本论文着重研究了带有Lp模球约束以及二次约束的多项式优化问题的计算复杂度以及关于此类问题的近似算法。在本论文中，利用张量对称化的技巧，我们首次证明了当P∈ [2 ，∞] ，任意高阶的带有Lp模球约束的多项式优化问题均为NP 困难。借助模的对偶性质，我们将这类优化问题转化为求解凸体半径的问题，从而使得我们获得了之前研究所无法使用的算法工具。具体来说，利用计算凸几何的算法工具，对于Lp模球约束的多项式优化问题，我们得到了近似比为[附圖]的确定性多项式时间近似算法，其中d为目标多项式的阶次， n 为问题的维度。使用随机算法，我们将近似比进一步提高为此类问题的己知最优值。[附圖]。此外，我们发展了计算凸几何当中对于凸体半径的计算方法，从而设计出了一种对二次约束多项式优化问题近似比为[附圖]的近似算法，其中m为问题的约束个数。我们的结果涵盖并提高了之前关于此类问题的研究结果。我们相信在本论文中使用的新的算法工具，将在今后的多项式优化问题研究中得到更广泛的应用。 / In this thesis, we present polynomial time approximation algorithms for solving various homogeneous polynomial optimization problems and their multilinear relaxations. Specifically, for the problems with Lp ball constraint, where P∈ [2 ，∞], by reducing them to that of determining the Lq-diameter of certain convex body, we show that they can be approximated to within a factor of [with formula] in deterministic polynomial time, where q = p=(p - 1) is the conjugate of p, n is the number of variables, and d is the degree of the polynomial. We further show that with the help of randomization, the approximation guarantee can be improved to [with formula], which is independent of p and is currently the best for the aforementioned problems. Moreover, we extend the argument of deterministic algorithm mentioned above to solve the quadratically constrained polynomial optimization problems. In particular, for any intersection of ellipsoids K, we can, in polynomial time, construct a random polytope P, which satisfies [with formula]. Then, by reducing the problem to that of evaluating the maximum polytopal norm [with formula] induced by P, over certain convex body, we can approximate the quadratically constrained problem within a factor of [with formula] in polynomial time. Our results unify and generalize those in the literature, which focus either on the quadratic case or the case where [with formula]. We believe that the wide array of tools used in this thesis will have further applications in the study of polynomial optimization problems. / Detailed summary in vernacular field only. / Hou, Ke. / On title page "p" is subscript. / Thesis (Ph.D.) Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 106-111). / Abstracts also in Chinese.
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_1077641 |
| Date | January 2013 |
| Contributors | Hou, Ke (author.), So, Anthony Man-Cho (thesis advisor.), Chinese University of Hong Kong Graduate School. Division of Systems Engineering and Engineering Management, (degree granting institution.) |
| Source Sets | The Chinese University of Hong Kong |
| Language | English, Chinese |
| Detected Language | English |
| Type | Text, bibliography, text |
| Format | electronic resource, electronic resource, remote, 1 online resource (ix, 111 leaves), computer, online resource |
| 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/) |
Page generated in 0.0028 seconds