(635e) Parametric & Adjustable Robust Optimization for the Integration of Design and Operation

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 [1]. 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 [2]. Such a formulation can be applied for the robust solution of the plant design and operation problem, where the design related decisions are treated as “here-and-now” variables, the operation or scheduling decisions are treated as “wait-and-see” variables and price and demand fluctuations are considered as uncertainties.

Solving ARO problems is challenging, therefore ways to reduce the computational effort have been proposed, with the most popular being the affine decision rules, where “wait-and-see” decisions are approximated as affine adjustments of the uncertainty [2, 3]; an approximation which can be overly conservative for general ARO problems. In this work we propose a novel method for the derivation of generalized affine decision rules for mixed-integer ARO problems through multi-parametric programming, that lead to the exact and global solution of the ARO problem. The problem is treated as a multi-level programming problem [4] and it is then solved using B-POP_®, a novel algorithm and toolbox for the exact and global solution of multi-level mixed-integer linear or quadratic programming problems [5, 6, 7]. The main idea behind the proposed approach is to solve the lower optimization level of the ARO problem parametrically, by considering “here-and-now” variables and uncertainties as parameters. This will result in a set of affine decision rules for the “wait-and-see” variables as a function of “here-and-now” variables and uncertainties optimal for their entire feasible space. A case study on the integration of plant design and operation is considered to illustrate the capacities of the proposed approach.

References:

[1] Ben-Tal, A., Ghaoui, L., Nemirovski, A. Robust optimization (2009) Princeton University Press, Princeton and Oxford.

[2] Ben-Tal, A., Goryashko, A., Guslitzer, E., Nemirovski, A. Adjustable robust solutions of uncertain linear programs (2004) Mathematical Programming, 99 (2), pp. 351-376.

[3] Kuhn, D., Wiesemann, W., Georghiou, A. Primal and dual linear decision rules in stochastic and robust optimization (2011) Mathematical Programming, 130 (1), pp. 177-209.

[4] Ning, C. and You, F. (2017) Data‐driven adaptive nested robust optimization: General modeling framework and efficient computational algorithm for decision making under uncertainty. AIChE J, 63, pp. 3790-3817.

[5] Avraamidou, S.; Pistikopoulos, E. N. (2019) B-POP: Bi-level Parametric Optimization Toolbox. Computers & Chemical Engineering, Available online.

[6] Avraamidou, S.; Pistikopoulos, E. N. (2019) A Multi-Parametric optimization approach for bilevel mixed-integer linear and quadratic programming problems. Computers & Chemical Engineering, 125, pp.98-113.

[7] Avraamidou, S.; Pistikopoulos, E. N. (2019) Multi-parametric global optimization approach for tri-level mixed-integer linear optimization problems. Journal of Global Optimization, Available online.