Nonlinear robust optimization for process design

Abstract

A novel robust optimization framework is proposed to address general nonlinear problems in process design. Local linearization is taken with respect to the uncertain parameters around multiple realizations of the uncertainty, and an iterative algorithm is implemented to solve the problem. Furthermore, the proposed methodology can handle different categories of problems according to the complexity of the problems. First, inequality-only constrained optimization problem as studied in most existing robust optimization methods can be addressed. Second, the proposed framework can deal with problems with equality constraint associated with uncertain parameters. In the final case, we investigate problems with operation variables which can be adjusted according to the realizations of uncertainty. A local affinely adjustable decision rule is adopted for the operation variables (i.e., an affine function of the uncertain parameter). Different applications corresponding to different classes of problems are used to demonstrate the effectiveness of the proposed nonlinear robust optimization framework.Next, the details for the feasibility test step are discussed. For the general case, to test the feasibility of the design variables, the following problem is solved

Robust Optimization for the Pooling Problem

“…Furthermore, even finding a feasible solution to Problem (1) requires solving this non-convex problem to global optimality. Problems in these categories can generally only be solved using semiinfinite programming techniques [46][47][48] , which are limited to small scale problems, or approximate robust optimization schemes [49][50][51][52] . Category 2 has received limited attention in the robust optimization and semiinfinite programming communities [53][54][55] .…”

A generalized cutting‐set approach for nonlinear robust optimization in process systems engineering

“…The flow sheet illustrating the reactor–separator system is shown in Figure 3a, and has been previously studied in Grossmann and Sargent, 1 Rooney and Biegler, 45 and Yuan et al 26 In this design problem, we are modeling the isothermal, liquid‐phase conversion of reactant A into a desired product C via a set of four first‐order chemical reactions, which are outlined in Figure 3b. Products D and E are undesirable side‐products in the reaction network.…”

“…To address this, Bertsimas et al 24 proposed a local search algorithm for identifying robust feasible solutions to uncertain optimization problems with non‐convex inequality constraints. Additionally, there have been recent advances in the development of novel methods and applications of RO methods to nonlinear process systems engineering models, including general nonlinear programming robust counterpart formulations, 25 robust counterparts with local linearization of nonlinear uncertain constraints and a novel sampling algorithm, 26 application to the pooling problem utilizing a cutting‐plane solution algorithm, 27 application to water treatment network operation, 28 robust counterpart derivation for the synthesis of fuel refineries under cost uncertainty, 29 and design and operation of process systems with resilience to disruptive events, 30 to name but a few.…”

A unified framework for adjustable robust optimization with endogenous uncertainty

“…For ease of exposition, we define new constraint and right-handside matrices such that we can express problem (19) in the following compact form:…”

“…However, traditional flexibility analysis focuses primarily on nonlinear problems, which may be the reason why it so far has not been recognized or noticed by the operations research community. In the last few years, the number of ARO-related works in PSE has increased rapidly, addressing diverse applications in process design, 19,20 planning and scheduling, [14][15][16][21][22][23][24] model predictive control, [25][26][27] supply chain optimization, 28,29 and so forth.…”