Optimal design of superstructures for placing units and streams with multiple and ordered available locations. Part I: A new mathematical framework

Abstract A new approach for the optimal design of superstructures in chemical engineering is proposed in this study. Contrary to most of the optimization techniques established in the literature, this approximation exploits the structure of a specific type of problem, i.e., the case where it is necessary to find the optimal location of a processing unit or a stream over a naturally ordered discrete set. The proposed methodology consists of reformulating the binary variables of the original Mixed-Integer Nonlinear Problem (MINLP) with a smaller set of integer variables referred to as external variables. Then, the reformulated optimization problem can be decomposed into a master Integer Program with Linear Constraints (master IPLC) and primal sub-problems in the form of Fixed Nonlinear Programs (FNLPs), i.e., Nonlinear Programs (NLPs) with integer variables fixed. The use of the Discrete-Steepest Descent Algorithm (D-SDA) is considered for the master IPLC, while the primal FNLPs are solved with existing Nonlinear Programming (NLP) solvers. The main features of this approach are discussed with an illustrative example: an isothermal Continuously Stirred Tank Reactor (CSTR) network with recycle and autocatalytic reaction. The new methodology does not guarantee global optimality; however, the results show that it can find a local solution in a short computational time.

[1]  Ignacio E. Grossmann,et al.  Systematic Methods of Chemical Process Design , 1997 .

[2]  C. Floudas.  Nonlinear and Mixed-Integer Optimization: Fundamentals and Applications , 1995 .

[3]  I. Grossmann,et al.  Optimal Design of Complex Distillation Columns Using Rigorous Tray-by-Tray Disjunctive Programming Models , 2000 .

[4]  P. Favati.  Convexity in nonlinear integer programming , 1990 .

[5]  Xiang Zhang,et al.  Rigorous design of reaction-separation processes using disjunctive programming models , 2018, Comput. Chem. Eng..

[6]  Ignacio E. Grossmann,et al.  An outer-approximation algorithm for a class of mixed-integer nonlinear programs , 1986, Math. Program..

[7]  Ignacio E. Grossmann,et al.  Mathematical programming approaches to the synthesis of chemical process systems , 1999 .

[8]  Ignacio E. Grossmann,et al.  Improving the performance of DICOPT in convex MINLP problems using a feasibility pump , 2020, Optim. Methods Softw..

[9]  I. Grossmann,et al.  Logic-based MINLP algorithms for the optimal synthesis of process networks , 1996 .

[10]  Jeff T. Linderoth,et al.  Algorithms and Software for Convex Mixed Integer Nonlinear Programs , 2012 .

[11]  Benjamin Müller,et al.  The SCIP Optimization Suite 5.0 , 2017, 2112.08872.

[12]  Edwin K. P. Chong,et al.  An Introduction to Optimization: Chong/An Introduction , 2008 .

[13]  I. Nowak.  Relaxation and Decomposition Methods for Mixed Integer Nonlinear Programming , 2005 .

[14]  Kazuo Murota,et al.  On Steepest Descent Algorithms for Discrete Convex Functions , 2003, SIAM J. Optim..

[15]  Zsolt Ugray,et al.  Dynamic filters and randomized drivers for the multi-start global optimization algorithm MSNLP , 2009, Optim. Methods Softw..

[16]  Michael C. Ferris,et al.  Grid-Enabled Optimization with GAMS , 2009, INFORMS J. Comput..

[17]  Arne Stolbjerg Drud,et al.  CONOPT - A Large-Scale GRG Code , 1994, INFORMS J. Comput..

[18]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[19]  Robin Smith,et al.  Chemical Process: Design and Integration , 2005 .

[20]  K. Murota.  Discrete convex analysis: A tool for economics and game theory , 2016, 2212.03598.

[21]  Michael R. Bussieck,et al.  General Algebraic Modeling System (GAMS) , 2004 .

[22]  Linus Schrage,et al.  The global solver in the LINDO API , 2009, Optim. Methods Softw..

[23]  Qi Chen,et al.  Recent Developments and Challenges in Optimization-Based Process Synthesis. , 2017, Annual review of chemical and biomolecular engineering.

[24]  Michel Gendreau,et al.  The Benders decomposition algorithm: A literature review , 2017, Eur. J. Oper. Res..

[25]  Kazuo Murota,et al.  Algorithms in discrete convex analysis , 2000 .

[26]  Nikolaos V. Sahinidis,et al.  Exploiting integrality in the global optimization of mixed-integer nonlinear programming problems with BARON , 2018, Optim. Methods Softw..

[27]  Lorenz T. Biegler,et al.  Nonlinear Optimization with Many Degrees of Freedom in Process Engineering , 2004 .

[28]  Juan P. Ruiz,et al.  Generalized Disjunctive Programming: A Framework for Formulation and Alternative Algorithms for MINLP Optimization , 2012 .

[29]  Satoru Iwata.  A Faster Scaling Algorithm for Minimizing Submodular Functions , 2003, SIAM J. Comput..

[30]  O. Nelles,et al.  An Introduction to Optimization , 1996, IEEE Antennas and Propagation Magazine.

[31]  Ignacio E. Grossmann,et al.  A review and comparison of solvers for convex MINLP , 2018, Optimization and Engineering.