A smooth quasi-exact penalty function for nonlinear programming (Q1203874)

This paper presents a new penalty function for solving nonlinear programming problems (P): minimize \(f(x)\) in a domain \(G=\{x\in R^ n| g_ i(x)\leq 0,\;i=1,\dots,m\}\), where \(f(x)\) and \(g_ i(x)\) have at least first derivatives. The penalty function presented here overcomes the difficulties of both smooth and non-smooth exact penalty functions by using a smooth function, which is derived by means of the maximum entropy principle described in another paper of the author [Computational Structural Mechanics and Appl. 8, 1-85 (1991) (in Chinese)], to replace the maximum function in an \(L_ \infty\) exact penalty function. It is shown that this penalty function can uniformly in the whole \(R^ n\) approximate and monotonically decrease to the \(L_ \infty\) penalty function when a controlling parameter tends to infinity. With these properties, it lends itself as a substitute to the \(L_ \infty\) exact penalty function in practical computations. The term ``quasi-exact'' is designated to a case of the parameter being finite, for which stronger penalties will be imposed than those given by the \(L_ \infty\) penalty function. Some examples are provided to demonstrate that a nonlinear programming problem can be accurately solved by only one iteration for a moderately large parameter.