博士論文著重研究關於多項式優化的理論，並討論其在風險管理及統計分析中的應用。我們主要研究的對象乃為在控制理論和穩健優化中常見的所謂S 引理。原始的S 引理最早由Yakubovich 所引入。它給出一個二吹多項式在另一個二吹多項式的非負域上為非負的等價條件。在本論文中，我們把S 引理推廣到一元高吹多項式。由於S 引理與穩健優化密切相關，所以我們的結果可廣泛應用於風險管理及統計分析，包括估算在高階矩約束下的非線性風險量度問題，以及利用半正定規劃來計算同時置信區域帶等重要課題。同時，在相關章節的末段，我們以數值實驗結果來引證有關的新理論的有效性和應用前景。 / In this thesis we study some structural results in polynomial optimization, with an emphasis paid to the applications from risk management problems and estimations in statistical analysis. The key underlying method being studied is related to the so-called S-lemma in control theory and robust optimization. The original S-lemma was developed by Yakubovich, which states an equivalent condition for a quadratic polynomial to be non-negative over the non-negative domain of other quadratic polynomial(s). In this thesis, we extend the S-Lemma to univariate polynomials of any degree. Since robust optimization has a strong connection to the S-Lemma, our results lead to many applications in risk management and statistical analysis, including estimating certain nonlinear risk measures under moment bound constraints, and an SDP formulation for simultaneous confidence bands. Numerical experiments are conducted and presented to illustrate the effectiveness of the methods. / Detailed summary in vernacular field only. / Wong, Man Hong. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 134-147). / Abstract also in Chinese. / Abstract --- p.i / 摘要 --- p.ii / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Meeting the S-Lemma --- p.5 / Chapter 3 --- A strongly robust formulation --- p.13 / Chapter 3.1 --- A more practical extension for robust optimization --- p.13 / Chapter 3.1.1 --- Motivation from modeling aspect --- p.13 / Chapter 3.1.2 --- Discussion of a more robust condition --- p.15 / Chapter 4 --- Theoretical developments --- p.19 / Chapter 4.1 --- Definition of several order relations --- p.19 / Chapter 4.2 --- S-Lemma with a single condition g(x)≥0 --- p.20 / Chapter 5 --- Confidence bands in polynomial regression --- p.47 / Chapter 5.1 --- An introduction --- p.47 / Chapter 5.1.1 --- A review on robust optimization, nonnegative polynomials and SDP --- p.49 / Chapter 5.1.2 --- A review on the confidence bands --- p.50 / Chapter 5.1.3 --- Our contribution --- p.51 / Chapter 5.2 --- Some preliminaries on optimization --- p.52 / Chapter 5.2.1 --- Robust optimization --- p.52 / Chapter 5.2.2 --- Semidefinite programming and LMIs --- p.53 / Chapter 5.2.3 --- Nonnegative polynomials with SDP --- p.55 / Chapter 5.3 --- Some preliminaries on linear regression and confidence region --- p.59 / Chapter 5.4 --- Optimization approach to the confidence bands construction --- p.63 / Chapter 5.5 --- Numerical experiments --- p.66 / Chapter 5.5.1 --- Linear regression example --- p.66 / Chapter 5.5.2 --- Polynomial regression example --- p.67 / Chapter 5.6 --- Conclusion --- p.70 / Chapter 6 --- Moment bound of nonlinear risk measures --- p.72 / Chapter 6.1 --- Introduction --- p.72 / Chapter 6.1.1 --- Motivation --- p.72 / Chapter 6.1.2 --- Robustness and moment bounds --- p.74 / Chapter 6.1.3 --- Literature review in general --- p.76 / Chapter 6.1.4 --- More literature review in actuarial science --- p.78 / Chapter 6.1.5 --- Our contribution --- p.79 / Chapter 6.2 --- Methodological fundamentals behind the moment bounds --- p.81 / Chapter 6.2.1 --- Dual formulations, duality and tight bounds --- p.82 / Chapter 6.2.2 --- SDP and LMIs for some dual problems --- p.84 / Chapter 6.3 --- Worst expectation and worst risk measures on annuity payments --- p.87 / Chapter 6.3.1 --- The worst mortgage payments --- p.88 / Chapter 6.3.2 --- The worst probability of repayment failure --- p.89 / Chapter 6.3.3 --- The worst expected downside risk of exceeding the threshold --- p.90 / Chapter 6.4 --- Numerical examples for risk management --- p.94 / Chapter 6.4.1 --- A mortgage example --- p.94 / Chapter 6.4.2 --- An annuity example --- p.97 / Chapter 6.5 --- Conclusion --- p.100 / Chapter 7 --- Computing distributional robust probability functions --- p.101 / Chapter 7.1 --- Distributional robust function with a single random variable --- p.105 / Chapter 7.2 --- Moment bound of joint probability --- p.108 / Chapter 7.2.1 --- Constraint (7.5) in LMIs --- p.112 / Chapter 7.2.2 --- Constraint (7.6) in LMIs --- p.112 / Chapter 7.2.3 --- Constraint (7.7) in LMIs --- p.116 / Chapter 7.3 --- Several model extensions --- p.119 / Chapter 7.3.1 --- Moment bound of probability of union events --- p.119 / Chapter 7.3.2 --- The variety of domain of x --- p.120 / Chapter 7.3.3 --- Higher moments incorporated --- p.123 / Chapter 7.4 --- Applications of the moment bound --- p.124 / Chapter 7.4.1 --- The Riemann integrable set approximation --- p.124 / Chapter 7.4.2 --- Worst-case simultaneous VaR --- p.124 / Chapter 7.5 --- Conclusion --- p.126 / Chapter 8 --- Concluding Remarks and Future Directions --- p.127 / Chapter A --- Nonnegative univariate polynomials --- p.129 / Chapter B --- First and second moment of (7.2) --- p.131 / Bibliography --- p.134
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_328521 |
| Date | January 2012 |
| Contributors | Wong, Man Hong., 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 (ix, 147 leaves) |
| 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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