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 這篇文章介紹了二次規劃(Quadratic Programming, QP)的基本概念和數值方法。二次規劃是一種優化問題,旨在最小化一個二次函數,並受到多面體約束。二次規劃可以應用於許多領域,例如機器學習、金融、控制工程等眾多領域。文章討論了各種數值方法,包括內點法、算子分裂方法(特別是交替方向乘子法, ADMM)和 active set methods。在這些方法中,active set methods 被認為是處理嚴格凸二次規劃高效的方法。該研究的目標是開發一種基於 active set method 的新型數值方法,且無需 proximal point iterations的外部迭代。
 The article introduces the basic concepts and numerical methods of Quadratic Programming(QP). Quadratic Programming is an optimization problem aimed at minimizing a quadratic function subject to polyhedral constraints. Quadratic programming can be applied in many fields such as machine learning, finance, control engineering, and many other fields. The article discusses various numerical methods, including interior point methods, operator splitting methods (specifically ADMM), and active set methods. Among these, active set methods are considered efficient for strictly convex quadratic programming. The objective of the research presented in the article is to develop a novel numerical method based on the active set method that does not require external proximal point iterations.
 Contents1 Introduction 12 Preliminary 32.1 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 KKT conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Active set method 103.1 Regularized Convex Quadratic Programs . . . . . . . . . . . . . . . . . 124 Algorithm 175 Numerical experiments 266 Appendix 316.1 Interior point methods[5] . . . . . . . . . . . . . . . . . . . . . . . . . . 316.2 Active set methods with proximal point iterations[1][2][3][4] . . . . . . . 326.3 Operator splitting methods[8] . . . . . . . . . . . . . . . . . . . . . . . 33
 [1] Y.C. Kuo, C.S. Liu, “An index search method based inner-outer iterative algorithm for solving nonnegative least squares problems,” Journal of Computational and Applied Mathematics, vol. 424, 2022.[2] D. Arnstr ̈om, A. Bemporad, and D. Axehill, “A Dual Active-Set Solver for Embedded Quadratic Programming Using Recursive LDLT Updates,” IEEE Transactions on Automatic Control, vol. 67, no.8, pp. 4362–4369, 2022.[3] A. Bemporad, “A quadratic programming algorithm based on nonnegative least squares with applications to embedded model predictive control,” IEEE Transactions on Automatic Control, vol. 61, no.4, pp. 1111–1116, 2016.[4] A. Bemporad, “A numerically stable solver for positive semidefinite quadratic programs based on nonnegative least squares,” IEEE Transactions on Automatic Control, vol. 63, no. 2, pp. 525–531, 2018.[5] S. Boyd and L. Vandenberghe, “Convex Optimization,” Cambridge UniversityPress, 2004.[6] B. Stellato, G. Banjac, P. Goulart, A. Bemporad, and S. Boyd, “OSQP : An Operator Splitting Solver for Quadratic Programs,” Mathematical Programming Computation, pp. 1–36, 2020.[7] S.C Fang and S. Puthenpura, “Linear Optimization and Extensions: Theory and Algorithms,” Prentice Hall, 1993.[8] Jean Gallier and Jocelyn Quaintance, “Linear Algebra and Optimization with Applications to Machine Learning : Volume II: Fundamentals of Optimization Theory with Applications to Machine Learning,” WSPC, 2020
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