(607b) Design Space Computation Via Restriction of Generalized Semi-Infinite Programs

Design space computation aims to determine a set of values for input to a process which yield process behavior within specifications. Often computation of the design space requires approximations. This can yield a set of input values which is not feasible in the specifications; i.e. the candidate design space contains inputs which do not yield desired process behavior. In many applications, such as pharmaceuticals manufacturing, such infeasible solutions are unacceptable. This work considers calculating a design space when feasibility is of primary importance.

Design space problems are a special instance of generalized semi-infinite programming (GSIP) called design centering. Typically these problems are only tractable if they can be reformulated as a semi-infinite program (SIP) or (finite) nonlinear program. However, the conditions under which these reformulations are possible are difficult to satisfy for many applications of interest.

Consequently, this work demonstrates a practical method to construct a restriction of the original design centering problem which is a GSIP that can be reformulated as a simpler problem. This GSIP can be solved to global optimality, and since it is a restriction, any solution must be feasible in the original design centering problem.

The construction of the restriction proceeds using nonsmooth convex relaxations (McCormick's). Replacing the lower-level problem of the GSIP with a convex relaxation results in a restriction of the overall GSIP. Since the relaxed lower-level problem is convex, arguments from duality theory can be used to reformulate the restricted GSIP as an equivalent SIP. Recent methods for the global solution of SIPs can then be used. The general framework is broadly applicable, and both interval and ellipsoidal design spaces can be determined. Furthermore, by parameterizing the convex relaxations of the lower-level problem by the design space parameters (the decision variables of the GSIP), tighter relaxations result, which yield a less conservative restriction of the GSIP.