Example 7.4.3

For the SIP algorithm, each constraint set was initialized as empty, ϵg,0 = 0.9, and

ϵtol = 10−4.

For Case 2, the implicit SIP algorithm was applied and the global optimal solution was obtained with f∗ = y∗ = 10.1794m3_{, p}∗ _{= (0.38 0.058 60)}T_{. Therefore, in order}

to produce at least 22kmol/h of chlorobenzene, taking into account uncertainty in the input flowrate and the reaction rate constants, the reactor volume must be 10.1794m3_.

Note that the worst-case realization of uncertainty is exactly what is to be expected; in order to have the least amount of chlorobenzene in the product stream, k1 should

be the smallest value it can take, k2 should be the largest it can take, and the least

amount of benzene should be fed to the reactor. For a value of r = 18, the algorithm converges in 7 iterations and 11.71 seconds. The performance of the algorithm for Case 2 can be found in Figure 7-5.

0 2 4 6 8 10 12 14 16 18 0 20 40 60 80 100 120 140 160 180 200 CP U T im e (s ec ) r 0 2 4 6 8 10 12 14 16 18 0 20 40 60 80 100 120 num be r of i te ra ti ons r

Figure 7-5: The computational effort in terms of the number of iterations the algo- rithm takes to solve Case 2 of Example 7.4.3 versus the reduction parameter r.

Similar to Case 1 of Example 7.4.2, a small value for r resulted in the implicit SIP algorithm taking many iterations to converge. As r was increased, the number of iterations required to converge, as well as the total solution time dropped drastically and plateaued. A parameter value of r = 18 reduced the solution time by two orders of magnitude over r = 1.1. For each example, the implicit SIP algorithm performed

very favorably converging after only a few iterations of the algorithm.

For this example, Case 1 failed to converge within 200 iterations of the algorithm. This result is simply a consequence of using PBUNL which only accepts affine con- straints. In this case, since the affine constraints are being constructed with reference to the midpoint of X, the solver apparently fails to ever return a point that is feasible in the original SIP.

7.6

Concluding Remarks

In this chapter, a class of bilevel programs that commonly arise in engineering design problems was reformulated as a semi-infinite program with implicit functions embed- ded. An algorithm for solving SIPs with implicit functions embedded was presented which is an adaptation of a recently developed algorithm for solving standard SIPs.

As a proof-of-concept, three numerical examples were presented that illustrate the global solution of implicit SIPs using this algorithm. The first example illustrated the solution of a simple numerical system that fits the implicit SIP form given in (7.5). The second example was an engineering problem of robust design under uncertainty, originally cast as a constrained max-min problem as in (2.3). It was then reformulated as an implicit SIP of the form in (7.5) and solved using the implicit SIP algorithm. The third example was an engineering problem of optimal design of a chemical reactor considering uncertainty in the kinetic parameters and formulated as an SIP.

A method was presented for reformulating equality-constrained bilevel programs as SIPs with embedded implicit functions, requiring that:

1. all functions involved are continuous, 2. all functions involved are factorable,

3. derivative information for the equality constraint functions is available and is factorable,

4. there exists at least one solution x to the system of equations in (7.2) for every (y, p)∈ Y × P , and

5. there exists a Slater point arbitrarily close to a SIP minimizer.

To solve the resulting implicit SIP, the global optimization algorithm developed by Mitsos [87] has been adapted. The algorithm relies on the ability to solve three non- convex implicit NLP subproblems to global optimality. This is performed utilizing the relaxation methods and the deterministic algorithm for global optimization of implicit functions which were developed in Chapter 4. Algorithm 4.1 relies on the ability to solve nonsmooth lower- and/or upper-bounding problems at each iteration. This can be done using any available nonsmooth optimization algorithm or using the calculated subgradient information to construct affine relaxations and transform the problem into a linear program and solved using any efficient LP optimization algo- rithm. For this chapter, the nonsmooth bundle solvers PBUN and PBUNL [83], were utilized. Note that the requirements (2) and (3) are only due to current limitations of the algorithm for global optimization of implicit functions. The requirements (4) and (5) imply that the SIP is feasible and (1) and (5) are required for guaranteed ϵ-optimal convergence of the original explicit SIP algorithm [87] after finitely many iterations. Altogether, these requirements guarantee ϵ-optimal convergence of Algorithm 7.1.

Due to the limitations of the PBUNL solver, only affine constraints could be used. Since the implicit semi-infinite constraint is almost surely nonlinear, affine relaxations must be constructed. For the numerical examples, two sets of experiments were conducted: one using a single reference point for constructing affine relaxations of the constraints and another using three reference points for constructing affine relaxations of the constraints and using them all simultaneously. The first method was hypothesized to be advantageous since it required less computational effort to calculate the constraints. Alternatively, the second method was hypothesized to be advantageous since using multiple reference points results in better approximations of the constraints, which in turn may speed up convergence of the overall algorithm. For each experiment, it was observed that Case 2 was superior to Case 1 in terms of total CPU time. For Example 7.4.3, Case 1 even failed to converge after 200 iterations. This was likely due to the affine relaxations of the semi-infinite constraint not being very tight, resulting in PBUNL failing to find a solution that is feasible in the original

SIP.

In the next chapter, worst-case design of subsea production facilities is addressed. A slightly modified version of Algorithm 7.1 is applied and the feasibility problem is solved for various cases.

Chapter 8

Robust Simulation and Design of

Subsea Production Facilities

In this chapter, the problem of designing subsea production facilities for the worst case is revisited. In particular, a model of a subsea separator process is presented that can act as a framework for modeling systems involving more complex unit operations models in future case studies. A complete implementation of the algorithm for robust simulation and design is presented which utilizes the developments of the previous chapters and elsewhere. Finally, the robust simulation SIP (2.5) is solved for this model using the robust simulation algorithm implementation.

8.1

Background

In solving SIPs with implicit functions embedded, meaningful global bounding in- formation for the semi-infinite constraint function g is required. Calculating this information is often a limiting step in the overall performance of the algorithm be- cause all that is known about the state variables initially are their natural bounds, which in turn lead to a prohibitively large initial bound on g from which no mean- ingful information can be deduced. Since interval-Newton-type methods, discussed in Chapter 3, often prove to be ineffective in refining sufficiently large initial intervals, this poses a serious problem for the algorithm. Although interval-Newton methods

are quite effective on smaller intervals, a method that can obtain meaningful bounds on the function g starting with large initial intervals on the state variables efficiently is necessary for the overall success and performance of the algorithm.

Interval analysis has been widely applied to many simulation and optimization applications in chemical engineering, e.g., [6, 80]. In [80] strategies for bounding the solution of interval linear systems were presented, which were solved in the con- text of the interval-Newton method. The authors reviewed several preconditioning techniques for the above mentioned method and proposed a new bounding approach based on the use of linear programming (LP) strategies. They demonstrated the performance of the proposed technique on global optimization problems such as pa- rameter estimation and molecular modeling. In [6], interval-based global optimization of modular process models is addressed. In their work, the authors explored the use of five different interval contraction methods to improve the performance of the interval optimization algorithm of [24]. The contraction methods used were: consistency tech- niques, constraint propagation, LP contractors, interval Gaussian elimination, and the interval-Newton contractor. Using a set of mathematical problems and chemical engineering flowsheet design problems such as the Haverly pooling problem, reac- tor flowsheet problem, and a reactor network problem, they compared the impact of various contraction methods on the overall performance of the interval optimization algorithm. Their computational experiments showed that the LP contractors per- formed the best while the constraint propagation and interval Gaussian elimination methods were ineffective.

In the context of interval contraction, there exist several methods developed by re- searchers outside of the process engineering community. For a detailed review of such methods, see [66]. In [125] the fundamentals of interval analysis on directed acyclic graphs (DAGs) for global optimization and constraint propagation were presented. The proposed framework overcomes the limitation of propagating each constraint in- dividually by taking into account the effects of any common subexpressions that are shared by many constraints. Later, the above framework was extended to perform adaptively forward evaluations and backward projections on only some select nodes

of a DAG [141]. The computational study showed that the adaptive framework per- forms at least one to two orders of magnitude faster than the other state-of-the-art interval constraint propagation techniques.

More recently, the adaptive DAG framework of [141] was used in a branch-and- prune algorithm to find multiple steady states in homogeneous azeotropic and ideal two-product distillation problems [5]. Their computational experiments showed some promising results from the application of constraint propagation techniques of [141]. In this work, a forward-backward constraint propagation technique, similar to the DAG framework of [125], will be discussed and exploited. The technique is used to obtain meaningful bounds on the implicit functions of (2.5). Thus, the goal is to expedite the above bounding procedure over a given large initial box using the con- straint propagation technique, and subsequently obtain rigorous, tight, and conver- gent bounds on implicit functions using the interval-Newton method. Combining the strengths of forward-backward constraint propagation and interval-Newton methods seems to be a promising approach to obtaining useful bounding information required for solving (2.5), and this will be the focus of the proposed solution framework.