POP â Parametric Optimization Toolbox - Industrial & Engineering

Article pubs.acs.org/IECR

POP − Parametric Optimization Toolbox Richard Oberdieck,†,‡ Nikolaos A. Diangelakis,†,‡ Maria M. Papathanasiou,†,‡ Ioana Nascu,‡ and Efstratios N. Pistikopoulos*,‡ †

Department of Chemical Engineering, Centre for Process Systems Engineering, Imperial College London, London, United Kingdom Artie McFerrin Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843, United States

‡

S Supporting Information *

ABSTRACT: In this paper, we describe POP, a MATLAB toolbox for parametric optimization. It features (a) efficient implementations of multiparametric programming problem solvers for multiparametric linear and quadratic programming problems and their mixed-integer counter-parts, (b) a versatile problem generator capable of creating random multiparametric programming problems of arbitrary size, and (c) a comprehensive library of multiparametric programming test problems featuring benchmark test sets for multiparametric linear, quadratic, mixed-integer linear, and mixedinteger quadratic programming problems. In addition, POP is equipped with a graphical user interface which enables the userfriendly use of all functionalities of POP and a link to the solvers of the Multi-Parametric Toolbox (MPT), as well as the ability to design explicit MPC problems. These features are demonstrated in detailed computational studies providing insights into the versatility and applicability of POP. Additionally, the example of a periodic chromatographic system is used to show the scalability of multiparametric programming in general and POP, in particular.

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INTRODUCTION In multiparametric programming, an optimization problem is solved as a function and for a range of certain parameters.1 After its first consideration in a master thesis in 1952,2 the most significant theoretical development occurred with the description of the Basic Sensitivity Theorem by Fiacco in 19763 (see refs 4−6 for a detailed overview). This provided a theoretical foundation for the development of new algorithms and applications, arguably the most influencial of which was presented in 2000, where it was shown that model-predictive control (MPC) problems can be solved offline via multiparametric programming.7,8 This led to high interest in the topic, with a special focus on areas related to explicit/ multiparametric MPC, such as multiparametric dynamic programming and robust explicit MPC. In Table 1, we present a list of theoretical contributions to multiparametric programming, while Table 2 shows a list of contributions where multiparametric programming has been applied. This range of applications in turn resulted in the development of software tools, most notably the Multi-Parametric Toolbox (MPT) in 2004,128 for which recently MPT 3.1 was released.31,129−131 It is a complete software tool able to solve multiparametric programming problems and perform key operations of linear algebraic geometry as well as to design explicit controllers very intuitively. Despite these features, MPT has three key limitations. First, it currently does not provide solvers for multiparametric mixed-integer programming prob© XXXX American Chemical Society

Table 1. Theoretical Contributions to Multiparametric Programming: Indicative List topic

contributions

multiparametric linear and quadratic programming multiparametric mixed-integer linear and quadratic programming multiparametric linear complementarity problem multiparametric dynamic programming inverse multiparametric programming multiparametric (mixed-integer) nonlinear programming postprocessing

refs 6, 8−14 refs 10, 15−27 refs refs refs refs refs

28−32 19, 33−37 38−46 47−59 60−79

lems. [The modeling tool YALMIP132 provides the ability to solve multiparametric mixed-integer programming problems via exhaustive enumeration and the solution of the multiparametric linear programming problem via MPT. However, no solution strategy beyond exhaustive enumeration is currently available.] Second, the interface of MPT is somewhat self-contained, which may not allow it to be easily connected to other software, such as PSE’s gPROMS133 or ASPEN. However, most high-fidelity models are represented in software such as gPROMS, and hence, the capability of manipulating and embedding the Received: May 19, 2016 Revised: July 18, 2016 Accepted: July 27, 2016

A

DOI: 10.1021/acs.iecr.6b01913 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

Definition 1. A function x(θ) with a polytopic feasible space Θ ∈ q , is piecewise affine if it is possible to partition Θ into convex polyhedral regions, CRi, and

Table 2. Application-Related Contributions Featuring Multiparametric Programming: Indicative List topic

contributions

explicit model predictive control multiparametric moving horizon estimation scheduling integration of design, scheduling and control bilevel programming multiparametric dynamic optimization multiparametric multiobjective optimization

refs 1, 8, 25, 33, 55, 80−92 refs 93−100 refs 30, 101−109 refs 92, 107, 109−111 refs 112−118 refs 119−121 refs 122−127

x(θ ) = K iθ + ri , ∀ θ ∈ CR i

(2)

Remark 1. The definition of piecewise quadratic is analogous. Theorem 1 (Properties of mp-QP solution8,10). Consider the mp-QP problem 1 and let Q be positive definite, Θ convex. Then, the set of feasible parameters Θf ⊆ Θ is convex, the optimizer x(θ) is continuous and piecewise affine, and the optimal objective function x(θ) is continuous and piecewise quadratic. Remark 2. In the case of mp-LP problems, Theorem 1 still holds; however, the optimal objective function z(θ) is continuous, convex, and piecewise affine.6 Additionally, we define the following lemma: Lemma 2 (Active set representation). Each critical region obtained from an mp-LP or mp-QP problem is uniquely defined by the optimal active set associated with it, and the solution of problem 1 can be represented as the set of all optimal active sets. Solution of mp-LP and mp-QP Problems. On the basis of Theorem 1 and Lemma 2, it is clear that the solution to problem 1 can be regarded either as a set of polytopes, which cover the feasible parameter space Θf or as a set of optimal active sets, which generate the critical regions based on the parametric solution x(θ),λ(θ). These considerations have given rise to three distinct types of solution approaches: a geometrical approach, a combinatorial approach, and a connected-graph approach. Remark 3. Other approaches for the solution of problem 1 involve vertex enumeration,134 graphical derivatives,135 or the reformulation as a multiparametric linear complementarity problem,28−30 which can be solved in a geometrical129 or combinatorial31 fashion. The Geometrical Approach. Possibly the most intuitive approach to solve mp-QP problems of type 1 is the geometrical approach. It builds upon the geometrical consideration and exploration of the parameter space Θ. The key idea is to fix a point θ0 ∈ Θ, solve the resulting QP, and obtain the parametric expressions x0(θ) and λ0(θ) alongside the corresponding critical region CR0. Then, a new, feasible point θ1 ∉ CR0 is fixed, and the same procedure is repeated until the entire parameter space has been explored. The different contributions differ in the way the parameter space is explored: In refs 8 and 10, the constraints of the critical region are reversed, yielding a set of new polytopes which are considered separately. As this introduces a large number of artificial cuts,12 the step-sized approach has gained importance, as it calculates a point on the facet of each critical region and steps away from it orthogonally.11,136 Additionally, researchers have proposed techniques to infer the active set of the adjacent critical region.12,13,136 Remark 4. Note that the geometrical approach presented in refs 11 and 136 is only guaranteed to provide the full parametric map if the so-called facet-to-facet property is fulfilled,13 i.e., the fact that every facet of a critical region corresponds to the facet of another region (see Definition 2). The Combinatorial Approach. As stated in Lemma 2, every critical region is uniquely defined by the corresponding optimal active set. Thus, a combinatorial approach has been suggested, which considers the fact that the possible number of active sets is finite and thus can be exhaustively enumerated. In order to make this approach computationally tractable, we state the following fathoming criteria:14

solution may be desired for the process systems engineering applications. Third, despite its widespread use, except for example problems, no comprehensive library of test problems exists for multiparametric programming. Such a library is key in order to provide an objective measure of efficiency and capabilities of the algorithms. This lack goes hand in hand with the capability of randomly generating multiparametric programming problems in an attempt towards testing and pushing the limits of the size of multiparametric programming problems, that can be solved. Motivated by these limitations, we have developed POP, the Parametric OPtimization toolbox, available for free at paroc. tamu.edu/Software, which features implementations of most state-of-the-art multiparametric programming solvers, a versatile problem generator, and a comprehensive problem library. In addition, the toolbox is equipped with an intuitive graphical user interface (GUI), which is seamlessly connected to all functionalities of the toolbox, as well as a link to the solvers of the MPT toolbox. Using as an example a periodic chromatographic system, the application of the POP toolbox to multiparametric MPC is shown to provide insights to the type of complex process systems POP is capable of solving. Additionally, a computational study featuring test sets with multiparametric linear programming (mp-LP), multiparametric quadratic programming (mp-QP), multiparametric mixedinteger linear programming (mp-MILP), and multiparametric mixed-integer quadratic programming (mp-MIQP) problems is presented, highlighting the capabilities of the solvers as well as looking in detail at the computational bottlenecks involved. In the case of mp-LP and mp-QP problems, the computational results are compared with the latest version of MPT.31,131

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THEORETICAL AND ALGORITHMIC BACKGROUND Multiparametric Linear and Quadratic Programming Problems. We consider the following mp-QP problem z*(θ ) = minimize (Qx + Hθ + c)T x x

subject to Ax ≤ b + Fθ x ∈ n θ ∈ Θ ≔ {θ ∈ q|CRAθ ≤ CR b} (1)

with Q ∈ n × n ≻ 0, H ∈ n × q, c ∈ n, A ∈ m × n, b ∈ m, F ∈ m × q, CRA ∈ r × q , CR b ∈ r , and Θ is compact (closed and bounded). Problem 1 has been widely studied in the open literature,8−10,14 which enables the statement of the main definition and theorem of the solution of mp-QP problems: B


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Industrial & Engineering Chemistry Research Let k1 and k2 be candidate active sets with k1 ⊂ k2. If problem 1 is not feasible for any θ when considering k1, then it will also not be feasible when considering k2. Thus, the following branch-and-bound approach has been presented:14 Step 1: Generate a tree consisting of all possible active sets. Step 2: Select the candidate active set with the lowest cardinality of the active set and check for feasibility. If it is infeasible, fathom that node and all its child nodes. Step 3: Obtain the parametric solution for the selected node accordingly and check whether the resulting region is nonempty. Step 4: If there are nodes to explore, go to Step 2. Otherwise terminate. The Connected-Graph Approach. Since the geometrical and combinatorial approach consider the same problem, it is clear that it should be possible to translate the concept of adjacency used in the geometrical approach to the combinatorial perspective. To that end, we consider the following theorems and definition: Definition 2 (Facet-to-facet property). Let CR1 and CR2 be two full-dimensional critical regions with int(CR1) ∩ int(CR2) = ⌀. Then, the facet-to-facet property is said to hold if F = CR1 ∩ CR2 is a facet of both CR1 and CR2. Theorem 3 (Active set of adjacent region12). Consider an active set k = {i1,i2,...,ik} and its corresponding critical region CR0 in minimal representation, i.e., with all redundant constraints removed. Additionally, let CRi be a full-dimensional neighboring critical region to CR0 and assume that the linear independent constraint qualification holds on their common facet F = CR0 ∩H, where H is the separating hyperplane. Moreover, assume that there are no constraints which are weakly active at the optimizer x(θ) for all θ ∈CR0. Then: Type I: If H is given by Aik+1x(θ) = bik+1 + Fik+1θ, then the optimal active set in CRi is {i1,...,ik,ik+1}. Type II: If H is given by λik(θ) = 0, then the optimal active set in CRi is {i1,...,ik−1}. Definition 3 (mp-QP Graph). Let each optimal active set k of a mp-QP problem be a node in : . Then the nodes k1 and k2 are connected if (a) there exists θ* ∈ Θf such that k1 and k2 are both optimal active sets and (b) the conditions of Theorem 3 are fulfilled on the facet or it is possible to pass from k1 to k2 by one step of the dual simplex algorithm. The resulting graph G is fully defined by the nodes : as well as all connections Γ, i.e., G = (:, Γ) Theorem 4 (Connected graph-based mp-QP solution). Consider the solution to a mp-QP problem and let θ1, θ2 ∈ Θf be two arbitrary feasible parameters and k1 ∈ : be given such that θ1 ∈CR1. Then, there exists a path {k1,...,kj} in the mpQP graph G = (:, Γ) such that k2 ∈CRj. On the basis of Theorem 4, it is possible to modify the combinatorial algorithm in order to only consider candidate active sets which fulfill Theorem 4. In essence, given an active set corresponding to a full-dimensional critical region, the next candidate active sets are obtained by considering the active set in conjunction with the irredundant constraints which form the critical region. These constraints, related to the geometrical consideration of the critical region, identify the active set for the adjacent critical regions, related to the combinatorial consideration. This decreases the number of candidate active sets to be considered, and thus yields a computationally more attractive algorithm.

Multiparametric Mixed-Integer Linear and Quadratic Programming Problems. We consider the following multiparametric mixed-integer quadratic programming (mp-MIQP) problem z*(θ ) = minimize (Qω + Hθ + c)T ω x ,y

subject to Ax + Ey ≤ b + Fθ x ∈ n , y ∈ {0, 1} p , ω = [xT yT ]T θ ∈ Θ ≔ {θ ∈ q|CRAθ ≤ CR b} (3) (n + p) × (n + p)

(n + p) × q

(n + p)

≻ 0, H ∈  where Q ∈  , c∈ , A ∈ m × n, E ∈ m × p, b ∈ m, F ∈ m × q, and Θ is compact (closed and bounded). The properties of the solution of mpMIQP problems of type 3 are given by the following theorem, lemma, and definition. Theorem 5 (Properties of mp-MIQP solution33). Consider the optimal solution of problem (3) with Q ≻ 0. Then, there exists a solution in the form xi(θ ) = K iθ + ri if θ ∈ CR i

(4)

where CRi,i = 1,...,M is a partition of the set Θf of feasible parameters θ, and the closure of the sets CRi has the following form CR i = {θ ∈ Θ|θ TGi , jθ + hiT, jθ ≤ wi , j , j = 1, ..., ti}

(5)

where ti is the number of constraints that describe CRi. Lemma 6 (Quadratic boundaries33). Quadratic boundaries arise from the comparison of quadratic objective functions associated with the solution of mp-QP problems for different feasible combinations of binary variables. Definition 4 (Envelope of solutions10). In order to avoid the nonconvex critical regions described by Lemma 6, an envelope of solutions is created where more than one solution is associated with a critical region. The envelope is guaranteed to contain the optimal solution, and a point-wise comparison procedure among the envelope of solutions is performed online. Solution of mp-MILP and mp-MIQP Problems. The most common approach for the solution of problem 3 is the decomposition algorithm, which can be summarized as follows: Step 1: Set the upper bound over the parameter space Θ to ∞, and add Θ to the list 5 . Step 2: If 5 is empty, the algorithm terminates. Select an element i from the list 5 , and solve the following mixed-integer nonlinear programming (MINLP) problem: i = minimize (Qω + Hθ + c)T ω zglobal x ,y,θ

subject to Ax + Ey ≤ b + Fθ (Qω + Hθ + c)T ω − zî (θ ) ≤ 0

∑ yj − ∑ yj ≤ |Ji | − 1 j ∈ Ji

j ∈ Ti n

x ∈  , y ∈ {0, 1} p , ω = [xT yT ]T θ ∈ CR i , (6)

where i = 1,...,v and v is the number of critical region that constitute the upper bound, ẑi(θ) is the objective function value C


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Industrial & Engineering Chemistry Research of the upper bound in the critical region CRi considered, and Ji and Ti are the sets containing the indices of the binary variables ŷi associated with the upper bound ẑi(θ) that attain the value 0 and 1 respectively, i.e.,

Ji = {j|yjî = 1}

(7a)

Ti = {j|yjî = 0}

(7b)

OPtimization toolbox, involving three key features: problem solution, problem generation, and problem library. Problem Solution. Solution of Problem 1. In POP, we have implemented the geometrical,11 a variation of the combinatorial14 and the connected-graph algorithm. These are accessible as functions in the Command Window: Solution = Geometrical(problem) Solution = Combinatorial(problem) Solution = ContactedGraph(problem) where problem is the structured array containing the mp-LP/ mp-QP problem to be solved. Additionally, POP provides an interface with the solver used in MPT: Solution = POPviaMPT(problem) Note that this requires the separate download of the MPT toolbox. Thus, POP features every major solution strategy for problems of type 1. These have been combined in a single wrapper: Solution = mpQP(problem) The option of which mp-LP/mp-QP solver to use is set in the OptionSet function, which contains all the adjustable settings of POP. The interested user is referred to the User Manual available at paroc.tamu.edu/Software and our YouTube channel “POP Toolbox”. Additionally, each solver provides statistical information about its performance, a feature which is also explained in detail in the User Manual. Solution of Problem 3. In POP, we have implemented a decomposition-based algorithm10 featuring four different comparison procedures discussed in Appendix A of the Supporting Information, as well as an exhaustive enumeration approach. The solver is available in the Command Window as Solution = mpMIQP(problem) and the options such as the choice of the integer handling technique and the comparison procedure are set in the OptionSet function. Additionally, the solver provides statistical information about its performance, a feature which is also explained in detail in the User Manual. Requirements and Validation. It is possible to use all functionalities of POP using only the built-in functionalities of MATLAB and its toolboxes. However, for speed and stability reasons, the use of commercial tools is encouraged. In particular, POP features links to CPLEX and NAG as LP and QP solvers, as well as CPLEX for the MI(N)LP problems. In Appendix B of the Supporting Information, we list the different settings available within POP. Remark 8. As CPLEX only provides MILP and MIQP solvers, in the case of mp-MIQP problems, the quadratic constraints in problem 6 are underestimated using a suitable set of envelopes of bilinear function by McCormick. Note that this guarantees correct execution of the algorithm. However, if no comparison procedure is employed (see Appendix A, Supporting Information), then the number of solutions per critical region might be higher than in the case where a MINLP solver is used. In order to validate the solution obtained from problems 1 or 3, POP features the function VerifySolution, which randomly seeds 5000 points in the parameter space Θ and solves the corresponding deterministic problem. While this does not provide a full certificate of guarantee, it is a strong indicator that a correct solution has been obtained. Problem Generation. The aim is to generate random, feasible problems with suitably defined constraints such that different active sets become optimal in different parts of the

Remark 5. Without loss of generality, it is assumed that CRi only features one upper bound ẑi(θ) in problem 6. Note that the introduction of suitable integer and parametric cuts to the MINLP ensures that previously considered integer combinations as well as solutions with a worse objective function value are excluded. If problem 6 is infeasible, select a new element from 5 . Otherwise, the solution of this problem yields a candidate solution for the binary variables, y*. Step 3: Fix y* in problem 3, and solve the resulting mp-QP problem. Step 4: Compare the obtained parametric solution to the upper bound for this region. This partitions the space into a series of regions, each of which is associated with one or multiple parametric solutions in the case where envelopes of solutions are present (see Definition 4). Return to Step 2. The main differences between the different decompositionbased algorithms presented in the literature10,26 lies in the comparison between a new solution and an upper bound over a polytope. In Step 4 of the algorithm, the solution obtained from the mp-QP problem is compared to the current best upper bound ẑ(θ) to form a new, tighter upper bound. This can be expressed as z(θ ) = min{z(̂ θ ), z*(θ )}

(8)

where z*(θ) denotes the piecewise quadratic, optimal objective function obtained by solving the mp-QP problem resulting by fixing the candidate solution of the binary variables obtained from the solution of problem 6. The solution of eq 8 requires in turn the comparison of the corresponding objective functions in each critical region, i.e., !

Δz(θ ) = z(̂ θ ) − zi*(θ ) = 0

(9)

where zi*(θ) denotes the objective function within the ith critical region of the solution of the mp-QP problem. Due to the quadratic nature of the objective functions, Δz(θ) might be nonconvex. The way eq 9 is handled distinguishes the different comparison procedures, which are discussed in detail in Appendix A in the Supporting Information. Remark 6. Without loss of generality, we assume in eq 9 that only one objective function is associated with each critical region and that no envelope of solutions is present (see Definition 4). Remark 7. There have been several approaches that proposed a branch-and-bound approach,24,25 especially for mp-MILP problems.20,21,137 However, while the scalability of the decomposition algorithm has been shown in several publications,22,23,105,138 only ref 25 presented the solution of randomly generated hybrid MPC problems which went beyond proof-ofconcept examples.

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POP On the basis of the background provided in the previous section, we now present the different aspects of POP, the Parametric D


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Figure 1. Problem statistics of the test sets “POP_mpLP1” and “POP_mpQP1”.

Figure 2. Problem statistics of the test sets “POP_mpMILP1” and “POP_mpMIQP1”.

parameter space, thus resulting in a partitioning of the parameter space into several critical regions. For the case of mp-QP problems, the development of such a generator can be decomposed into the following steps: Step 1 − Objective Function: In order to define the objective function, Q, H and c according to problem 1 need to be defined. While for H and c no specific criterion apply, Q needs to be symmetric positive definite. This is achieved by randomly generating a diagonal matrix featuring positive entries. Step 2 − Constraints: The two criteria for the generation of constraints for multiparametric programming problems are (i) feasibility and (ii) tightness in the sense that different solutions should be optimal in different parts of the parameter space. The algorithm to generate appropriate constraints is shown in Algorithm 1.

R = minimize − t θ ,t

subject to Aθ ≤ (b − A 2 t ) θ ∈ Θ, t ∈ 

(10)

where ∥·∥2 denotes the row-wise 2-norm and R is the radius of the largest Euclidian ball enclosed in the defined feasible set Ax ≤ b. If problem 10 is feasible and R > tol, where tol is a prescribed tolerance, the feasible set Ax ≤ b is guaranteed to be nonempty and not lower-dimensional. • The parameter space Θ is by default defined as Θ = {θ ∈ q| − 10 ≤ θl ≤ 10, l = 1, ..., q}. • The coefficients in Algorithm 1 are randomly generated for each problem instance in order to make the generation procedure as random as possible. Within POP, the problem generator is accessible from the Command Window as problem = ProblemGenerator(Type,Size,options) where Type is “mpLP”, “mpQP”, “mpMILP”, or “mpMIQP” and Size is a structured array featuring the desired dimensions of the optimization variables, parameters, and constraints. Additionally, the options input specifies settings which are discussed in detail in the User Manual. In particular, it is possible to generate more than one problem directly, which enables the seamless generation of problem libraries and test sets. Problem Library. The third key feature of POP is its problem library, currently featuring the four randomly generated test sets “POP_mpLP1”, “POP_mpQP1”, “POP_mpMILP1”, and “POP_mpMIQP1” containing 100 randomly generated mp-LP, mp-QP, mp-MILP, and mp-MIQP problems, respectively (see Figures 1 and 2). These problem libraries are later used on to analyze the performance of the different solvers and options available in POP. These test problems represent to our knowledge the first ever comprehensive library of test problems in multiparametric programming. Within POP, each problem is stored in the folder “Library”, which contains a folder for each test set, which in return contains all the individual problems as “.mat” files. These files can be loaded into MATLAB and the corresponding problem

Remark 9. We make the following comments regarding Algorithm 1: • Algorithm 1 also applies to multiparametric mixed-integer programs. • As the generator described is random, the feasibility of the generated problem cannot be guaranteed by default. Thus, in order to ensure a nonempty feasible set for any generated problem, it is solved for the Chebyshev center, i.e., E


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Figure 3. Structure of the graphical user interface (GUI) of POP.

comprises two chromatographic columns operating in countercurrent mode and alternated between batch and interconnected state. Here, we focus on the purification of a monoclonal antibody (mAb) from a ternary mixture composed by weak impurities (W), the product (P), and strong adsorbing impurities (S). As described in ref 141, at the beginning of I1 phase, column 2 starts empty and equilibrated. During this step, the outlet flow of column 1 enters column 2 mixed with an additional fraction of adsorbing eluent (E). This helps the recycling of the impure fraction of the weak impurities and the product. After the completion of I1, the two columns enter B1 phase, where the feed (F) is introduced to column 2 and the product is eluted from column 1. In the I2 phase, the recycling stream containing the impure fraction of product and strong impurities exits column 1 and enters column 2. By the end of the I2 phase, column 2 starts eluting pure W (B2 phase). The B2 phase finishes when the overlapping region of W and P reach the end of column 2. At this point, the two columns switch positions. Therefore, column 1 will go through the recycling and feeding tasks as described above, while column 2 will continue with the gradient elution. The MCSGP process is described by a PDAE model capturing the events taking place during the chromatographic separation.139,140 The model is based on first principles and follows a lumped-kinetic approach comprising 4119 equations with highly nonlinear terms (after spatial discretization, using 50 collocation points). For a detailed approach on the model development, the reader is referred to refs 142 and 143. Each chromatographic column can be approximated by a Single Input-Multiple Output (3 × 1 SIMO) linear state space model that is used for the formulation of the mp-QP

can be solved. Additionally, it is possible to use the Graphical User Interface (GUI, see next section) to perform statistical analysis as well as to create customized test sets which can be exported and solved directly. Graphical User Interface (GUI). In order to facilitate its use, POP is equipped with a GUI which can be launched from the Command Window using POP It enables direct access to the different functions of POP including postprocessing and exporting automatically generated code. The main screens of the interface are shown in Figure 3, i.e., the welcome screen, and the solver, library, and generator interfaces. Note that in order to maintain a user-friendly approach, some of the options available in POP are set to defaults when the interface is used. More information on the interface can be found in the User Manual.

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APPLICATION OF POP TO PERIODIC CHROMATOGRAPHIC SEPARATION Remark 10. The computations were carried out on a 4-core machine with an Intel Core i5-4200 CPU at 2.50 GHz and 8 GB of RAM, MATLAB R2014a, and IBM ILOG CPLEX Optimization Studio 12.6.1. In order to highlight the applicability of POP as well as its scaling capabilities, we consider the challenging problem of optimally controlling a periodic chromatographic separation system.92,139,140 The twin-column Multicolumn Countercurrent Solvent Gradient Purification Process (MCSGP) is an ionexchange, semicontinuous chromatographic separation process used for the purification of several biomolecules.141 The setup F


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Figure 4. Computational results for the optimal control of a periodic chromatographic separation system. In (a), the computational time using the combinatorial algorithm for different control and output horizons is shown, while (b) presents the number of problems solved as a function of time required. The different problems result from the consideration of different output and control horizons.

problems.143 The model is derived using system identification in MATALB and its formulation is given below:

min .xNT PxN u

N−1

x(t + Ts) = Ax(t ) + Bu(t ) + Dd(t )

(11a)

y(t ) = Cx(t )

(11b)

+

∑ (xkTQ kxk + (yk

− ykR )T QR k(yk − ykR ))

k=1 M−1

+

where x, u, and y are the states, inputs, and outputs, respectively, t corresponds to the time, Ts is the sample time, and A, B, C, and D represent the matrices of the state space model, i.e.,

∑ ((uk − ukR)T R k(uk − ukR) + ΔukT R1k Δuk) k=0

s. t. xk + 1 = Axk + Buk + Cdk yk = Dxk + Euk + e

A= ⎡ 0.9998 0.0003667 0.0004885 −9.137 × 10−5 ⎤ ⎢ ⎥ 0.9971 − 0.003154 − 0.001599 ⎥ ⎢ 0.0009448 ⎢−0.001027 0.0003536 0.999 0.001881 ⎥⎥ ⎢ ⎣ 0.000969 0.001559 0.0005033 ⎦ 0.9976

umin ≤ uk ≤ umax Δumin ≤ Δuk ≤ Δumax xmin ≤ xk ≤ xmax ymin ≤ yk ≤ ymax ,

(12a)

(13)

⎡−6.252 × 10−6 ⎤ ⎢ ⎥ ⎢ 4.514 × 10−5 ⎥ B=⎢ ⎥ ⎢ 6.337 × 10−5 ⎥ ⎢ ⎥ ⎣−3.396 × 10−5 ⎦

(12b)

⎡ 545.6 101.1 17.69 7.422 ⎤ ⎥ ⎢ C = ⎢ 2925 660.2 102.6 −67.49 ⎥ ⎢⎣ 331.9 78.42 17.12 − 11.23 ⎥⎦

(12c)

where xk are the state variables; uk and uRk are the control variables and their respective set points; Δuk are the differences between two consecutive control actions; yk and yRk denote the outputs and their respective set points; dk denote the measured disturbances; Qk, Rk, R1k, and QRk are the corresponding weights in the quadratic objective function; P is the stabilizing term derived from the Riccatti Equation for discrete systems; N and M are the output horizon and control horizon, respectively; k is the time step; A, B, C, D, E are the matrices of the discrete linear state space model; e denotes the mismatch between the actual system output and the predicted output at initial time. This quadratic programming problem is reformulated into a mpQP problem of type 1 according to the principles presented in ref 8. The correlation of the state variables at time step k to the uncertain parameters and the control variables uk is derived by the reverse substitution of the states in the linear state space model. The states at the initial time (x0), the set points (uRk and yRk ), the initial output mismatch, the previous control actions in Δuk, and the disturbances (dk) are treated as uncertain parameters denoted by the parameter vector θ. In Figure 4, we present the computational performance for different output and control horizons N OH and N CH , respectively. The problem size thereby varies from 13

⎡ 1.557 × 10−5 8.449 × 10−5 9.616 × 10−6 ⎤ ⎢ ⎥ ⎢−9.018 × 10−5 − 0.0004893 − 5.57 × 10−5 ⎥ D=⎢ ⎥ ⎢−3.487 × 10−5 − 0.0001892 − 2.154 × 10−5 ⎥ ⎢ ⎥ ⎣ 3.698 × 10−5 0.0002007 2.284 × 10−5 ⎦ (12d)

The state space model is validated against the mathematical, process model and resulted into 94.88%, 94.93%, and 93.06% fits for the three outputs, respectively. On the basis of this state space model, the general MPC problem formulation is considered as follows: G


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Figure 5. Performance of the geometrical, combinatorial, and connected-graph algorithm as well as MPT v3.1 over the test sets “POP_mpLP1” and “POP_mpQP1”.

Figure 6. Analysis of the computational effort spent on different aspects of the algorithm for the geometrical, combinatorial, and connected-graph algorithm for the test set “POP_mpLP1”.

Figure 7. Analysis of the computational effort spent on different aspects of the algorithm for the geometrical, combinatorial, and connected-graph algorithm for the test set “POP_mpQP1”.

Figure 8. Performance of the decomposition algorithm for the different comparison procedures available in POP for the test sets “POP_mpMILP1” and “POP_mpMIQP1”.

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Figure 9. Analysis of the computational effort spent on different comparison procedures for the test sets “POP_mpMILP1” for the three different comparison procedures evaluated. Note that due to the linear objective function the affine comparison yields the exact solution, which removes the need to use the “Exact” comparison used for mp-MIQP problems.

Figure 10. Analysis of the computational effort spent on different comparison procedures for the test sets “POP_mpMIQP1” for the four different comparison procedures evaluated.

None: No comparison procedure is used. MinMax: The sole solution of eq 9 Affine: The linearization of Δz(θ) Exact: The calculation of the exact solution for mp-MIQP problems. Note that for mp-MILP problems, the “Affine” comparison already yields the exact solution (i.e., no envelope of solutions) due to the linear nature of the objective function.

parameters, 2 optimization variables, and 20 constraints up to 62 parameters, 10 optimization variables, and 240 constraints.

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COMPUTATIONAL PERFORMANCE FOR TEST SETS

Remark 11. The time out of the algorithms are set to 600 s, i.e., 10 min. Thus, the percentage of problems solved is directly linked to the ability of the algorithm to solve the problem given this time constraint. Note that this setting is user-defined and completely arbitrary. Continuous Problems. In Figure 5, the time versus the percentage of problems of the test set solved is shown for “POP_mpLP1” and “POP_mpQP1”. Additionally, Figures 6 and 7 highlight the computational effort spent on different aspects of each algorithm for the test sets “POP_mpLP1” and “POP_mpQP1”, respectively. For the geometrical algorithm, the three aspects considered are (a) solution of the QP problem, (b) removal of redundant constraints, and (c) identification of a new point θ0. For the combinatorial and the connected-graph algorithm, the different aspects are (a) validation whether the selected active set was already considered or can be discarded as infeasible, (b) establishing feasibility, and (c) establishing optimality. Mixed-Integer Problems. In Figure 8, the time versus the percentage of problems of the test set solved for “POP_mpMILP1” and “POP_mpMIQP1” is shown. Additionally, Figures 9 and 10 highlight the computational effort spent on the different aspects of the algorithm, namely the integer handling, the mp-QP solution, and the comparison procedure for the different comparison procedures considered, i.e.,

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DISCUSSION The application to the MCSGP system as well as the solution of the test set problems highlights the full capabilities of POP. On the basis of the random problem generator, comprehensive test sets are developed which are used to compare different algorithms and investigate their computational behavior. This enables the identification of areas for future developments as well as an objective measure regarding the efficiency and stability of various algorithms. For the continuous case, the computational efficiency of the geometrical and connected-graph algorithms are substantial, especially for the mp-QP case. However, the results for the MCSGP system suggest that this is not generally true, as the combinatorial approach outperforms all the other algorithms, even though it was the least efficient in the test sets. The key bottlenecks which limit the additional speedup of these algorithms seem to be the solution of the QP problem and ensuring optimality for the geometrical and connected-graph algorithm, respectively. For the combinatorial approach, the validation procedure seems to be the most demanding, especially for larger scale problems. I


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Ogumerem, Justin Katz, Baris Burnak). More information about the group can be found at parametric.tamu.edu

For the mixed-integer case, the computational efficiency of using no, a min-max, or an affine comparison procedure are very similar. This is due to the fact that the main computational effort is spent in the solution of the mp-QP problem. This is surprising, as eq 9 is nonconvex, and thus, its solution could be potentially limiting. However, as we used an approximate algorithm without strict error tolerance requirements, this did not cause computational limitations. In addition, it appears that the increased number of partitions resulting from the use of an affine comparison procedure (see ref 26) does not impact the computational performance significantly. However, the use of the exact algorithm resulted in an increased computational expense. In addition, the calculation of the exact solution for mp-MIQP problems requires the solution of a quadratically constrained feasibility problem. In numerous cases, this led to numerical tolerance issues, as the convergence of the algorithm (the MATLAB in-built fmincon) was sometimes not guaranteed.

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CONCLUSION In this paper, we describe POP, the Parametric Optimization toolbox for MATLAB, which is available for free at paroc.tamu.edu/Software. It features an array of powerful multiparametric programming problem solvers, a random problem generator as well as a comprehensive problem library. The capabilities of the new toolbox are highlighted in extensive computational studies, and present a state-of-the-art analysis of the capabilities of multiparametric programming. In future, we aim at developing the toolbox beyond the MATLAB environment, as well as integrating the ability to design model predictive controllers for hybrid discrete-time systems. In addition, we aim to design problem libraries from real-world applications in order to further highlight the abilities of POP. Lastly, we will investigate in detail the different computational aspects of the algorithms, especially for the mixed-integer case, and identify how the different options of POP impact the solution time and complexity.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.6b01913. Appendix A: On the different comparison procedures for the solution of problem 3. Appendix B: List of options in POP. (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from the EPSRC (EP/M027856/1, EP/ M028240/1), Texas A&M University and Texas A&M Energy Institute is gratefully acknowledged. Special thanks to (i) Martina Wittmann-Hohlbein and Nikolaos A. Bozinis their important contributions to earlier versions of POP and (ii) the other members Multi-Parametric Optimization and Control group at Texas A& M (Mario E. Villanueva, Amit M. Manthanwar, Muxin Sun, Styliani Avraamidou, Gerald S. J


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