Distributionally robust stochastic variational inequalities and applications

Pao Yue-kong Library Electronic Theses Database

Distributionally robust stochastic variational inequalities and applications

 

Author: Zhang, Yanfang
Title: Distributionally robust stochastic variational inequalities and applications
Degree: Ph.D.
Year: 2012
Subject: Variational inequalities (Mathematics)
Hong Kong Polytechnic University -- Dissertations
Department: Dept. of Applied Mathematics
Pages: xii, 99 p. : ill. ; 30 cm.
Language: English
InnoPac Record: http://library.polyu.edu.hk/record=b2615868
URI: http://theses.lib.polyu.edu.hk/handle/200/6970
Abstract: This thesis focuses on the stochastic variational inequality (VI). The stochastic VI has been used widely in engineering and economics as an effective mathematical model for a number of equilibrium problems involving uncertain data. For a class of stochastic VIs, we present a new residual function defined by the gap function in Chapter 2. The expected residual minimization (ERM) formulation is a nonsmooth optimization problem with linear constraints. We prove the Lipschitz continuity and semismoothness of the objective function and the existence of minimizers of the ERM formulation. We show various desirable properties of the here and now solution, which is a minimizer of the ERM formulation. In Chapter 3, we propose a globally convergent (a.s.) smoothing sample average approximation (SSAA) method for finding a minimizer of the ERM formulation. We show that the SSAA problems of the ERM formulation have minimizers in a compact set, and any cluster point of minimizers (stationary points) of the SSAA problems is a minimizer (a stationary point) of the ERM formulation (a.s.) as the sample size N →∞ and the smoothing parameter μ ↓ 0. We discuss the ERM formulation for the stochastic linear VI in Chapter 4, which is convex under some mild conditions. We apply the Moreau-Yosida regularization to present an equivalent smooth convex minimization problem. To have the convexity of the sample average approximation (SAA) problems of the ERM formulation, we adopt the Tikhonov regularization. We show that any cluster point of minimizers of the Tikhonov regularized SAA problems is a minimizer of the ERM formulation as the sample size N →∞ and the Tikhonov regularization parameter ε → 0. Moreover, we prove that the minimizer is the least l2-norm solution of the ERM formulation. We also prove the semismoothness of the gradients of the Moreau-Yosida and Tikhonov regularized SAA problems. In Chapter 5, we discuss the distributionally robust stochastic linear VI based on the ERM formulation. We introduce the CVaR formulation defined by the ERM formulation and establish the relationship between the CVaR formulation and the ERM formulation. For a wide range of cases, we show that the two formulations have the same minimizers. Moreover, we derive the gradient consistency for the smoothing CVaR formulation. We employ the sublinear expectation to consider the distributionally robust CVaR formulation for the stochastic linear VI, and prove the existence of minimizers of the robust CVaR formulation. We provide applications arising from traffic flow problems for stochastic VI in Chapter 6. We show the conditions and assumptions imposed in this thesis hold in such applications. Moreover, numerical results illustrate that the solutions, efficiently generated by the ERM formulation, have desirable properties.

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