Semi-Infinite Optimization with Implicit Functions

Transcription

1 ubs.acs.org/iecr Semi-Infinite Otimization with Imlicit Functions Matthew D. Stuber and Paul I. Barton* Process Systems Engineering Laboratory, Det. of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States ABSTRACT: In this work, equality-constrained bilevel otimization roblems, arising from engineering design, economics, and oerations research roblems, are reformulated as an equivalent semi-infinite rogram (SIP) with imlicit functions embedded, which are defined by the original equality constraints that model the system. Using recently develoed theoretical tools for bounding imlicit functions, a recently develoed algorithm for global otimization of imlicit functions, and a recently develoed algorithm for solving standard SIPs with elicit functions to global otimality, a method for solving SIPs with imlicit functions embedded is resented. The method is guaranteed to converge to ϵ-otimality in finitely many iterations given the eistence of a Slater oint arbitrarily close to a minimizer. Besides the Slater oint assumtion, it is assumed only that the functions are continuous and factorable and that the model equations are once continuously differentiable. INTRODUCTION Many engineering design feasibility and reliability roblems give rise to otimization rograms whose feasible sets are arametrized. This is because it is often of great interest to study erformance and/or safety of engineering systems under arametric uncertainty. Particularly, it is imortant to study the erformance and/or safety in the face of the worst case, giving rise to equality-constrained bilevel rograms of the following form: f*= min f( ) s.t.0 ma g(, y, ) y, s.t. h (, y, ) = 0 X = { : } P = { : } ny y D y It is assumed that the objective function f : D and the inequality constraint function g: D Dy D are continuous and are factorable in the sense that they are comosed from elementary arithmetic oerations and transcendental functions. The equality constraints of eq 1 are the system of equations reresenting a steady-state model of the system of interest: h (, y, ) = 0 (2) n with h: D Dy D y, also assumed factorable and continuously differentiable on its domain with n n D, D oen. Due to the comleity of the models to be considered in this roblem, the bilevel formulation eq 1 is intractable, or even imossible to solve. It is roosed in this work that, under some relatively mild assumtions, the equality constraints are used to eliminate y and eq 1 is reformulated as an equivalent semi-infinite rogram (1) (SIP) without equality constraints. If a y eists that satisfies eq 2 for each (,) X P D D, then it defines an imlicit function of (,), eressed as y(,). It must be assumed that there eists at least one imlicit function y:x P Y such that h(,y(,),) =0, (,) X P with Y D y. Conditions guaranteeing uniqueness of y Y are given by the semilocal imlicit function theorem. 1 Given the eistence of an imlicit function y (and its uniqueness in Y), the equality constraints can be eliminated and the rogram eq 1 can be eressed as min f ( ) s.t.0 ma g(, y(, ), ) X = { : } P = { : } Furthermore, the inner maimization rogram can be eressed as ma g(, y(, ), ) 0 g(, y(, ), ) 0, P P where the latter constraint is referred to as the (imlicit) semiinfinite constraint. The following SIP, equivalent to the original bilevel rogram in eq 1, can then be formulated: f*= min f( ) s.t. g(, y(, ), ) 0, P X = { : } P = { : } Received: July 21, 2014 Revised: Setember 26, 2014 Acceted: December 11, 2014 Published: December 11, 2014 (3) (4) (5) 2014 American Chemical Society 307

2 For a chemical engineering alication, y may reresent internal state variables, such as comosition, determined by an equation of state or some other hysics, the variables may reresent design variables such as chemical reactor dimensions or some ie lengths, and may reresent uncertain model arameters such as reaction rate constants. In this case, f may reresent some economic objective related to sizing and g may reresent a critical erformance and/or safety constraint. The global solution (if one eists) will corresond to the worst-case realization of uncertainty and address the question of otimal design under uncertainty. Alternatively, may reresent uncertainty in the system or environment and may reresent the controls. The function f may then reresent some metric of uncertainty and g may again reresent a erformance and/or safety constraint. In that case, the global solution (if one eists) will corresond to the worst-case realization of uncertainty for which there eists a control setting such that the system meets secification. This formulation addresses the question of feasibility of the design as well as the determination of the maimum allowable uncertainty realization such that the design remains feasible. In the net section, a background and literature review of semi-infinite rogramming is given. Then, the revious work on standard SIPs is etended to SIPs with imlicit functions embedded including the full statement of the algorithm for solving such SIPs. Following the resentation of the algorithm, three numerical eamles are resented and analyzed. BACKGROUND It has long been known that bilevel rograms and SIPs can be utilized to address roblems that are inherently uncertain. Kwak and Haug 2 addressed the question of otimal design under uncertainty using the bilevel formulation in eq 1 with first-order (affine) aroimations of f, g, and h. A stochastic version of the rogram in eq 1 was considered by Malik and Hughes 3 for otimal design of chemical rocesses under uncertainty. In Halemane and Grossmann, 4 the equality constraints were maniulated to give an elicit function of the control and uncertainty variables and a secial case of the SIP in eq 5, known as a ma min, min ma, or minima roblem. This aroach was elored further by Swaney and Grossmann 5 with alication to addressing the feasibility roblem. Two algorithms were resented by Swaney and Grossmann 6 for solving the roblem under certain conveity requirements. This formulation was further elored by Stuber and Barton, 7 with general nonconve g as in eq 5, in order to address the question of robust feasibility of engineering designs. A rigorous finite ϵ- convergent deterministic global otimization algorithm was resented that was based on the algorithm of Bhattacharjee et al. 8 The algorithm relies on the ability to solve the equality constraints aroimately for an imlicit function of the control and uncertainty variables using the successive-substitution fied-oint iteration 7 and so is limited to certain systems. Halemane and Grossmann 4 elained that the SIP formulation leads to an otimization roblem that is not necessarily differentiable. In order to maintain differentiability, a secial case of the bilevel formulation (eq 1), equivalent to that given by Halemane and Grossmann, 4 was further elored by Floudas et al. 9 The authors resented a rigorous deterministic algorithm, relying on twice-differentiability of h and g. 9 Ostrovsky and co-workers develoed algorithms for bounding solutions of the feasibility roblem as formulated by Halemane and Grossmann 4 and Swaney and Grossmann. 5 The algorithms rely on iteratively solving nonlinear rogramming roblems under similar certain conveity requirements. Therefore, the alicability is limited to such systems considered by Halemane and Grossmann 4 and Swaney and Grossmann. 5 Kwak and Haug 2 briefly discussed the relationshi between the min ma roblem formulation 3 7,9 and the bilevel formulation. In Mitsos et al., 13 the authors resented an algorithm to address the bilevel formulation with a nonconve inner rogram. However, their algorithm was limited in that it could only handle inequality constrained roblems and so cannot be alied to eq 1. This aer will focus on solving the general SIP in eq 5, without relying on aroimations of f, g, and h, such as in Kwak and Haug, 2 with alication to the min ma roblems that show u in economics and design feasibility roblems considered in the aforementioned articles, 3 7,9 only requiring continuity of f and g, once-differentiability of h, the eistence of a Slater oint arbitrarily close to a minimizer, and the eistence of a unique imlicit function y Y such that h(,y(,),) =0, (,) X P. Solving SIPs with elicit functions (e.g., without imlicit functions embedded as in eq 5), referred to as elicit SIPs herein, has been an active area of research for many years. An overview of the revious alication of elicit SIPs to realworld roblems with theoretical results and available methods can be found in revious works Contributions that have secific relevance to this work are summarized below. Blankenshi and Falk 17 resent a cutting-lane algorithm for aroimating solutions to elicit SIPs, which amounts to solving two nonlinear rograms (NLPs), to global otimality in the general case, at each iteration. Their algorithm generates a sequence of (not necessarily feasible) oints that converge to the solution of the SIP in the limit. 17 Under aroriate conveity assumtions, their algorithm converges finitely 17 to a feasible solution. Their method is alicable to SIPs in general and they make secific mention of the alication to the ma min roblem. The ma min roblem is further elored by Falk and Hoffman 18 for general nonconve functions. The cutting-lane algorithm relies on the techniques of discretization and what is called local reduction, which is a technique for theoretically describing (locally) the SIP feasible region with finitely many constraints. 19 Most SIP algorithms emloy these techniques in various ways. 16 Zuhe et al. 20 resented a method based on interval analysis for solving elicit min ma roblems, again, which are secial instances of elicit SIPs. Their method is alicable to minma roblems with twice continuously differentiable elicit functions. Interval analysis was used to dynamically eclude regions of the search sace guaranteed not to contain solutions. 20 It was suggested that, using the roerties of interval analysis and generalized bisection, their method converges in finitely many iterations. 20 Bhattacharjee et al. 21 alied interval analysis to the general case of elicit SIPs in order to construct what is called the inclusion-constrained reformulation, which is a valid restriction of the original elicit SIP. This idea was used further in the first algorithm for generating SIP-feasible oints finitely, that relies on the inclusion-constrained reformulation. 8 A lower-bounding rocedure that relies on McCormick s conve and concave relaations 22 and discretization was introduced. 8 Together with the inclusion-constrained reformulation and the branchand-bound (B&B) framework, Bhattacharjee et al. were able to solve SIPs to global otimality with guaranteed finite ϵ-otimal 308

3 convergence. 8 As reviously mentioned, this algorithm was emloyed by Stuber and Barton 7 to solve imlicit ma-min roblems cast as imlicit SIPs. Due to the overestimation of inclusion functions and the fact that the size of the uer- and lower-bounding roblems grow raidly with deth in the branch-and-bound tree, 8 this algorithm can be ineffective at solving imlicit SIPs modeling more comle rocesses. Stein and Still 23 solved elicit SIPs, with g conve, as a Stackelberg game using an interior-oint method. By conveity of g, they were able to eloit the first-order otimality conditions to characterize the solution set of the inner rogram and solve equivalent finite nonlinear rograms. 23 Floudas and Stein 24 used a similar idea and constructed concave relaations of g on P using αbb. 25 They then relaced the inner rogram with its KKT otimality conditions and solved the resulting finite nonlinear rogram with comlementarity constraints. 24 By doing so, the resulting rogram is a restriction of the original elicit SIP, and therefore, uon solution, it generates SIPfeasible oints. 24 This idea was concurrently discussed by Mitsos et al., 26 where they also considered a technique closely related to the inclusion-constrained reformulation 8,21 but instead used interval analysis to further construct Mc- Cormick-based concave relaations 22 of g on P to restrict the inner rogram and generate SIP-feasible oints finitely. More recently, Mitsos 27 develoed an algorithm based on the ideas of Blankenshi and Falk 17 that relies on a new restriction technique for the uer-bounding rocedure that requires the right-hand side of the semi-infinite constraint to be erturbed from zero. This formulation results in solving at least three NLP subroblems to global otimality at each iteration, in the general case, and the comutational results reorted 27 are quite romising. The key contribution is the novel uer-bounding rocedure that is guaranteed to generate SIP-feasible oints after finitely many iterations. It is stated elicitly that the algorithm only requires continuity of f and g and the eistence of a Slater oint arbitrarily close to a SIP minimizer, rovided the functions can be handled by the NLP solver. 27 Therefore, this algorithm could be alied to solve the SIP in eq 5 while handling the equality constraints directly, without requiring the introduction of the imlicit function, by formulating each nonconve subroblem as an equality-constrained global otimization roblem. However, this strategy is not advisable since the algorithm would then require the number of variables in the uer- and lower-bounding subroblems to increase with each iteration. Thus, these subroblems become increasingly more eensive to solve with each iteration. However, this algorithm is a romising candidate for the global solution of SIPs with imlicit functions embedded; the focus of this aer. With the ecetion of Stuber and Barton, 7 all of the aforementioned methods were develoed to solve elicit SIPs (or elicit min ma rograms). For clarity, it is worth mentioning that all of the reviously develoed methods that are guaranteed to generate a rigorous SIP-feasible oint in finitely many iterations rely on the eistence of a Slater oint or sequence of Slater oints arbitrarily close to an SIP minimizer. If one were to simly reformulate eq 1 as an SIP and handle the equality constraints, h, directly as a series of inequality constraints, as is commonly done in global otimization, the Slater oint assumtion would be violated. In other words, the rigorous methods are simly not alicable to rograms of the form of eq 1. The major comlication with formulating the bilevel rogram in eq 1 as the SIP in eq 5, is that an imlicit function y, which may not have a closed algebraic form, becomes embedded within the semi-infinite constraint g. Therefore, y (and g) cannot be evaluated directly but instead must be aroimated using a numerical method, such as Newton s method or some other fied-oint iteration. In order to modify reviously develoed methods that rely on relaations of the inner rogram, it must be ossible to construct relaations of g(,y(,),) onx, P. However, in order to rela g(,y(,),) onx, P, conve and concave relaations of the imlicit function y(,) on X must be calculable. As reviously mentioned, this has been achieved for roblems in which the imlicit function y could be aroimated using the successive-substitution fied-oint iteration. 7 The theoretical details of these relaations were resented by Scott et al. 28 This work will imrove on the revious results of Stuber and Barton 7 and consider solving SIPs with more general imlicit functions embedded that can be aroimated using any available method, such as Newton s method, instead of being restricted to the successivesubstitution case. This work will make use of a modified version of the algorithm develoed by Mitsos, 27 where the solution of each of the (imlicit) subroblems will be erformed using the novel relaation techniques and global otimization algorithm develoed in a recently ublished article. 29 In that article, 29 theoretical develoments were made to construct conve and concave relaations of more general imlicit functions. The construction of these relaations are analogous in many ways to how interval bounds can be calculated for imlicit functions using (arametric) interval Newton-tye methods. By alying (arametric) interval Newton-tye methods 1,7,30,31 to a function h, under certain conditions, an interval can be calculated that bounds a unique root, y, ofh over the set X P. Taking these bounds as initial relaations of y, they can be iteratively refined using the methods develoed by Stuber et al. 29 to roduce conve and concave relaations of y on X P. As a result, global otimization of imlicit functions was develoed. 29 It should be noted that although global otimization of imlicit functions will be relied uon in this aer, the intricate theoretical details of constructing relaations of imlicit functions and the workings of the B&B algorithm will not be necessary as these develoments will be called uon simly as art of the eternal otimization subroutines. In the net section, the global otimization algorithm for SIPs with imlicit functions embedded is discussed. The alication to min ma and ma min roblems is made elicit, immediately following the statement of the algorithm. Finally, three numerical eamles are given that illustrate the solution of imlicit SIPs to global otimality. GLOBAL SOLUTION OF SIPS WITH IMPLICIT FUNCTIONS EMBEDDED The global otimization algorithm for imlicit SIPs is based entirely on the cutting-lane algorithm resented by Mitsos, 27 which itself is based on the algorithm develoed by Blankenshi and Falk 17 but with a novel uer-bounding rocedure. The algorithm, as alied to elicit SIPs is guaranteed to roduce SIP-feasible oints after finitely many iterations under the assumtion that there eists a Slater oint arbitrarily close to a minimizer. 27 As reviously mentioned, the algorithm relies on the ability to solve three nonconve NLP subroblems to global otimality at each iteration. The three 309

4 subroblems are discussed below secialized to the case of imlicit SIPs. Lower-Bounding Problem. The lower-bounding rocedure comes from a simle relaation technique based on the adative discretization rocedure originally described by Blankenshi and Falk. 17 The SIP is reduced to an imlicit NLP, again by considering only a finite number of constraints corresonding to realizations of P LBD with P LBD P a finite set. The lower-bounding roblem is formulated as f LBD = min f( ) s.t. g(, y(, ), ) 0, P X = { : } LBD In order to guarantee f LBD f*, the lower-bounding roblem must be solved to global otimality. Inner Program. The inner rogram, stated elicitly in eqs 3 and 4, which is equivalent to the semi-infinite constraint, defines the SIP feasible region. Thus, given a candidate X, feasibility can be determined by solving the inner (in general nonconve) rogram: g ( ) = ma g(, y(, ), ) P (7) The oint is feasible in the original SIP given in eq 5 if g ( ) 0. Therefore, in order to determine feasibility of a candidate, the inner rogram (eq 7) must be solved to global otimality for the general case. Uer-Bounding Problem. The uer-bounding roblem comes from erturbing the right-hand side of the semi-infinite constraint away from zero by a arameter ϵ g > 0, referred to as the restriction arameter, 27 and reducing the SIP to an imlicit NLP by only considering a finite number of constraints corresonding to realizations of P UBD, where P UBD P is a finite set. The uer-bounding roblem is formulated as f UBD = min f( ) s.t. g(, y(, ), ) ϵ, P X = { : } g UBD As mentioned by Mitsos, 27 the uer-bounding roblem (eq 8) must be solved to global otimality in order for the algorithm to solve the original SIP (eq 5) to global otimality. However, a valid uer bound, f UBD f*, can be obtained by solving eq 8 locally for and verifying that it is feasible in the original SIP (eq 5). Algorithm. In this section, the algorithm used for solving globally SIPs with imlicit functions embedded to guaranteed ϵ-otimality is given. Again, as resented, this algorithm is an adatation of the algorithm given by Mitsos 27 to SIPs with imlicit functions embedded. Finite convergence of the algorithm for elicit SIPs was reviously roven. 27 The results roven by Mitsos 27 etend directly to the imlicit SIP algorithm rovided that finite convergence of each imlicit NLP subroblem can be guaranteed. The latter result was roven by Stuber et al. 29 The assumtions on which the SIP algorithm relies uon are stated elicitly in the following. Assumtion 1. (a) The functions f : D and g: D Dy D are factorable 28 and continuous on their domains. (6) (8) (b) Derivative information y h i, i = 1,...,n y is available and is factorable, say by automatic differentiation. 32,33 (c) There eists y:x P D y such that h(,y(,),) =0, (,) X P, and an interval Y D y is available such that y(x,p) Y and y(,) is unique for every (,) X P. (d) A matri Ψ n y n y is known such that A ΨJ y (X,Y,P) satisfies 0 A ii for all i, where J y is an inclusion monotonic interval etension of the Jacobian matri of h, J y,onx Y P. (e) There eists a oint S X with g( S,y( S,),)<0, P such that f( S ) f* < ϵ tol. Assumtion 1(a d) are essentially required by the eternal subroutines for constructing conve and concave relaations for global otimization of imlicit functions. 29 Assumtion 1(c) can be satisfied by alying arametric interval-newton methods 1,7,30,31 and their key theoretical results. For Assumtion 1(d), the matri Ψ is a reconditioning matri and has been the focus of many research articles. The alication to interval-newton methods is discussed by Kearfott, 34 among others. The interval-valued matri A can be calculated efficiently by taking natural interval etensions 1,35 and thus satisfying Assumtion 1(d). Assumtion 1(e) is the ϵ tol -otimal SIP-Slater oint condition, which guarantees that a sequence of feasible oints can be generated by the algorithm. Altogether, satisfying Assumtion 1 guarantees that the following SIP algorithm terminates in finitely many iterations with a certificate of otimality and a rigorous ϵ tol -otimal feasible oint. 27,29 The algorithm for semi-infinite otimization with imlicit functions embedded is resented in the following. Algorithm 1 (Global Otimization Algorithm for Imlicit SIPs) 1. (Initialization) (a) Set LBD =, UBD = +, ϵ tol >0,k := 0. (b) Set initial arameter sets P LBD = P LBD,0, P UBD = P UBD,0. (c) Set initial restriction arameter ϵ g,0 > 0 and r >1. 2. (Termination) Check UBD LBD ϵ tol. (a) If true, f* := UBD, terminate. (b) Else k := k (Lower-Bounding Problem) Solve the lower-bounding roblem (eq 6) to global otimality. (a) Set LBD := f LBD, set equal to the otimal solution found. 4. (Inner Program) Solve the inner rogram (eq 7) to global otimality. (a) If g(, y(, ), ) =g ( ) 0, set * :=, UBD := f( ), terminate algorithm. (b) Else, add to P LBD. 5. (Uer-Bounding Problem) Solve the uer-bounding roblem (8) to global otimality. (a) If feasible: i. Set equal to the otimal solution found and solve the inner rogram (eq 7) to global otimality. ii. If g ( ) <0: A. If f( ) UBD, set UBD := f( ), * :=. B. Set ϵ g,k+1 := ϵ g,k /r, goto2. iii. Else (g ( ) 0), add to P UBD,goto2. (b) Else (infeasible), set ϵ g,k+1 := ϵ g,k /r, goto2. 310

5 Industrial & Engineering Chemistry Research Figure 1. Flowchart for Algorithm 1. It should be noted that the subroblems can only be solved finitely to within some chosen tolerances. In order to guarantee that the SIP algorithm is rigorous, the convergence tolerances for the subroblems must be set such that they are lower than ϵ tol. In order to better visualize Algorithm 1, the flowchart is given in Figure 1. APPLICATION TO MAX MIN AND MIN MAX PROBLEMS Constrained min ma roblems of the form: min ma G(, y, ) X P, y Y s.t. h (, y, ) = 0 and constrained ma min roblems of the form: ma min G(, y, ) X P, y Y s.t. h (, y, ) = 0 (9) (10) can also be solved using Algorithm 1. The min ma case results in solving the imlicit rogram: G*= minma G(, y(, ), ) X P (11) which can be reformulated as an imlicit SIP by introducing a variable η H, with H a comact interval, and writing: min X, η s.t. η ma G(, y(, ), ). P (12) Using the relationshi given by eq 4, and setting g(,y(,),,η) = G(,y(,),) η, the following SIP can be written: min X, η s.t. g(, y(, ),, η) 0, P (13) which is equivalent to the imlicit SIP in eq 5. The imlicit SIP algorithm can be alied directly to this roblem without any modification by setting n := n + 1 and treating η as the n +1 comonent of. Here, an otimal solution value of η* 0 imlies that G* 0, and alternatively, η* > 0 imlies G* >0. The constrained ma min roblem reformulation is slightly different. This case results in solving G*= mamin G(, y(, ), ) X P (14) Again, the variable η H is introduced and eq 14 is written as ma X, η s.t. η min G(, y(, ), ) P which can be written as ma X, η s.t. η G(, y(, ), ), P or equivalently as min X, η s.t. g(, y(, ),, η) 0, P (15) (16) (17) by using the identity g(,y(,),,η) =η G(,y(,),). Now, the imlicit SIP algorithm can be alied without modification by again setting n :=n + 1 and treating η as the n +1 comonent of. Now, analogous to the min ma case, an otimal solution value of η* 0 imlies that G* 0 and η* > 0 imlies G* >0. 311

6 Figure 2. Objective function and imlicit semi-infinite constraint for Eamle 1. NUMERICAL EXAMPLES fluctuations in τ, K i is the vaor liquid equilibrium coefficient Eamle 1. Consider the following illustrative eamle with n for the i th chemical comonent, and z i is the mole-fraction of = n = n y =1: chemical comonent i in the feed. Solving h(τ,α,) = 0 for α defines the cut fraction as an imlicit function of temerature f( ) = ( 3.5) 5( 3.5) 2( 3.5) and ressure, α:t P Y. Any value α [0,1] is nonhysical so the interval Y = [0,1] was considered. For this system, the + 15( 3.5) vaor liquid equilibrium coefficient can be calculated as 3 5 h (, y, ) = y ( /6 + /120)/ y = 0 sat () τ i Ki( τ, ) = g (, y, ) = y+ cos( /90) 0, P X = [0.5, 8.0] for each chemical comonent i with P = [80, 120] sat Bi log ( τ) = A 10 i i C The objective function and imlicit semi-infinite constraint i + τ are shown in Figure 2. An interval Y = [68.8,149.9], guaranteed with τ in C and sat i in torr. The Antoine coefficients A i, B i, C i to contain a unique imlicit function y:p X Y was obtained are available in Table 1. using the arametric interval-newton method. 1,7 This illustrative eamle was chosen because the SIP feasible set and the Table 1. Antoine Coefficients for the Ternary Miture in objective function are obviously nonconve and the objective Eamle 2 36 function has a subotimal local minimum that is SIP-feasible. Eamle 2. Consider the robust design of an isothermal flash e. two antoine coefficients searator under uncertainty. We wish to verify robust oeration i A i B i C i tem. range ( C) of a roosed design in the face of the worst-case realization of 1: n-butane uncertainty. The flash searator is designed to searate a 2: n-entane ternary miture of n-butane, n-entane, and n-heane, with 3: n-heane molar fractions of 0.5, 0.4, and 0.1, resectively. The searator is designed to create a vaor roduct stream with no more than 0.05 mol-fraction of n-heane. To do so, it is designed to oerate at 85 C and a ressure no greater than 5100 torr (6.80 bar). It is eected that during oeration, the vessel temerature, error in the thermocoule reading, or both, may vary by as much as ±5 C. For this system, there are si For robust design roblems, one must consider the worstcase realization of uncertainty and eamine if there eists a control setting that allows the design to still meet the erformance and/or safety secification. This roblem can be formulated mathematically as a ma min roblem: unknowns: the comositions of the vaor and liquid streams. ma min G( τ, α( τ, ), ) Three secies balance equations and three hase-behavior τ T P equations can be written, resulting in a dimensionality n y =6. However, an alternative, and equivalent, model formulation T = [80, 90] with n y = 1 can be formulated by writing the stream P = [4400, 5100] comosition model equations in terms of the cut fraction α : h τ τ α = z i( K i(, ) 1) (,, ) ( K( τ, ) 1) α + 1 = 0 i i where τ will be the temerature (uncertain) variable, the cut fraction, α, isdefined as the fraction of the feed that leaves in the vaor stream (internal state variable), is the vessel ressure which can be controlled in order to mitigate which, if G(τ*,α(τ*,*),*) 0, the design is robustly feasible, or simly, for the worst-case realization of uncertainty, there eists a control setting such that the system meets secification. The lower bound on the control variable comes from a requirement that there are two hases resent in the searator at all times (i.e., any lower ressure will flash all of the liquid into the vaor hase). According to the revious discussion, this roblem can be reformulated as an imlicit SIP: 312

7 min τ T, η s.t. η G( τ, α( τ, ), ) 0, P with H =[ 1,1]. The erformance secification can be written as G τ τ α τ zk 3 3(, ) (, (, ), ) = ( K ( τ, ) 1) α( τ, ) which comes from material balances on the system. Figure 3 shows G lotted against τ. Figure 3. Design constraint function for Eamle 2. Eamle 3. Consider the otimal design of a continuousstirred tank reactor (CSTR) for the chlorination of benzene, shown in Figure 4. The reactions taking lace are k CH + Cl CHCl+ HCl k C H Cl + Cl C H Cl + HCl Figure 4. Continuous-stirred tank reactor for Eamle 3. where the rate constants k 1 and k 2 (h 1 ), as well as the feed flow rate F 1 (kmol/h), will be considered as uncertain arameters, =(k 1 k 2 F 1 ) T. The design variable will be the reactor volume (m 3 ), = v. The reaction kinetics can be considered to be firstorder with resect to benzene and chlorobenzene and the reactions are irreversible. 37 For simlicity, A will denote C 6 H 6, B will denote C 6 H 5 Cl, and C will denote C 6 H 4 Cl 2. Therefore, there are a total of four unknowns: the comosition (molefractions) of the roduct stream and the roduct stream flow rate in terms of A, B, and C, y = (y A y B y C F 2 ) T. In this formulation, n =1,n y = 4, and n = 3. Note that F 1 and F 2 are the flow rates (kmol/h) in terms of the chemical secies A, B, and C only. The model equations are then y y y r A, y y y + ( r r ) = B, h(, y, ) = 0 y y y + r C, y y y (18) with y i,1 as the mole-fraction of chemical secie i in the feed stream, and the reaction rates r 1 and r 2 are given by r = y /( yv + yv + yv ) A 2 B 3 C r = y /( yv + yv + yv ) A 2 B 3 C with V i as the molar volumes of chemical secie i: V A = m 3 /kmol, V B = m 3 /kmol, V C = m 3 /kmol. The feed was taken to be ure benzene. For this articular system, the design objective is to minimize the reactor volume while satisfying the erformance constraint that at least 22 kmol C 6 H 5 Cl/h is roduced: min X s.t. 22 y (, ) y (, ) 0, P 2 4 The uncertainty interval will be P = [0.38,0.42] [0.053,0.058] [60,70], the design interval will be X = [10,20]. From the arametric interval-newton method, 1,7 an interval Y = [0.15,0.85] [0.3,0.65] [0.0,0.12] [60,70] was calculated that encloses the imlicit function y:x P Y such that h(,y(,),) =0, (,) X P. EXPERIMENTAL CONDITIONS AND RESULTS Algorithm 1 was imlemented in C++. Each NLP subroblem was solved using the algorithm for global otimization of imlicit functions, 29 which was also imlemented in C++ and utilizes the library MC The algorithm for global otimization of imlicit functions relies on the ability to solve conve nonsmooth subroblems. This is because conve and concave relaations of imlicit functions 29 are in general nonsmooth. For this task, the nonsmooth bundle solvers PBUN and PBUNL 39 were utilized with default settings for the NLP lower-bounding roblems, and the objective function was evaluated at NLP feasible oints to obtain valid uer bounds on the NLP. Since the constrained bundle solver (PBUNL) can only handle affine constraints, affine relaations of the conve constraints with resect to reference oints must be calculated. In other words, once a nonconve rogram is conveified by constructing relaations of the nonconve objective function and nonconve constraints, the newly constructed conve constraints must be further relaed by constructing affine underestimating functions that PBUNL can handle. The hierarchy of information flow for global otimization of imlicit functions is shown in Figure 5. Two sets of eeriments were conducted. Case 1: A single reference oint taken as the midoint of X was used to construct affine relaations of constraints. Case 2: Three reference oints the lower bound, the midoint, and the uer bound of X were used to construct affine relaations of the constraints and used simultaneously. 313

8 Figure 5. Hierarchy of information flow for global otimization of imlicit functions. 29 The numerical eeriments were erformed using a PC with an Intel Core2 Quad 2.66 GHz CPU oerating Linu. For each eamle, absolute and relative convergence tolerances of 10 7 and 10 5, resectively, were used for the NLP subroblems unless otherwise noted. Eamle 1. For the SIP algorithm, each constraint set was initialized as emty, ϵ g,0 = 0.9, r = 2.0, and ϵ tol =10 4. For each set of eeriments, the imlicit SIP algorithm was alied and an aroimate global otimal solution with an objective function value of f* = at * = Convergence was observed in three iterations for each case and took s for Case 1 and s for Case 2. For this eamle, the algorithm terminates after the lower-bounding roblem furnishes a SIP-feasible oint (Ste 4a of the algorithm) and so the arameter r does not affect the erformance of the imlicit SIP algorithm. In an effort to elore the behavior of the algorithm, the NLP subroblem algorithm absolute and relative convergence tolerances were relaed to 10 6 and 10 4, resectively. Interestingly, for both cases, the SIP algorithm does not terminate with the lower-bounding roblem furnishing an SIPfeasible oint but instead terminates at Ste 2 of the algorithm. This suggests that sending more time solving the NLP subroblems to higher accuracy benefits the total algorithm runtime by heling to accelerate locating a global otimal SIPfeasible oint. The convergence results are lotted in Figure 6. Qualitatively, the algorithm erformed similarly to the elicit SIP algorithm by Mitsos 27 where the solution time and number of iterations raidly decreased with increasing reduction arameter. This behavior readily lateaued after a value of around r = 8. As eected, Case 2 ehibited higher comutational cost without reducing the overall solution time or number of iterations. Eamle 2. For the imlicit SIP algorithm, each constraint set was initialized as emty, ϵ g,0 = 0.9, r = 2.0, and ϵ tol =10 4. For both Case 1 and Case 2, the imlicit SIP algorithm was alied and an aroimate global otimal solution was obtained with η* = , τ* =90 C, * = 5100 torr. For both cases, the algorithm terminates in three iterations after the lower-bounding roblem furnishes an SIP-feasible oint. Thus, as reviously mentioned, the arameter r has no effect on the erformance of the algorithm. Relaing the NLP convergence tolerances did not have the same effect on the behavior of the algorithm as it did in Eamle 1. Case 1 converged just after s whereas Case 2 took s. This suggests that the choice of the reference oint for linearizing the constraints for the NLP subroblems was not very imortant and choosing more than one reference oint just increased the comutational comleity and, therefore, the solution time. Returning to the idea of robust design, since η* > 0, the flash searator design is not robust. However, as can be seen from Figure 3, if the design can be imroved such that the temerature (or thermocoule reading) may only vary by ±4 C, the design aears to be robust. This result was verified by the imlicit SIP algorithm converging after three iterations and s to an aroimate otimal solution with η* = Eamle 3. For the SIP algorithm, each constraint set was initialized as emty, ϵ g,0 = 0.9, and ϵ tol =10 4. For Case 2, the imlicit SIP algorithm was alied and an aroimate global otimal solution was obtained with f* = * = m 3, * = ( ) T. Therefore, in order to roduce at least 22 kmol/h of chlorobenzene, taking into account uncertainty in the inut flow rate and the reaction rate constants, the reactor volume must be m 3. The reader is reminded that, similar to Eamle 2, this eamle is looking to solve the roblem of robust design under uncertainty. However, there is a key difference in the formulation, between the two eamles. Eamle 2 is seeking a yes or no answer to whether or not Figure 6. Comutational effort in terms of the solution time (left) and the number of iterations (right) the algorithm takes to solve Eamle 1 versus the reduction arameter r. Note that the number of iterations is the same for both cases. 314

9 Figure 7. Comutational effort in terms of the number of solution time (left) and the number of iterations (right) the algorithm takes to solve Case 2 of Eamle 3 versus the reduction arameter r. the rocess is robust to the worst-case realization of uncertainty. Alternatively, this eamle is seeking to identify the actual size of the reactor such that the rocess will erform as desired in the face of the worst-case realization of uncertainty. Note that the worst-case realization of uncertainty is eactly what is to be eected; in order to have the least amount of chlorobenzene in the roduct stream, k 1 should be the smallest value it can take, k 2 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 s. The erformance of the algorithm for Case 2 can be found in Figure 7. Similar to Eamle 1, a small value for r resulted in the imlicit 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 droed drastically and lateaued. A arameter value of r = 18 reduced the solution time and number of iterations by 94% over r = 1.1. Again, qualitatively, this behavior is similar to what Mitsos 27 demonstrated. In his elicit SIP eamles, the number of iterations was reduced from about 80 to between 5 and 20, deending on the eamle. For this eamle, Case 1 failed to converge within 200 iterations of the algorithm. This result is simly a consequence of using PBUNL which only accets affine constraints. In this case, since the affine constraints are being constructed with reference to the midoint of X, the solver aarently fails to ever return a oint that is feasible in the original SIP. CONCLUSION A method was resented for reformulating equality-constrained bilevel rograms as SIPs with embedded imlicit functions, requiring that (a) all functions involved are continuous and factorable, (b) derivative information on the equality constraint function is available and is factorable, (c) there eists at least one solution y to the system of equations in eq 2 for every (,) X P, and an interval Y can be found that bounds an isolated solution, (d) an aroriate matri for reconditioning the interval- Jacobian can be calculated such that their roduct has nonzero diagonal elements, and (e) there eists a Slater oint arbitrarily close to a SIP minimizer. To solve the resulting imlicit SIP, the global otimization algorithm develoed by Mitsos 27 has been adated. The algorithm relies on the ability to solve three nonconve imlicit NLP subroblems to global otimality. This is erformed utilizing the relaation methods and the deterministic algorithm for global otimization of imlicit functions which were develoed by Stuber et al. 29 The algorithm develoed by Stuber et al. 29 relies on the ability to solve nonsmooth lowerand/or uer-bounding roblems at each iteration. This can be done using any available nonsmooth otimization algorithm or using the calculated subgradient information to construct affine relaations and transform the roblem into a linear rogram and solved using any efficient LP otimization algorithm. For this aer, the nonsmooth bundle solvers PBUN and PBUNL 39 were utilized. Note that the requirements (b) and (c) are only due to current limitations of the algorithm for global otimization of imlicit functions. The requirements (d) and (e) imly that the SIP is feasible and (a) and (e) are required for guaranteed ϵ-otimal convergence of the original elicit SIP algorithm 27 after finitely many iterations. Altogether, these requirements guarantee ϵ-otimal convergence of Algorithm 1. As a roof-of-concet, three numerical eamles were resented that illustrate the global solution of imlicit SIPs using this algorithm. The first eamle illustrated the solution of a simle numerical system that fits the imlicit SIP form given in eq 5. This roblem is interesting because it is easy to visualize the nonconveity of the functions and identify the SIPfeasible subotimal local minimum. The second eamle was an engineering roblem of robust design under uncertainty, originally cast as a constrained ma-min roblem. It was then reformulated as an imlicit SIP of the form in eq 5 and solved using the imlicit SIP algorithm. Under the original design conditions and uncertainty interval, the design was not robustly feasible. After altering the design such that the uncertainty interval was reduced, the design was found to be robust. The third eamle was an engineering roblem of otimal design of a chemical reactor considering uncertainty in the kinetic arameters and feed rate which was formulated as an SIP. The eamles chosen offered varying levels of comleity as well as size which allowed various features and behavior of the imlicit SIP algorithm to be elored. Due to the limitations of the PBUNL solver, only affine constraints could be used. Since the imlicit semi-infinite constraint is almost surely nonlinear, affine relaations must be 315

10 constructed. For the numerical eamles, two sets of eeriments were conducted: one using a single reference oint for constructing affine relaations of the constraints and another using three reference oints for constructing affine relaations of the constraints and using them all simultaneously. The first method was hyothesized to be advantageous since it required less comutational effort to calculate the constraints. Alternatively, the second method was hyothesized to be advantageous since using multile reference oints results in better aroimations of the constraints, which in turn may seed u convergence of the overall algorithm. For Eeriments 1 and 2, it was observed that Case 2 offered no benefit over Case 1 and only added comutational comleity. However, for Eamle 3, Case 1 failed to converge after 200 iterations. This was likely due to the affine relaations of the semi-infinite constraint not being very tight, resulting in PBUNL failing to find a solution that is feasible in the original SIP. It was found that for this roblem, Case 2 and its multile reference oints for calculating affine relaations were clearly suerior with the algorithm converging after 7 iterations and s. Lastly, the generalization of eq 5 with the semi-infinite n constraint g to g g will be discussed. For this case, eq 5 would take the form f*= min f( ) s.t. g(, y(, ), ) 0, j = 1,..., n, P j X = { : } P = { : } g (19) The elicit case was discussed by Mitsos. 27 The inner rogram would of course need to be relaced with n g inner rograms: one for each j = 1,...,n g and instead of a single constraint inde set for the uer- and lower-bounding roblems there should be a set P UBD j and P LBD j for each j = 1,...,n g constraints, resectively. 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