Quantitative stability of mixed-integer two-stage quadratic stochastic programs

For our introduced mixed-integer quadratic stochastic program with fixed recourse matrices, random recourse costs, technology matrix and right-hand sides, we study quantitative stability properties of its optimal value function and optimal solution set when the underlying probability distribution is perturbed with respect to an appropriate probability metric. To this end, we first establish various Lipschitz continuity results about the value function and optimal solutions of mixed-integer parametric quadratic programs with parameters in the linear part of the objective function and in the right-hand sides of linear constraints. The obtained results extend earlier results about quantitative stability properties of stochastic integer programming and stability results for mixed-integer parametric quadratic programs.     

A gaussian upper bound for gaussian multi-stage stochastic linear programs

This paper deals with two-stage and multi-stage stochastic programs in which the right-hand sides of the constraints are Gaussian random variables. Such problems are of interest since the use of Gaussian estimators of random variables is widespread. We introduce algorithms to find upper bounds on the optimal value of two-stage and multi-stage stochastic (minimization) programs with Gaussian right-hand sides. The upper bounds are obtained by solving deterministic mathematical programming problems with dimensions that do not depend on the sample space size. The algorithm for the two-stage problem involves the solution of a deterministic linear program and a simple semidefinite program. The algorithm for the multi-stage problem invovles the solution of a quadratically constrained convex programming problem.     

Marginal permutation invariant covariance matrices with applications to linear models

The goal of the present paper is to perform a comprehensive study of the covariance structures in balanced linear models containing random factors which are invariant with respect to marginal permutations of the random factors. We shall focus on model formulation and interpretation rather than the estimation of parameters. It is proven that permutation invariance implies a specific structure for the covariance matrices. Useful results are obtained for the spectra of permutation invariant covariance matrices. In particular, the reparameterization of random effects, i.e., imposing certain constraints, will be considered. There are many possibilities to choose reparameterization constraints in a linear model, however not every reparameterization keeps permutation invariance. The question is if there are natural restrictions on the random effects in a given model, i.e., such reparameterizations which are defined by the covariance structure of the corresponding factor. Examining relationships between the reparameterization conditions applied to the random factors of the models and the spectrum of the corresponding covariance matrices when permutation invariance is assumed, restrictions on the spectrum of the covariance matrix are obtained which lead to “sum-to-zero” reparameterization of the corresponding factor.     

基于列随机矩阵的逐次差分代换与正半定型的机械化判定

本文选择列随机平均矩阵T_n作为基本代换矩阵,建立了基于T_n的逐次差分代换方法.获得了R_+~n上正半定型,不定型判定的充要条件.并进一步证明了:正定型的差分代换集序列正向终止.根据这些结果编写的Maple程序TSDS3,能够自动证明代数型不等式,对不成立的不等式总能输出反例.该程序虽可能不停机,但大量的应用实例证实了该方法的实用性.     

Limiting optimal adaptive filtering with unknown disturbance covariance

The accuracy of estimating the variance of the Kalman-Bucy filter depends essentially on disturbance covariance matrices and measurement noise. The main difficulty in filter design is the lack of necessary statistical information about the useful signal and the disturbance. Filters whose parameters are tuned during active estimation are classified with adaptive filters. The problem of adaptive filtering under parametric uncertainty conditions is studied. A method for designing limiting optimal Kalman-Bucy filters in the case of unknown disturbance covariance is presented. An adaptive algorithm for estimating disturbance covariance matrices based on stochastic approximation is described. Convergence conditions for this algorithm are investigated. The operation of a limiting adaptive filter is exemplified.     

Estimation of Wishart mean matrices under simple tree ordering

For Wishart density functions, we study the risk dominance problems of the restricted maximum likelihood estimators of mean matrices with respect to the Kullback-Leibler loss function over restricted parameter space under the simple tree ordering set. The results are directly applied to the estimation of covariance matrices for the completely balanced multivariate multi-way random effects models without interactions.     

Stochastic wave equation simulating the behavior of random variables whose mean values obey a system of ordinary first-order differential equations

We suggest a theory of stochastic waves that describe the behavior of random vectors satisfying a set of ordinary first-order differential equations. The equation for the stochastic waves is given for the case where the mean values are described by a differential model. The relationship between this equation and the Liouville equation is considered and analogue of the Ehrenfest theorem is proved. For the covariance of a component of a random vector, we obtain an ordinary first-order differential equation. Interpretation of Planck’s constant is discussed. Conditions are formulated under which wave packets propagate with increasing or decreasing covariance. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 111, No. 3, pp. 356–368, June, 1997.     

Growth Curve Model with Hierarchical Within-Individuals Design Matrices

This paper deals with some inferential problems under an extended growth curve model with several hierarchical within-individuals design matrices. The model includes the one whose mean structure consists of polynomial growth curves with different degrees. First we consider the case when the covariance matrix is unknown positive definite. We derive a LR test for examining the hierarchical structure for within individuals design matrices and a model selection criterion. Next we consider the case when a random coefficients covariance structure is assumed, under certain assumption of between-individual design matrices. Similar inferential problems are also considered. The dental measurement data (see, e.g., Potthoff and Roy (1964, Biometrika, 51, 313-326)) is reexamined, based on extended growth curve models.     

On a model selection problem from high-dimensional sample covariance matrices

Modern random matrix theory indicates that when the population size p is not negligible with respect to the sample size n, the sample covariance matrices demonstrate significant deviations from the population covariance matrices. In order to recover the characteristics of the population covariance matrices from the observed sample covariance matrices, several recent solutions are proposed when the order of the underlying population spectral distribution is known. In this paper, we deal with the underlying order selection problem and propose a solution based on the cross-validation principle. We prove the consistency of the proposed procedure.     

Continuity and stability of fully random two-stage stochastic programs with mixed-integer recourse

In order to derive continuity and stability of two-stage stochastic programs with mixed-integer recourse when all coefficients in the second-stage problem are random, we first investigate the quantitative continuity of the objective function of the corresponding continuous recourse problem with random recourse matrices. Then by extending derived results to the mixed-integer recourse case, the perturbation estimate and the piece-wise lower semi-continuity of the objective function are proved. Under the framework of weak convergence for probability measure, the epi-continuity and joint continuity of the objective function are established. All these results help us to prove a qualitative stability result. The obtained results extend current results to the mixed-integer recourse with random recourse matrices which have finitely many atoms.     

IIS branch-and-cut for joint chance-constrained stochastic programs and application to optimal vaccine allocation

We present a new method for solving stochastic programs with joint chance constraints with random technology matrices and discretely distributed random data. The problem can be reformulated as a large-scale mixed 0–1 integer program. We derive a new class of optimality cuts called IIS cuts and apply them to our problem. The cuts are based on irreducibly infeasible subsystems (IIS) of an LP defined by requiring that all scenarios be satisfied. We propose a method for improving the upper bound of the problem when no cut can be found. We derive and implement a branch-and-cut algorithm based on IIS cuts, and refer to this algorithm as the IIS branch-and-cut algorithm. We report on computational results with several test instances from optimal vaccine allocation. The computational results are promising as the IIS branch-and-cut algorithm gives better results than a state-of-the-art commercial solver on one class of problems.     

Symbolic and numeric scheme for solution of linear integro-differential equations with random parameter uncertainties and Gaussian stochastic process input

The paper describes a theoretical apparatus and an algorithmic part of application of the Green matrix-valued functions for time-domain analysis of systems of linear stochastic integro-differential equations. It is suggested that these systems are subjected to Gaussian nonstationary stochastic noises in the presence of model parameter uncertainties that are described in the framework of the probability theory. If the uncertain model parameter is fixed to a given value, then a time-history of the system will be fully represented by a second-order Gaussian vector stochastic process whose properties are completely defined by its conditional vector-valued mean function and matrix-valued covariance function. The scheme that is proposed is constituted of a combination of two subschemes. The first one explicitly defines closed relations for symbolic and numeric computations of the conditional mean and covariance functions, and the second one calculates unconditional characteristics by the Monte Carlo method. A full scheme realized on the base of Wolfram Mathematica and Intel Fortran software programs, is demonstrated by an example devoted to an estimation of a nonstationary stochastic response of a mechanical system with a thermoviscoelastic component. Results obtained by using the proposed scheme are compared with a reference solution constructed by using a direct Monte Carlo simulation.     

Estimation of systems with statistically-constrained inputs

This paper discusses the estimation of a class of discrete-time linear stochastic systems with statistically-constrained unknown inputs (UI), which can represent an arbitrary combination of a class of un-modeled dynamics, random UI with unknown covariance matrix and deterministic UI. In filter design, an upper bound filter is explored to compute, recursively and adaptively, the upper bounds of covariance matrices of the state prediction error, innovation and state estimate error. Furthermore, the minimum upper bound filter (MUBF) is obtained via online scalar parameter convex optimization in pursuit of the minimum upper bounds. Two examples, a system with multiple piecewise UIs and a continuous stirred tank reactor (CSTR), are used to illustrate the proposed MUBF scheme and verify its performance.     

Shift permutation invariance in linear random factor models

The objective of this paper is to consider shift invariance, a specific type of exchangeability, of random factors in linearmodels. The randomfactors are described via their covariance matrices and it is shown that shift invariance implies circular Toeplitz covariancematrices and marginally shift invariance implies block circular Toeplitz covariance matrices. In order to get interpretable linear models reparametrization is performed. It is shown that by putting restrictions on the spectrum of the shift invariant covariance matrices natural reparametrization conditions for the corresponding factors are obtained which then, among others, can be used to obtain unique parametrizations under shift invariance.      

Problem-based optimal scenario generation and reduction in stochastic programming

Scenarios are indispensable ingredients for the numerical solution of stochastic programs. Earlier approaches to optimal scenario generation and reduction are based on stability arguments involving distances of probability measures. In this paper we review those ideas and suggest to make use of stability estimates based only on problem specific data. For linear two-stage stochastic programs we show that the problem-based approach to optimal scenario generation can be reformulated as best approximation problem for the expected recourse function which in turn can be rewritten as a generalized semi-infinite program. We show that the latter is convex if either right-hand sides or costs are random and can be transformed into a semi-infinite program in a number of cases. We also consider problem-based optimal scenario reduction for two-stage models and optimal scenario generation for chance constrained programs. Finally, we discuss problem-based scenario generation for the classical newsvendor problem.

     

Linear-quadratic efficient frontiers for portfolio optimization

Finding portfolios with given mean return and minimal lower partial mean or variance, two risk criteria of interest in the theory of optimal portfolio selection, is a stochastic linear-quadratic program that can be converted to a large-scale linear or quadratic program when the asset returns are finitely distributed. These efficient frontiers can be computed on presently available platforms for problems of reasonable size; we discuss our experience with a problem involving one thousand assets. Asymptotic statistics for stochastic programs can be applied to justify sampling as a means to approximate continuous distributions by finite distributions.     

New Risk-Averse Control Paradigm for Stochastic Two-Time-Scale Systems and Performance Robustness

This work is concerned with the optimal control of stochastic two-time-scale linear systems with performance measure in a finite-horizon integral-quadratic form. Nature, modeled by stationary Wiener processes whose mean and covariance statistics are known, malevolently affects the state dynamics and output observations of the control problem class. With particular focus on the system performance robustness, the use of higher-order statistics or cumulants associated with the performance measure of chi-squared random variable type makes it possible to restate the stochastic control problem as the solution of a deterministic one, which subsequently allows disregarding all sample-path realizations by Nature acting on the original problem.     

Behavior of Stochastic Shear Beams Under Random Loading via Stochastic Variational Principles

In this study we derive, apparently for the first time in the literature, some exact solutions for the shear beams with stochastic flexibility, when these beams are acted upon by the random loading. The importance of these solutions lies in the fact that they can serve as benchmark solutions, to which the approximate solutions of various nature can be compared. Then we formulate stochastic variational principles, the first principle governs the behavior of the mean displacement, whereas the second principle is satisfied by the displacements covariance function. These variational principles allow us to formulate approximate techniques for the cases in which the exact solution is presently unavailable. In particular, we develop a stochastic version of the Rayleigh–Ritz method. Several examples are evaluated to shed more light on the probabilistic behavior of randomly excited structures possessing random flexibility.     

Spatial Localization for Nonlinear Dynamical Stochastic Models for Excitable Media

Nonlinear dynamical stochastic models are ubiquitous in different areas. Their statistical properties are often of great interest, but are also very challenging to compute. Many excitable media models belong to such types of complex systems with large state dimensions and the associated covariance matrices have localized structures. In this article, a mathematical framework to understand the spatial localization for a large class of stochastically coupled nonlinear systems in high dimensions is developed. Rigorous \linebreak mathematical analysis shows that the local effect from the diffusion results in an exponential decay of the components in the covariance matrix as a function of the distance while the global effect due to the mean field interaction synchronizes different components and contributes to a global covariance. The analysis is based on a comparison with an appropriate linear surrogate model, of which the covariance propagation can be computed explicitly. Two important applications of these theoretical results are discussed. They are the spatial averaging strategy for efficiently sampling the covariance matrix and the localization technique in data assimilation. Test examples of a linear model and a stochastically coupled FitzHugh-Nagumo model for excitable media are adopted to validate the theoretical results. The latter is also used for a systematical study of the spatial averaging strategy in efficiently sampling the covariance matrix in different dynamical regimes.     

Sample Covariance Matrix for Random Vectors with Heavy Tails

We compute the asymptotic distribution of the sample covariance matrix for independent and identically distributed random vectors with regularly varying tails. If the tails of the random vectors are sufficiently heavy so that the fourth moments do not exist, then the sample covariance matrix is asymptotically operator stable as a random element of the vector space of symmetric matrices.