Optimality conditions and solution procedures for nondegenerate dual-response systems

This paper investigates the dual-response problem in the case where the response functions are nonconvex (nonconcave) quadratics and the independent variables satisfy a radial bound. Sufficient conditions for a global optimum are established and shown to generalize to the multi-response case. It is then demonstrated that the sufficient conditions may not hold if the problem is ‘degenerate’. However, if the problem is nondegenerate, it is shown that the sufficient conditions are necessarily satisfied by some stationary point. In this case, a specialized algorithm (DRSALG) is shown to locate the global optimum in a finite number of steps. DRSALG will also identify the degenerate case and pinpoint the location where degeneracy occurs. The algorithm is easy to implement from the standpoint of code development, and we illustrate our elementary version on a well-studied dual-response example from quality control.