proposed algorithms, especially in high dimensional cases. And even if the optimal control \(\tilde{u}\) in the maximum condition may not be solved explicitly, our algorithms can still deal with the stochastic optimal control problem. An important application of our proposed method is to calculate the sub-linear expectations, which correspond to a kind of fully nonlinear Partial Differential Equations (PDEs).

A weak-intrusive stochastic finite element method for stochastic structural dynamics analysis

This paper presents a weak-intrusive stochastic finite element method for solving stochastic structural dynamics equations. In this method, the stochastic solution is decomposed into the summation of a series of products of random variables, spatial vectors and temporal functions. An iterative algorithm is proposed to compute each triplet of the random variable, spatial vector and temporal function one by one. The original stochastic dynamics problem is firstly transformed into spatial–temporal coupled problems (i.e. deterministic structural dynamics equations), which can be solved efficiently by existing FEM solvers. Based on the solution of the spatial–temporal coupled problem, the original problem is then transformed into stochastic-temporal coupled problems (i.e. one-dimensional second-order stochastic ordinary differential equations), which are solved by a proposed sampling method. All random sources are embedded into the stochastic-temporal coupled problems. The proposed sampling method can solve the stochastic-temporal problems of hundreds of dimensions with low computational costs. Thus the curse of dimensionality in high-dimensional stochastic spaces is avoided with great success. Three numerical examples, including low- and high-dimensional stochastic problems, are used to demonstrate the good accuracy and the high efficiency of the proposed method.

An intelligent stochastic optimization approach for stochastic order allocation problems with high-dimensional order uncertainties

Effective order allocation has a great impact on improving the efficiency of supply chain operations. This paper investigates a high-dimensional stochastic order allocation problem that assigns uncertain customer orders to heterogeneous parallel machines. Considering the arrival uncertainty of customer orders in the real-world production environment, a mathematical model of the investigated problem is established with the objective of maximizing the expected profit of order processing. A novel intelligent stochastic optimization approach is proposed by combining a modified adaptive large neighborhood search algorithm with a scenario generation technique. In this approach, the modified adaptive large neighborhood search algorithm suitable for the generalized allocation problem is proposed to seek the best solutions, while the scenario generation technique is used to generate the scenarios needed to evaluate the solutions of the high-dimensional stochastic optimization problem. Experimental results demonstrate that the proposed approach outperforms the compared stochastic optimization method in terms of optimization efficiency and optimization ability.

Delta family approach for the stochastic control problems of utility maximization

In this paper, we propose a new approach for stochastic control problems arising from utility maximization. The main idea is to directly start from the dynamical programming equation and compute the conditional expectation using a novel representation of the conditional density function through the Dirac Delta function and the corresponding series representation. We obtain an explicit series representation of the value function, whose coefficients are expressed through integration of the value function at a later time point against a chosen basis function. Thus we are able to set up a recursive integration time-stepping scheme to compute the optimal value function given the known terminal condition, e.g. utility function. Due to tensor decomposition property of the Dirac Delta function in high dimensions, it is straightforward to extend our approach to solving high-dimensional stochastic control problems. The backward recursive nature of the method also allows for solving stochastic control and stopping problems, i.e. mixed control problems. We illustrate the method through solving some two-dimensional stochastic control (and stopping) problems, including the case under the classical and rough Heston stochastic volatility models, and stochastic local volatility models such as the stochastic alpha beta rho (SABR) model.