Analysis of the local well‐posedness of optimization‐constrained differential equations by local optimality conditions. Issue 10 (23rd July 2020)

• Record Type: Journal Article
• Authors: Zhao, Xiao; Ploch, Tobias; Noack, Stephan; Wiechert, Wolfgang; Mitsos, Alexander; von Lieres, Eric
• Is Part Of: AIChE journal. Volume 66:Issue 10(2020)
• Page Start: n/a
• Page End: n/a
• Publication Date: 2020-07-23
• Subjects: complementarity system -- KKT conditions -- optimization‐constrained differential equations -- parametric optimization; Chemical engineering -- Periodicals; Génie chimique -- Périodiques; 660.28
• DOI: 10.1002/aic.16548
• Languages: English
• ISSNs: 0001-1541

Abstract

Abstract: Optimization‐constrained differential equations (OCDE) are a class of mathematical problems where differential equations are constrained by an embedded algebraic optimization problem. We analyze the well‐posedness of the local solutions of OCDE based on local optimality. By assuming linear independence constraint qualification and applying the Karush‐Kuhn‐Tucker optimality conditions, an OCDE is transformed into a complementarity system (CS). Under second‐order sufficient condition we show that (a) if strict complementary condition (SCC) holds, the local solution of OCDE is well‐posed, which corresponds to a mode of the derived CS; (b) at points where SCC is violated, a local solution of OCDE exists by sequentially connecting the local solutions of two selected modes of the derived CS. We propose an event‐based algorithm to numerically solve OCDE. We illustrate the approach and algorithm for microbial cultivation, single flash unit and contrived numerical examples.