Title: Multi-level Optimization by multi-parametric programming & its use for the solution of Mixed Integer Adjustable Robust Optimization Problems
Speaker: Prof. Stratos Pistikopoulos, FREng
Affiliation: Artie McFerrin Department of Chemical Engineering, Texas A&M University
Location: Room 217 Huxley Building
Abstract. Optimization problems involving multiple decision makers at different decision levels are referred to as multi-level programming problems. We are considering bi-level (two decision levels) and tri-level (three decision levels) programming problems. Multi-level programming problems are very challenging to solve even when considering just two linear decision levels. For classes of problems where the lower level problems also involve discrete variables, this complexity is further increased, typically requiring global optimization methods for its solution. Solution approaches for mixed integer bi-level problems with discrete variables in both levels mainly include reformulation approaches, branch and bound techniques or genetic algorithms, all of which result in approximate solutions.
In this work, we present novel algorithms for the exact, global and parametric solution of two classes of multi-level programming problems, namely (i) bi-level mixed-integer linear or quadratic programming problems (B-MILP or B-MIQP) and (ii) tri-level mixed-integer linear or quadratic programming problems (T-MILP or T-MIQP) containing both integer and continuous variables at all optimization levels. Based on multi-parametric theory and our earlier results for bi-level programing problems [5, 6], the main idea is to recast the lower levels of the multi-level programming problem as multi-parametric programming problems, in which the optimization variables of all the upper level problems, both continuous and integer, are considered as parameters for the lower level problems.
This novel algorithm can be then used for the exact and global solution of adjustable robust optimization problems. Classical robust optimization (RO) is an approach for incorporating uncertainty in optimization problems, and traditionally assumes that all decisions must be made before the realization of uncertainty (referred to as “here-and-now” decisions), a strategy which may be overly conservative. A more realistic approach is adjustable robust optimization (ARO) which involves recourse decisions (i.e. reactive actions after the realization of the uncertainty, “wait-and-see”) as functions of the uncertainty, typically posed in a two-stage stochastic setting. We propose a novel method for the derivation of generalized affine decision rules for linear/quadratic/nonlinear and mixed-integer ARO problems through multi-parametric programming. The problem is treated as a multi-level programming problem that can be then solved using the presented algorithm. A set of illustrative numerical examples are provided to demonstrate the potential of the proposed novel approach.
About the speaker. Professor Pistikopoulos is TEES Distinguished Research Professor in the Artie McFerrin Department of Chemical Engineering at Texas A&M University. He was a Professor of Chemical Engineering at Imperial College London, UK (1991-2015) and the Director of its Centre for Process Systems Engineering (2002-2009). At Texas A&M, he is the Interim Co-Director & Deputy Director of the Texas A&M Energy Institute, the Course Director of the Master of Science in Energy, the Director of the Gulf Coast Regional Manufacturing Centre, and the Texas A&M Principal Investigator of the RAPID Institute on process intensification, co-leading the Modeling & Simulation Focus Area.
He holds a Ph.D. degree from Carnegie Mellon University and he worked with Shell Chemicals in Amsterdam before joining Imperial. He has authored or co-authored over 400 major research publications in the areas of modelling, control and optimization of process, energy and systems engineering applications, 12 books and 2 patents. He is a Fellow of IChemE and AIChE, and the Editor-in-Chief of Computers & Chemical Engineering. He is the current Chair of the Computing and Systems Technology (CAST) Division of AIChE and he serves as a trustee of the Computer Aids for Chemical Engineering (CACHE) Organization. In 2007, Prof. Pistikopoulos was a co-recipient of the prestigious MacRobert Award from the Royal Academy of Engineering. In 2012, he was the recipient of the Computing in Chemical Engineering Award of CAST/AIChE. He received the title of Doctor Honoris Causa in 2014 from the University Politehnica of Bucharest, and from the University of Pannonia in 2015. In 2013, he was elected Fellow of the Royal Academy of Engineering in the UK.