Implicitly defined optimization problems (Q1184525)

This paper considers the solution of the general nonlinear implicity defined problems (GM), composed of a problem \(\text{G}: \inf_ y f[y,x(y)]\), s.t. \(y\in\overline R[y,x(y)]\subseteq E^ k\), \(y\in Y\), where \(Y\) is a set of vectors for which \(x(y)\) is the unique solution of the optimization problem \(\text{M}: \min_ x F(x,y)\), s.t. \(x\in R(x,y)\subseteq E^ n\). The authors give a survey of the relevant results from nonlinear sensitivity analysis theory. Their principal contribution is the use of the above theory to explicitly state the derivative information during the optimization of the problem (G) and the use of a penalty function approximation in (M) to resolve the generally unavoidable problems of nondifferentiability or noncontinuity of the implicitly defined problem (G). The derivative can be used to solve problem (G) by including it in iterative nonlinear programming algorithms which are appropriate for the structure of the problem (G). Other appropriate iterative nonlinear programming algorithms can be applied to problem (M). A higher-level iteration was employed between the solutions of problem (G) and problem (M). A proof of overall convergence is not provided.