近年来，随着机会约束规划被广泛应用以及凸分析和概率论的新进展，如何有效的处理机会约束成为一个炙手可热的研究方向。其中，一个成功的解决方法就是考虑其安全可解近似，也就是说将机会约束转化成一组方便处理的确定性约束，并且保持原机会约束在新的约束下成立。目前这样的方法主要应用于带有独立分布的数据扰动的机会约束规划，或者已知扰动的协方差矩阵的情况。同时，带有相关数据扰动的机会约束下的锥不等式广泛应用于供应链管理、金融、控制以及信号处理等学科，而现有的优化理论却极少涵盖。 / 在这篇论文中我们主要研究机会约束下的线性矩阵不等式，并假设扰动分布不必相互独立，其仅有的相关性信息只由一系列子扰动的独立关系结构提供。通过推导矩阵值随机变量的大偏差上界，我们得出这一类条件约束的安全可解近似。我们随后考虑了基于条件风险价值度量的机会约束规划问题, 以及带多项式扰动的机会约束优化问题。另外，通过构造相应的鲁棒对等式的不确定集合，我们把机会约束规划转换成鲁棒优化问题。由于这种近似可以表示为一组线性矩阵不等式，因而可以使用现成的优化软件方便地求解。最后，我们把该安全可解近似方法运用到一个控制理论问题，以及一个带风险价值约束的投资组合优化问题中。 / The wide applicability of chanceconstrained programming, together with advances in convex optimization and probability theory, has created a surge of interest in finding efficient methods for processing chance constraints in recent years. One of the successes is the development of so-called safe tractable approximations of chance-constrained programs, where a chance constraint is replaced by a deterministic and efficiently computable inner approximation. Currently, such an approach applies mainly to chance-constrained linear inequalities, in which the data perturbations are either independent or define a known covariance matrix. However, its applicability to the case of chanceconstrained conic inequalities with dependent perturbations--which arises in supply chain management, finance, control and signal processing applications--remains largely unexplored. / In this thesis, we consider the problem of processing chance-constrained affinely perturbed linear matrix inequalities, in which the perturbations are not necessarily independent, and the only information available about the dependence structure is a list of independence relations. Using large deviation bounds for matrix-valued random variables, we develop safe tractable approximations of those chance constraints. Extensions to the Matrix CVaR (Conditional Value-at-Risk) risk measure and general polynomials perturbations are also provided separately. Further more, we show that the chanceconstrained linear matrix inequalities optimization problem can be converted to a robust optimization problem by constructing the uncertainty set of the corresponding robust counterpart. A nice feature of our approximations is that they can be expressed as systems of linear matrix inequalities, thus allowing them to be solved easily and efficiently by off-the-shelf optimization solvers. We also provide a numerical illustration of our constructions through a problem in control theory and a portfolio VaR (Value-at-Risk) optimization problem. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Wang, Kuncheng. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 94-101). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese. / Abstract --- p.i / Acknowledgement --- p.vi / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Motivations and Philosophy --- p.1 / Chapter 1.2 --- Background --- p.2 / Chapter 1.3 --- Literature Review --- p.4 / Chapter 1.4 --- Contribution --- p.7 / Chapter 2 --- Preliminaries --- p.10 / Chapter 2.1 --- Probabilistic Inequalities --- p.10 / Chapter 2.2 --- Exact Proper Fractional Covers --- p.12 / Chapter 2.2.1 --- Exact Proper Fractional Cover of Quadratic Perturbations --- p.15 / Chapter 3 --- Large Deviations of Sums of Dependent Random Matrices --- p.18 / Chapter 3.1 --- The Matrix Exponential Function and Its Properties --- p.18 / Chapter 3.2 --- Main Theorem --- p.19 / Chapter 4 --- From Large Deviations to ChanceConstrained LMIs --- p.26 / Chapter 4.1 --- General Results --- p.26 / Chapter 4.2 --- Application to ChanceConstrained Quadratically Perturbed Linear Matrix Inequalities --- p.30 / Chapter 4.3 --- Bounding the Matrix Moment Generating Functions --- p.31 / Chapter 4.4 --- Iterative Improvement of the Proposed Approximations --- p.42 / Chapter 5 --- Computational Studies --- p.49 / Chapter 5.1 --- Application to Control Problems --- p.49 / Chapter 5.2 --- Application to Value-at-Risk Portfolio Optimization --- p.57 / Chapter 6 --- ChanceConstrained LMIs with CVaR Risk Measure --- p.64 / Chapter 6.1 --- Matrix CVaR Risk Measure --- p.65 / Chapter 6.2 --- Some Useful Inequalities --- p.68 / Chapter 6.3 --- From Matrix CVaR to ChanceConstrained LMIs --- p.69 / Chapter 6.3.1 --- Bound π¹(A₀, · · · ,A[subscript m]) --- p.70 / Chapter 6.3.2 --- Bound π²(A₀, · · · ,A[subscript m]) --- p.71 / Chapter 6.3.3 --- Bound π³(A₀, · · · ,A[subscript m]) --- p.72 / Chapter 6.3.4 --- Convex Approximation of π[superscript i](A0, · · · ,Am) --- p.73 / Chapter 7 --- Extension to Polynomials Perturbations --- p.75 / Chapter 7.1 --- Decoupling Theory --- p.75 / Chapter 7.2 --- Safe Tractable Approximation by SecondOrder Cone Programming --- p.77 / Chapter 8 --- Construct Uncertainty Set for Chance Constraints --- p.81 / Chapter 8.1 --- Problem Statement --- p.82 / Chapter 8.2 --- Fractional Cover for Quartic Perturbations --- p.83 / Chapter 8.3 --- Probabilistic Guarantees --- p.85 / Chapter 8.3.1 --- Probabilistic Bound Based on Large Deviations --- p.85 / Chapter 8.4 --- The Value of Ω for Bounds --- p.88 / Chapter 8.5 --- Computational Study --- p.89 / Chapter 8.5.1 --- Independent Standard Normal Perturbations --- p.89 / Chapter 8.5.2 --- Independent Bounded Quadratic Perturbations --- p.91 / Chapter 9 --- Conclusion --- p.93 / Bibliography --- p.94
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_328175 |
| Date | January 2012 |
| Contributors | Wang, Kuncheng., Chinese University of Hong Kong Graduate School. Division of Systems Engineering and Engineering Management. |
| Source Sets | The Chinese University of Hong Kong |
| Language | English, Chinese |
| Detected Language | English |
| Type | Text, bibliography |
| Format | electronic resource, electronic resource, remote, 1 online resource (xi, 101 leaves) : ill. (some col.) |
| Rights | Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/) |
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