Full metadata record
| DC Field | Value | Language |
|---|---|---|
| dc.contributor | Department of Applied Mathematics | en_US |
| dc.contributor.advisor | Chen, Xiaojun (AMA) | en_US |
| dc.creator | Zha, Xiao | - |
| dc.identifier.uri | https://theses.lib.polyu.edu.hk/handle/200/14465 | - |
| dc.language | English | en_US |
| dc.publisher | Hong Kong Polytechnic University | en_US |
| dc.rights | All rights reserved | en_US |
| dc.title | Robust solutions to a system of stochastic vertical linear complementarity problems and applications in portfolio selections | en_US |
| dcterms.abstract | The stochastic vertical linear complementarity problem (SVLCP) provides a mathematical model encompassing the optimality conditions of stochastic linear and quadratic programs, as well as stochastic bimatrix games. Various formulations and variants of the SVLCP have been proposed and analyzed in the literature. This thesis introduces a new stochastic minimization formulation designed to obtain robust solutions for systems of SVLCPs. The proposed formulation minimizes a risk function subject to stochastic vertical linear complementarity constraints. The content is mainly divided into the following two aspects. | en_US |
| dcterms.abstract | The first part of this thesis addresses the case where the underlying probability space consists of finitely many scenarios. We reformulate the proposed formulation with a finite support set as a linearly constrained piecewise smooth minimization problem by a penalty method. We prove the existence of exact penalty parameters regarding global and local minimizers. We define a smoothing function of the piecewise smooth objective function and show that the smoothing function satisfies the Kurdyka-Lojasiewicz (KL) property. Moreover, we propose a smoothing block coordinate descent (SBCD) algorithm, and prove that the sequence generated by the algorithm globally converges to an ϵ-Clarke stationary point of the penalty problem by the KL property for any ϵ > 0. Finally, we apply our formulation and algorithm to portfolio selection problems with real data. Numerical results demonstrate the robustness of our approach. | en_US |
| dcterms.abstract | The second part of the thesis investigates the sample average approximation (SAA) of the stochastic minimization formulation. We begin by introducing an implicit reformulation of the original problem, which is then approximated through the SAA scheme. The convergence properties of this approximation are analyzed in two folds. First, as the sample size tends to infinity, the optimal value of the SAA problem converges to the counterpart in the true problem. Second, as the sample size approaches infinity, an optimal solution of the SAA problem becomes an optimal solution of the true problem with probability one (w.p. 1). Moreover, we prove the uniform exponential convergence of the objective values and further show that this result implies exponential convergence of the SAA solutions toward the solution set of the original problem. Finally, the theoretical findings are complemented by extensive numerical experiments on portfolio selection problems, which confirm the convergence behavior of the SAA method. | en_US |
| dcterms.extent | xx, 79 pages : color illustrations | en_US |
| dcterms.isPartOf | PolyU Electronic Theses | en_US |
| dcterms.issued | 2026 | en_US |
| dcterms.educationalLevel | Ph.D. | en_US |
| dcterms.educationalLevel | All Doctorate | en_US |
| dcterms.accessRights | open access | en_US |
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