Solving nonconvex nonlinear programming problems via a new aggregate constraint homotopy method
By introducing \(C^2\) mappings \(\xi_i(x,z_i),\;i=1,2,\dots, m\) and using the idea of the aggregate function method, which replaces the exact aggregation \(\max_{i=1,2,\dots,m} g_i(x)\) by a smooth asymptotically exact aggregation \(g(x,\rho)=\rho\ln\sum_{j=1}^m \exp(g_i(x)/\rho), \rho\downarrow +0\), a new aggregate constraint homotopy method is proposed to solve the Karush-Kuhn-Tucker (KKT) conditions on nonlinear programming problems. With this mapping \(\xi\) the usual normal cone condition is weakened and the region for determining a starting point is greatly enlarged which may improve the computational efficiency of reduced predictor-corrector algorithms. The main results are proven and demonstrated with to examples. It remains open how to find the mappings \(\xi_i\) satisfying the weak normal cone conditions \(C_1,\dots, C_4\). Once the \(\xi_i\) are known then \(\xi=\sum_{i=1}^m \xi_i(x,y_i(x,\rho))z\) where \(\nabla_x g(x,\rho)=\sum_{i=1}^m y_i(x,\rho)\nabla g_i(x)\). Using \(\nabla_x g_i(x,\theta \mu)\) and \(\xi\) they construct a homotopy for KKT-conditions where for \(\mu=1\) the initial points are the solution und for \(\mu\rightarrow +0\) the solution of the original problems is given. The Euler-Newton procedure is used.
scientific article; zbMATH DE number 5774968