arXiv subjects

Huifu Xu

Publications and source records attributed to Huifu Xu.

At least 19 recordsLinked to original sources

arXiv2026

Risk-Averse Bayesian Games with an Unknown Type Distribution: Bayesian Learning, Equilibrium Analysis, and Finite-Sample Guarantees

Classical Bayesian games assume that the joint distribution of players' types is common knowledge, an assumption that rarely holds in practical applications. We address this issue by studying a group of myopic players who repeatedly interact under the Bayesian Nash conjecture and learn a parametric joint type distribution from type profiles observed over time within a Bayesian learning framework. Players may hold heterogeneous priors and may be risk averse toward both epistemic uncertainty about the distributional parameter and aleatoric uncertainty in their rivals' types. We propose two models: a Bayesian Nash equilibrium among risk-averse Bayesian learners (BNE-RABL), in which epistemic and aleatoric risks are evaluated separately, and a BNE based on Bayesian predictive distributions (BNE-BPD), in which the two sources of uncertainty are integrated into a Bayesian predictive distribution and evaluated through a single risk measure. Under suitable conditions, we establish the existence and uniqueness of both equilibria, and derive non-asymptotic convergence rates of the equilibrium sequences toward the corresponding oracle BNE under the true type distribution as the game is repeatedly played and more type profiles are observed. These results further show that the discrepancy between the BNE-RABL and BNE-BPD strategies vanishes as the sample size grows, which illustrates that the two models can be regarded as effective approximations to each other. We apply the proposed models to a price competition problem and numerically illustrate the theoretical results.

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arXiv2026

Maximum Utility Split Method for Utility Preference Elicitation

In this paper, we propose a new approach, called maximum utility split (MUS) scheme, which is built on random utility split (RUS) scheme but with a notable difference: one lottery is designed with two fixed outcomes but with varying probability, and the other has a deterministic outcome specifically chosen at the point where the range between the largest and smallest possible utility values is maximized. Consequently, the probability of random lottery is set such that the range of the ambiguity set of utility functions is reduced by half at the point. Under moderate conditions, we show that MUS can successively generate a sequence of such questionnaires and effectively reduce the ambiguity set, eventually converging to the true utility function as the number of questionnaires increases. The main challenge is to effectively identify the point with the largest utility range for a given ambiguity set constructed from preference information. Based on the structure of the ambiguity set, we propose an interval-based algorithm which identifies each certain-outcome lottery by solving a sequence of linear programs. Moreover, to deal with the case where elicitation terminates before the ambiguity set reduces to a singleton, we demonstrate how to figure out a nominal utility function by solving optimization programs. These identify the smallest and largest utility functions under the Kantorovich metric within the ambiguity set, after which we identify a nominal utility function located in the middle of them. Finally, numerical results demonstrate the efficiency of the MUS method and the performance of a robo-advisor system based on MUS-type queries and the nominal utility elicitation. While the main discussions focus on concave utility functions, we also demonstrate how the MUS approach can be extended to accommodate general non-concave utility functions, particularly S-shaped ones.

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arXiv2026

Coordinate-wise Polyhedral Method for Eliciting Multivariate Linear Utility and Univariate Nonlinear Utility Functions

In this paper, we propose a coordinate-wise polyhedral method (CPM) for cutting polyhedra with theoretical guarantees of convergence. Unlike the existing polyhedral method, which designs pairwise comparison queries by solving coupled optimization problems and performs cuts subsequently, CPM specifies coordinate-wise cuts in advance and then designs corresponding pairwise comparison queries by solving a linear system of equations. Under this framework, we show that CPM reduces the diameter of the polyhedron at a linear convergence rate. Moreover, we extend CPM to a univariate piecewise-linear utility function by representing it with its increment vector over consecutive breakpoints. We show that the Kantorovich distance between the true utility function and the estimated one obtained by CPM decreases at a linear convergence rate. Further, we extend the CPM to general nondecreasing Lipschitz continuous utility functions by piecewise-linear approximation (PLA). We introduce an adaptive-breakpoint strategy to avoid direction errors caused by PLA. We prove that the set of piecewise-linear utility functions corresponding to the ambiguity set of increment vectors converges to the true utility function as the number of queries increases, and derive an explicit bound for the approximation. Finally, to evaluate the performance of CPM, we conduct a series of numerical experiments. The results demonstrate comparable convergence to the standard polyhedral method in the case of linear multivariate utility functions. For nonlinear univariate utility functions, CPM achieves stable convergence to the true utility function, in line with the theoretical findings.

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arXiv2026

Adaptive Distributionally Robust Optimal Control with Bayesian Ambiguity Sets

In stochastic optimal control (SOC), uncertainty may arise from incomplete knowledge of the true probability distribution of the underlying environment, which is known as Knightian or epistemic uncertainty. Distributionally robust optimal control (DROC) models are subsequently proposed to tackle this source of uncertainty. While such models are effective in some practical applications, most existing DROC models are offline and can be overly conservative when data are scarce. Moreover, they cannot be applied to the case when samples are generated episodically. Motivated by the Bayesian SOC framework recently proposed by Shapiro et al.~\cite{shapiro2025episodic}, we propose an adaptive DROC model in which the ambiguity set is updated via Bayesian learning from new data. Under some moderate conditions, we derive a tractable risk-averse reformulation, establish consistency of the optimal value function and optimal policy for an infinite-horizon SOC and establish a finite-sample posterior credibility guarantee for the policy value induced by the proposed episodic Bayesian DROC model. We also study the stability and statistical robustness of the proposed model with respect to sample perturbations that often arise in data-driven environments. To solve the episodic Bayesian DROC model, we propose a Bellman-operator cutting-plane (BOCP) algorithm that is computationally efficient and provably convergent. Numerical results on an inventory control problem demonstrate the effectiveness, adaptivity, and robust performance of the proposed model and algorithm.

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arXiv2026

Necessary Optimality Conditions for Integrated Learning and Optimization Problem in Contextual Optimization

Integrated learning and optimization (ILO) is a framework in contextual optimization which aims to train a predictive model for the probability distribution of the underlying problem data uncertainty, with the goal of enhancing the quality of downstream decisions. This framework represents a new class of stochastic bilevel programs, which are extensively utilized in the literature of operations research and management science, yet remain underexplored from the perspective of optimization theory. In this paper, we fill the gap. Specifically, we derive the first-order necessary optimality conditions in terms of Mordukhovich limiting subdifferentials. To this end, we formulate the bilevel program as a two-stage stochastic program with variational inequality constraints when the lower-level decision-making problem is convex, and establish an optimality condition via sensitivity analysis of the second-stage value function. In the case where the lower level optimization problem is nonconvex, we adopt the value function approach in the literature of bilevel programs and derive the first-order necessary conditions under stochastic partial calmness conditions. The derived optimality conditions are applied to several existing ILO problems in the literature. These conditions may be used for the design of gradient-based algorithms for solving ILO problems.

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arXiv2025

Quantification of Errors of the Performance Estimators in the Linear-Quantized Precoding Models for Massive MIMO Systems

Massive MIMO (Multiple-Input Multiple-Output) is a key enabler for 5G and future wireless systems, boosting channel capacity, energy efficiency, and spectral efficiency. However, high power consumption and hardware costs of Digital-to-Analog Converters (DACs) in massive MIMO create practical challenges. To mitigate these, recent work proposes low-resolution DACs-restricting transmitted signals to finite voltage levels-to cut power and costs. This requires studying quantized precoding: signals are processed via a linear precoding matrix, then quantized by DACs. In this paper, we explore the linear-quantized precoding model and its statistically or asymptotically equivalent variants. We derive error bounds for two key metrics:Signal-to-Interference-plus-Noise Ratio (SINR) and Symbol Error Probability (SEP), based on the linear-quantized model and its equivalent counterparts. We also formulate and analyze the SINR maximization problem in both asymptotic and finite-dimensional scenarios. Our analysis shows that as system dimensions scale, finite-dimensional problem solutions/values converge to their asymptotic equivalents-underscoring the practical value of asymptotic insights with stability guarantees. These findings theoretically support robust precoding design under hardware constraints, enabling efficient massive MIMO implementation with low-resolution DACs. Beyond validating asymptotic predictions in finite regimes, our framework offers practical optimization guidelines for real-world systems, linking theory and applications.

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arXiv2025

Existence and Uniqueness Theorem of Continuous and Monotone Bayesian Nash Equilibrium and Stability Analysis

Since the seminal work by Meirowitz, there has been growing attention on the existence and uniqueness of continuous Bayesian Nash equilibria. In the existing literature, existence is typically established using Schauder's fixed-point theorem, relying on the equicontinuity of players' best response functions. Uniqueness, on the other hand, is usually derived under additional monotonicity conditions. In this paper, we revisit the issues of existence and uniqueness, and advance the literature by establishing both simultaneously using the Banach fixed-point theorem under a set of moderate conditions. Furthermore, we analyze the stability of such equilibria with respect to perturbations in the joint probability distribution of type parameters, offering theoretical support for the application of Bayesian Nash equilibrium models in data-driven contexts.

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arXiv2025

SCOPE: Spectral Concentration by Distributionally Robust Joint Covariance-Precision Estimation

We propose a distributionally robust formulation for simultaneously estimating the covariance matrix and the precision matrix of a random vector.The proposed model minimizes the worst-case weighted sum of the Frobenius loss of the covariance estimator and Stein's loss of the precision matrix estimator against all distributions from an ambiguity set centered at the nominal distribution. The radius of the ambiguity set is measured via convex spectral divergence. We demonstrate that the proposed distributionally robust estimation model can be reduced to a convex optimization problem, thereby yielding quasi-analytical estimators. The joint estimators are shown to be nonlinear shrinkage estimators. The eigenvalues of the estimators are shrunk nonlinearly towards a positive scalar, where the scalar is determined by the weight coefficient of the loss terms. By tuning the coefficient carefully, the shrinkage corrects the spectral bias of the empirical covariance/precision matrix estimator. By this property, we call the proposed joint estimator the Spectral concentrated COvariance and Precision matrix Estimator (SCOPE). We demonstrate that the shrinkage effect improves the condition number of the estimator. We provide a parameter-tuning scheme that adjusts the shrinkage target and intensity that is asymptotically optimal. Numerical experiments on synthetic and real data show that our shrinkage estimators perform competitively against state-of-the-art estimators in practical applications.

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arXiv2025

Stability Analysis of an Integrated Multistage Stochastic Programming and Markov Decision Process Problem

In this paper, we consider an integrated MSP-MDP framework which captures features of Markov decision process (MDP) and multistage stochastic programming (MSP). The integrated framework allows one to study a dynamic decision-making process that involves both transition of system states and dynamic change of the stochastic environment affected respectively by potential endogenous uncertainties and exogenous uncertainties. The integrated model differs from classical MDP models by taking into account the effect of history-dependent exogenous uncertainty and distinguishes itself from standard MSP models by explicitly considering transition of states between stages. We begin by deriving dynamic nested reformulation of the problem and the Lipschitz continuity and convexity of the stage-wise optimal value functions. We then move on to investigate stability of the problem in terms of the optimal value and the set of optimal solutions under the perturbations of the probability distributions of the endogenous uncertainty and the exogenous uncertainty. Specifically, we quantify the effects of the perturbation of the two uncertainties on the optimal values and optimal solutions by deriving the error bounds in terms of Kantorovich metric and Fortet-Mourier metric of the probability distributions of the respective uncertainties. These results differ from the existing stability results established in terms of the filtration distance \cite{heitsch2009scenario} or the nested distance \cite{pflug2012distance}. We use some examples to explain the differences via tightness of the error bounds and applicability of the stability results. The results complement the existing stability results and provide new theoretical grounding for emerging integrated MSP-MDP models.

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arXiv2025

A Bayesian Composite Risk Approach for Stochastic Optimal Control and Markov Decision Processes

Inspired by Shapiro et al.~\cite{shapiro2023episodic}, we consider a stochastic optimal control (SOC) and Markov decision process (MDP) where the risks arising from epistemic and aleatoric uncertainties are assessed using Bayesian composite risk (BCR) measures (Qian et al.~\cite{qian2019composite}). The time dependence of the risk measures allows us to capture the decision maker's (DM) dynamic risk preferences opportunely as increasing information about both uncertainties is obtained. This makes the new BCR-SOC/MDP model more flexible than conventional risk-averse SOC/MDP models. Unlike \cite{shapiro2023episodic} where the control/action at each episode is based on the current state alone, the new model allows the control to depend on the probability distribution of the epistemic uncertainty, which reflects the fact that in many practical instances the cumulative information about epistemic uncertainty often affects the DM's belief about the future aleatoric uncertainty and hence the DM's action \cite{strens2000bayesian}. The new modeling paradigm incorporates several existing SOC/MDP models including distributionally robust SOC/MDP models and Bayes-adaptive MDP models and generates so-called preference robust SOC/MDP models. Moreover, we derive conditions under which the BCR-SOC/MDP model is well-defined, demonstrate that finite-horizon BCR-SOC/MDP models can be solved using dynamic programming techniques, and extend the discussion to the infinite-horizon case. By using Bellman equations, we show that under some standard conditions, asymptotic convergence of the optimal values and optimal actions as the episodic variable goes to infinity is achieved. Finally, we carry out numerical tests on a finite horizon spread betting problem and an inventory control problem and show the effectiveness of the proposed model and numerical schemes.

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arXiv2025

Robust Data-Driven Quasiconcave Optimization

We investigate a data-driven quasiconcave maximization problem where information about the objective function is limited to a finite sample of data points. We begin by defining an ambiguity set for admissible objective functions based on available partial information about the objective. This ambiguity set consists of those quasiconcave functions that majorize a given data sample, and that satisfy additional functional properties (monotonicity, Lipschitz continuity, and permutation invariance). We then formulate a robust optimization (RO) problem which maximizes the worst-case objective function over this ambiguity set. Based on the quasiconcave structure in this problem, we explicitly construct the upper level sets of the worst-case objective at all levels. We can then solve the resulting RO problem efficiently by doing binary search over the upper level sets and solving a logarithmic number of convex feasibility problems. This numerical approach differs from traditional subgradient descent and support function based methods for this problem class. While these methods can be applied in our setting, the binary search method displays superb finite convergence to the global optimum, whereas the others do not. This is primarily because binary search fully exploits the specific structure of the worst-case quasiconcave objective, which leads to an explicit and general convergence rate in terms of the number of convex optimization problems to be solved. Our numerical experiments on a Cobb-Douglas production efficiency problem and a fair resource allocation problem demonstrate the tractability of our approach.

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arXiv2025

Efficiently Computing the Quasiconcave Envelope with Incomplete Information

In this paper, we study the approximation of an unknown quasiconcave function based on limited partial information. Available information includes lower bounds on the values of the target function at a specified set of points, as well as some functional properties including monotonicity, Lipschitz continuity, ranking, and permutation invariance. We consider the class of admissible quasiconcave functions that dominate these lower bounds and satisfy these functional properties. We then compute the smallest quasiconcave function among the class of admissible quasiconcave functions. Specifically, we show how to efficiently compute the quasiconcave envelope (QCoE) of a data sample of points, subject to the additional functional properties. The solution procedure takes two steps. First, a value problem is solved to determine the values of the QCoE on the given data sample. Second, an interpolation problem is solved to compute the values of the QCoE on other points. Both the value problem and the interpolation problem introduce some theoretical and computational challenges, as they are non-convex and large-scale. The MILP reformulations of both problems require an exponential number of linear programs (LPs) to be solved in the worst-case. As our main contribution, we solve the value problem with only a polynomial number of LPs, and then solve the interpolation problem for any candidate point with only a logarithmic number of LPs. Some preliminary numerical tests show that the proposed approach is efficient and proper.

q-fin.RM↗

arXiv2025

Modified Polyhedral Method for Elicitation of Shape-Free Utility and Conservatism Reduction in Robust Optimization

In this paper, we propose a modified polyhedral method to elicit a decision maker's (DM's) nonlinear univariate utility function, which does not rely on explicit information about the shape structure, Lipschitz modulus, and the inflection point of the utility. The method is inspired by Toubia et al. (2004) for elicitation of the linear multi-variate utility and the success of the modification needs to overcome two main difficulties. First, we use the continuous piecewise linear function (PLF) to approximate the nonlinear utility and represent the PLF in terms of the vector of increments of linear pieces. Subsequently, elicitation of the nonlinear utility corresponds to reducing the polyhedral feasible set of the vectors of increments. Second, we reduce the size of the polyhedron by successive hyperplane cuts constructed by adaptively generating new queries (pairwise comparison lotteries) where the parameters of the lotteries are obtained by solving some optimization problems. In this reduction procedure, direction error of the cut hyperplane may occur due to the PLF approximation error. To tackle the issue, we develop a strategy by adding the support points of new lotteries to the set of breakpoints of the PLF. As an application, we use all the responses to the queries to construct an ambiguity set of utility functions which allows one to make decisions based on the worst-case utility and apply the modified polyhedral method in a preference robust optimization problem with proper conservatism reduction scheme. The preliminary numerical test results show that the proposed methods work very well.

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arXiv2025

Risk-averse Decision Making with Contextual Information: Model, Sample Average Approximation, and Kernelization

We consider risk-averse contextual optimization problems where the decision maker (DM) faces two types of uncertainties: problem data uncertainty (PDU) and contextual uncertainty (CU) associated with PDU, the DM makes an optimal decision by minimizing the risk arising from PDU based on the present observation of CU and then assesses the risk of the optimal policy against the CU. A natural question arises as to whether the nested risk minimization/assessment process is equivalent to joint risk minimization/assessment against CU and PDU simultaneously. First, we demonstrate that the equivalence can be established by appropriate choices of the risk measures and give counter examples where such equivalence may fail. One of the interesting findings is that the optimal policies are independent of the choice of the risk measure against the CU under certain conditions. Second, by using the equivalence, we propose computational method for solving the risk-averse contextual optimization problem by solving a one-stage risk minimization problem. The latter is particularly helpful in data-driven environments. We consider a number of risk measures/metrics to characterize the DM's risk preference for PDU and discuss the computational tractability for the resulting risk-averse contextual optimization problem. Third, when the risk-averse contextual optimization problem is defined in the reproducing kernel Hilbert space, we show consistency of the optimal values obtained from solving sample average approximation problems. Some numerical tests, in newsvendor problem and portfolio selection problem, are performed to validate the theoretical results.

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arXiv2025

Generalized Bayesian Nash Equilibrium with Continuous Type and Action Spaces

Bayesian game is a strategic decision-making model where each player's type parameter characterizing its own objective is private information: each player knows its own type but not its rivals' types, and Bayesian Nash equilibrium (BNE) is an outcome of this game where each player makes a strategic optimal decision according to its own type under the Nash conjecture. In this paper, we advance the literature by considering a generalized Bayesian game where each player's action space depends on its own type parameter and the rivals' actions. This reflects the fact that in practical applications, a firm's feasible action is often related to its own type (e.g. marginal cost) and the rivals' actions (e.g. common resource constraints in a competitive market). Under some moderate conditions, we demonstrate existence of continuous generalized Bayesian Nash equilibria (GBNE) and uniqueness of such an equilibrium when each player's action space is only dependent on its type. In the case that each player's action space is also dependent on rivals' actions, we give a simple example to show that uniqueness of GBNE is not guaranteed under standard monotone conditions. To compute an approximate GBNE, we restrict each player's response function to the space of polynomial functions of its type parameter and consequently convert the GBNE problem to a stochastic generalized Nash equilibrium problem (SGNE). To justify the approximation, we discuss convergence of the approximation scheme. Some preliminary numerical test results show that the approximation scheme works well.

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arXiv2024

Bayesian Distributionally Robust Nash Equilibrium and Its Application

Inspired by the recent work by Shapiro et al. [45], we propose a Bayesian distributionally robust Nash equilibrium (BDRNE) model where each player lacks complete information on the true probability distribution of the underlying uncertainty represented by a random variable and subsequently determines the optimal decision by solving a Bayesian distributionally robust optimization (BDRO) problem under the Nash conjecture. Unlike most of the DRO models in the literature, the BDRO model assumes (a) the true unknown distribution of the random variable can be approximated by a randomized parametric family of distributions, (b) the average of the worst-case expected value of the objective function with respect to the posterior distribution of the parameter, instead of the worst-case expected value of the objective function is considered in each player's decision making, and (c) the posterior distribution of the parameter is updated as more and more sampling information of the random variable is gathered. Under some moderate conditions, we demonstrate the existence of a BDRNE and derive asymptotic convergence of the equilibrium as the sample size increases. Moreover, we propose to solve the BDRNE problem by Gauss-Seidel-type iterative method in the case when the ambiguity set of each player is constructed via Kullback-Leibler (KL) divergence. Finally, we apply the BDRNE model to a price competition problem under multinomial logit demand. The preliminary numerical test results show that the proposed model and computational scheme perform well.

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arXiv2024

Multistage Robust Average Randomized Spectral Risk Optimization

In this paper, we revisit the multistage spectral risk minimization models proposed by Philpott et al.~\cite{PdF13} and Guigues and Römisch \cite{GuR12} but with some new focuses. We consider a situation where the decision maker's (DM's) risk preferences may be state-dependent or even inconsistent at some states, and consequently there is not a single deterministic spectral risk measure (SRM) which can be used to represent the DM's preferences at each stage. We adopt the recently introduced average randomized SRM (ARSRM) (in \cite{li2022randomization}) to describe the DM's overall risk preference at each stage. To solve the resulting multistage ARSRM (MARSRM) problem, we apply the well-known stochastic dual dynamic programming (SDDP) method which generates a sequence of lower and upper bounds in an iterative manner. Under some moderate conditions, we prove that the optimal solution can be found in a finite number of iterations. The MARSRM model generalizes the one-stage ARSRM and simplifies the existing multistage state-dependent preference robust model \cite{liu2021multistage}, while also encompassing the mainstream multistage risk-neutral and risk-averse optimization models \cite{GuR12,PdF13}. In the absence of complete information on the probability distribution of the DM's random preferences, we propose to use distributionally robust ARSRM (DR-ARSRM) to describe the DM's preferences at each stage. We detail computational schemes for solving both MARSRM and DR-MARSRM. Finally, we examine the performance of MARSRM and DR-MARSRM by applying them to an asset allocation problem with transaction costs and compare them with standard risk neutral and risk averse multistage linear stochastic programming (MLSP) models.

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arXiv2024

Distributional stability of sparse inverse covariance matrix estimators

Finding an approximation of the inverse of the covariance matrix, also known as precision matrix, of a random vector with empirical data is widely discussed in finance and engineering. In data-driven problems, empirical data may be ``contaminated''. This raises the question as to whether the approximate precision matrix is reliable from a statistical point of view. In this paper, we concentrate on a much-noticed sparse estimator of the precision matrix and investigate the issue from the perspective of distributional stability. Specifically, we derive an explicit local Lipschitz bound for the distance between the distributions of the sparse estimator under two different distributions (regarded as the true data distribution and the distribution of ``contaminated'' data). The distance is measured by the Kantorovich metric on the set of all probability measures on a matrix space. We also present analogous results for the standard estimators of the covariance matrix and its eigenvalues. Furthermore, we discuss several applications and conduct some numerical experiments.

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