求解一类约束优化信赖域方法的子问题

１引言设Ｈ为一给定的ｎ×ｎ对称矩阵，ｃＲ＂，本文考虑如｝的约束优化问题这里ａ＞０为给定的参数，Ｃ＝｛ｘＲｎｘ＜ａ是Ｒ”中的一个球体，Ｋ是一个简单凸闭集．当Ｋ＝Ｒｎ时，问题（Ｐ）便是无约束优化的信赖域子问题．当Ｋ＝｛ｘＲｎμ≤ｘ≤υ５，（μ１，μ２，…，μｎ）Ｔ，υ＝（υ１，υ２…，υｎ）Ｔ，且—∞＜μｉ＜υｉ＜ｖ＜＋∞，ｉ＝１，２，…，ｎ时，问题（Ｐ）便是用信赖域方法求解带上下界约束的优化问题时遇到的子问题．对于无约束信赖域方法的子问题已经有了比较成熟的算法［８，１２－１３，１５－１６］．Ｋ＝Ｒ…

SOLVING A SUBPROBLEM IN TRUST REGION APPROACH FOR A CLASS OF CONSTRAINED OPTIMIZATION PROBLEMS

In this paper, we present a method for solving tile basic problem in the trust region approach for optimization problem with simple bounds constraints. This method exploits the advantages of Gabay's decomposition method and the Projection and Contraction (PC) method. In fact. when the treated problem is solved by Gabay's decomposition method. the main work at each iteration is to solve a 'wellconditioned' linear variational inequality, which can be solved by the Projection and Contraction method efficiently. Some numerical tests show that the method is applicable.