熵函数法的数学理论

１．引言考虑复合函数其中ｇ；：Ｒ"－Ｒ，ｉ＝１，２，...；。连续可微．因的ｘ）的不可微性，涉及的ｘ）的优化问题，如极大极小问题Ｉｒｌｍｌｎｏｔｘｊ．ｆｉｊＺＦＲ"通常属不可微优化范畴．文山借助最大嫡原理推导出一类一致逼近一（Ｘ）的可微函数（称之为妨函数）Ｏ。ｉｌｌ--一ｉｆｌｙｌｌＸＤＣＱ．ｌｏｌｌ．ＩＪｊ容易证明ｔｉｍ人一中且对任意ｘＥＲ"，ＣＭ＋ＯＯｇｂｃｌ（ｌ＞ｏｆｃ。（ｌ，ＶＣＺ＞ＣＩ＞０，（４ｉｌｌｆｆｉ０＜ｕｌ。（）ｄｉ（］＜．（５基于该性质，文山一【４］通过一次取定较大有限值Ｃ。＞０，将…

MATHEMATICAL THEORY FOR ENTROPYFUNCTION METHOD

The convergence of the entropy function method for convex nonlinear min-maxproblems is proved. By analyzing the eigenvalue structure of the Hessian matrix,it is found that for high values of the approximation controlling parameter c thedifferentiable optimization problem involved in the entropy function method becomes ill-conditioned and hence difficult to solve. Furthermore, it is shown thatthe entropy function method is indeed equivalent to the simple exponential penaltymethod and hence can be further discussed in the framework of penalty functionmethods. Based on this discovery, in the convex case, it is proved that the entropyfunction method involving Lagrange multiplier (i.e. exponential multiplier penaltymethod) is convergence for ally finite parameter c and hence the ill-condition encountered in the original method can be completely avoided.