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Reduced order optimization of large-scale nonlinear
systems with nonlinear inequality constraints using
steady state simulators
Panagiotis Petsagkourakis†, Ioannis Bonis†, and Constantinos Theodoropoulos *†
*Corresponding author. Tel.: +44 1612004386; fax: +44 1612367439. E-mail address:
[email protected] (C. Theodoropoulos).
†School of Chemical Engineering and Analytical Science, University of Manchester, Sackville
St, Manchester M13 9PL, UK
Keywords: Model reduction-based optimization, reduced Hessian, nonlinear inequality
constraints, black-box simulator, large-scale optimization.
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Abstract
Technological advances have led to the widespread use of computational models of increasing
complexity, in both industry and everyday life. This helps to improve the design, analysis and
operation of complex systems. Many computational models in the field of engineering consist of
systems of coupled nonlinear Partial Differential Equations (PDEs). As a result, optimization
problems involving such models may lead to computational issues because of the large number
of variables arising from the spatio-temporal discretization of the PDEs. In this work, we present
a methodology for steady-state optimization, with non-linear inequality constraints of complex
large-scale systems, for which only an input/output steady-state simulator is available. The
proposed method is efficient for dissipative systems and is based on model reduction. This
framework employs a two-step projection scheme followed by three different approaches for
handling the nonlinear inequality constraints. In the first approach, partial reduction is
implemented on the equality constraints, while the inequality constraints remain the same. In the
second approach an aggregation function is applied in order to reduce the number of inequality
constraints and solve the augmented problem. The final method applies slack variables to replace
the one aggregated inequality from the previous method with an equality constraint without
affecting the eigenspectrum of the system. Only low-order Jacobian and Hessian matrices are
employed in the proposed formulations, utilizing only the available black-box simulator. The
advantages and disadvantages of each approach are illustrated through the optimization of a
tubular reactor where an exothermic reaction takes place. It is found that the approach involving
the aggregation function can efficiently handle inequality constraints while significantly reducing
the dimensionality of the system.
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1. Introduction
The optimization of large-scale systems is not a trivial task and has received significant
attention over the years. Advances in technology have allowed the evolution of available
simulators that can model accurately physical systems with high complexity, like COMSOL1 and
OpenFOAM2. However, in most cases, these simulators cannot perform optimization tasks. In
addition, many of these simulators do not offer access to the underlying modelling equations.
Distributed parameter systems (DPS)3, consisting of partial differential equations (PDEs)4, which
can express the physical behaviour of many engineering systems such as supercapacitor
manufacturing5, thermal-fluid process6 or convection-diffusion–reaction systems7. The most
common way to treat PDEs is to discretise them over a computational mesh producing large
systems of nonlinear (dynamic) equations. The resulting large-scale models cannot easily be
used for optimization and control applications8 as these applications require the repeated solution
of the system at real time. Computing reduced versions of the (large-scale) system gradients can,
therefore significantly enhance the applicability of deterministic optimization and control
algorithms for large-scale systems. Moreover, when commercial simulators are employed, the
systems’ gradients are not usually explicitly available to the user and need to be computed
numerically. Automatic differentiation9,10,11 can be utilised for the computation of numerical
derivatives in an efficient manner.
Reduced Hessian methods (rSQP)12–15 have been developed based on sequential quadratic
programming (SQP) and can be used for large-scale DPS with relatively few degrees of freedom.
The advantage of these algorithms is that a low-order projection of the Hessian matrix is used, so
less computational effort is required. The main idea is that suitable bases are constructed and
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used to project the large-scale system onto the low-dimensional subspace of the system’s
independent variables, thus effectively reducing the dimensionality of the original system.
However, these methods still require the construction (and inversion) of large-scale Jacobians
and Hessians, hence requiring significant computational effort. To side-step these issues, rSQP
methods have been combined with equation-free model reduction methodologies16, when a
(black-box) dynamic simulator of the system is available, significantly enhancing the
computational efficiency of large-scale optimization problems.
Equality constraints regularly represent the physical model of the system, for which the
optimization algorithm has to find a feasible solution. In engineering practice, there are many
limitations, either physical or technical, such as bounds of the system, of (dependent and
independent) variables, and of properties. In addition, there are economic limitations, as well as
limitations due to safety considerations (e.g. temperature bounds in the case of exothermic
reactions where sudden temperature rise can lead to runaways).
There are two main approaches for handling inequality constraints within the SQP
framework17: The first approach involves the sequential solution of inequality‐constrained QP
sub-problems (IQP) and the second involves the sequential solution of equality constrained ones
(EQP)17,18. Following the IQP rationale, at every iteration of the SQP (termed outer or major
iteration) the nonlinear inequality constraints are linearized and included in the QP sub-problem,
which in turn is solved using an active set approach. Conversely, in the EQP formulation, at
every major iteration an estimation of the active subset of the inequality constraints is identified
using estimates of the Lagrange multipliers and passed on to the QP as a working‐set, which
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leads to only equality‐constrained QP. This method has the advantage of lower computational
cost and the utilization of simpler algorithms for quadratic programming. Both approaches have
advantages and disadvantages, however none of the two can effectively handle the nonlinear
inequality and equality constraints produced by a large-scale black box simulator for solution at
real-time, as they require the full system gradients.
In the framework of rSQP, the inequality constraints cannot be introduced directly as is the
case in IQP. As a result Schulz19 proposed a variant of rSQP, the so-called Partially Reduced
SQP (PRSQP), which combines the strong properties of SQP and rSQP. The main idea of this
approach is to exploit the structure of the null space of the equality constraints (or some of them)
and handle the inequality constraints as in the SQP method. One widely-used approach to handle
inequality constraints is the use of an aggregation function like the Kreisselmeier-Stainhauser
(KS)20. This method can reduce the number of inequality constraints to just one inequality and it
can be combined with SQP20. The KS function has been also used as a barrier function21 in
chemical vapor deposition (CVD) applications in order to remove the nonlinear inequality
constraints use them as a penalty term in the objective function through the KS function. Another
interesting technique is the use of slack variables22, to turn inequality constraints into equalities,
taking into account only the active inequality constraints. This method has some disadvantages
as the number of the equality constraints may change from iteration to iteration. Therefore rSQP
may fail to produce feasible solutions. Another interesting approach to solve the optimization
problem with nonlinear inequality constraints is the barrier method, also known as interior point
method22,23, which solves an unconstrained problem by introducing a barrier function in the
objective function. The most common barrier function is the logarithmic function. Thus, when
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one inequality is close to zero, the value of the barrier function increases exponentially. Hence,
this method may produce sub-optimal results, as the inequality constraints can never be active.
However, techniques exist that help the convergence of the algorithm to an active inequality
taking advantage of central path methodologies24–26.
A new optimisation technique was recently presented for large-scale dissipative systems27
based on equation-free methods28, which exploits their dissipative nature for model order
reduction (MOR). Dissipativity is expressed as separation of eigenvalues in the spectrum of the
linearized system and therefore as a separation of system modes (or scales) to slow and fast
ones29,30. This separation has been used in various ways within the MOR context, leading to
different formulations27,31–33. Nevertheless, none of the above MOR-based methods has dealt with
problems that include nonlinear constraints.
In this paper, the aforementioned model reduction technique27 was exploited in conjunction
with three different approaches to handle large-scale systems with nonlinear constraints. The first
approach combines PRSQP with equation-free model reduction, to reduce the dimensionality of
equality constraints only. The second approach adds the feature of constraint aggregation34,
where all the inequality constraints are replaced by a single KS function. This way large-scale
inequality constraints can be effectively handled. In the last approach, a slack variable is
employed to turn the aggregated inequality into an additional equality constraint. Slack variables
could potentially produce difficulties in the model reduction step; however we provide a proof
that the aggregation of the inequalities helps to avoid such issues.
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The rest of the paper is organized as follows: Section 2 presents the background for this work
including a brief overview of the Partial reduced Sequential Quadratic Programming method, the
constraint aggregation method and the equation-free model reduction framework.
In section 3 we present the new methodology developed in this work for handling large-scale
nonlinear optimisation problems with nonlinear inequalities. The proposed schemes are
discussed alongside with proofs of equivalence of the computed optima. We apply the 3 schemes
developed to an illustrative case study, the optimization of a tubular reactor, in section 4. Finally,
a comparison of the three schemes along with relevant conclusions is presented in section 5.
2. Background
2.1 Partial Reduced Sequential Quadratic Programming
Partial Reduced Sequential Quadratic Programming (PRSQP) was introduced19 in order to
extend reduced Hessian methods for problems with “additional” nonlinear equality and
inequality constraints. PRSQP reduces the space of (some) equality constraints and the rest of the
equality and inequality constraints are treated in a similar manner as in SQP. Thus, the extra
constraints (both inequality and equality constraints) are passed to the QP sub-problem. The
problem formulation is described as follows:
min f ( x )
s . t .G ( x )=0h ( x )≤0xL≤ x≤ xU (1)
Here f ( x ) is the objective function, x∈ RN+dof is the vector of (dependent (u) and independent (
xdof )) variables. G :RN+dof→RN represents the N❑ equality constraints, and h :RN+dof→RN ¿ the
N ¿ inequality constraints. Additionally, the Lagrange function, L ,is defined as follows
L ( x )=f ( x )+λΤG ( x )+ λ¿T h (x) (2)
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where the Lagrange multipliers, λ❑ ,that correspond to equality constraints and the ones
corresponding to inequality constraints, λ¿❑, are computed directly from the solution of the QP
sub-problem. As mentioned above, large-scale problems, i.e. systems with large number of
equality and inequality constraints still require the construction (and inversion) of large full-scale
Jacobians and Hessians, compromising the computational efficiency of the optimization method.
For cases with large number of inequality constraints, aggregation methods can be applied.
2.2. Constraints aggregation
The KS function was first presented by G. Kreisselmeier and R. Steinhauser20. The function
contains an ‘aggregation parameter’,ρ which is equivalent to the penalty factor in penalty
methods. This formulation was first used to combine multiple objectives and constraints into a
single function and has been utilised in a wide variety of applications such as CVD
optimization21 and structural optimization20,34. The KS function can be used to aggregate
inequality constraints and is described as follows:
KS (hi )=1ρ
ln ¿ (3)
An equivalent expression is as follows:
KS (hi )=M+ 1ρ
ln ¿ (4)
where ρ and M are design parameters. The second expression provides better behaviour when
one or more inequalities are positive, due to numerical difficulties that may be caused by the
exponential term. The design parameterM is suggested21 to be the maximum value of the
inequality constraints. Properties of the KS function can be found in Raspanti et al21.
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2.3. Equation-Free Model Reduction
Equation-free model reduction16,31 has been successfully combined with rSQP, in the RSPQP
methodology24 so that the dissipative nature of the system can be exploited. The system’s
dissipativity can be expressed as a gap in the spectrum of the eigenvalues of the linearized
problem (Jacobian of the equality constraints). A (usually) small number of eigenvalues is
clustered near the imaginary axis (red dots in Figure 1). These eigenvalues correspond to the
slow and/or unstable modes of the system. The rest of the eigenvalues beyond the gap (blue dots
in Figure one) correspond to the fast modes. This idealized spectrum is illustrated in Figure 1.
Figure 1. Idealized separation of scales for the eigenvalues of a dissipative system.
The eigenvalues affect the stability of the static states19,30. In fact, the rightmost, slow modes in
the idealized eigenspectrum (Figure 1) enslave the rest and determine the system’s stability35.
The number, m, of the slow modes depends on the separation of scales whose existence has been
proven for parabolic PDEs36–38. A basis spanning the dominant subspace of the system, of sizem,
can efficiently be computed using subspace iterations or Krylov subspace-based algorithms like
Arnoldi iterations35. The size of the basis can be heuristically derived or adaptively computed as
in the case of the recursive projection method (RPM)39. These techniques follow the matrix-free
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concept as only evaluation of matrix-vector products is needed. As a result, even though the
systems’ equations and/or Jacobians may not be explicitly available, the calculation of the basis
is efficient and feasible.
Assume thatG=0 represents the system equations, as in eq. 1 above, contained within a
(steady-state) black box simulator, G :RN+dof→RN being Lebesque integrable. If P is the
dominant sub-space of the system and Q its orthogonal complement then:
P⊕Q=RN (5)
An orthonormal basis Zϵ RN×m for the subspace P and a projector P are defined as
P=Z ZT (6)
ZT Z=I (7)
As discussed above, an approximation, Z, of Z is computed through Krylov or Anrnoldi
iterations. The vector of system states, u , can be replaced with its low-dimensional projection, υ,
ontoP:υ=ZT u. The reduced Jacobian, Hϵ Rm×m,is then computed by the restriction of the full-
scale Jacobian, J, onto Z:
H=ZT J Z (8)
side-stepping the need to calculate the full Jacobian. The reduced Jacobian is efficiently
computed through m numerical perturbations for ε>0:
J Z j=1
2 ε (G (u+ε Z j )−G (u−ε Ζ j )) , j=1 ,…,m (9)
This scheme follows the matrix-free concept, to reduce mainly memory requirements and has
roots in the Recursive Projection Method30. Multiplying ZTwith J Zproduces the desired reduced
Jacobian21,27. The selection of the m slow modes is crucial for efficient model reduction and more
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details can be found in Bonis & Theodoropoulos27. It should be noted that only the directional
derivatives need to be computed as is the case of automatic differentiation (AD) approaches. As
a result, AD can be combined with our model reduction approach to enhance even further the
computational capabilities of our methodology.
3. MethodologyOur proposed methodologies combining equation-free model reduction with PRSQP to handle
large-scale optimisation problems are presented below. Three different ways to handle (non-
linear) inequality constraints are examined.
3.1. Equation-Free Reduced PRSQP (EF-PRSQP)
Here, equation-free model reduction is employed to project the full-scale system onto the low-
dimensional space of the slow modes, taking advantage of the separation of scales, effectively
reducing the dimensionality of the state variables given by the system equality constraints. The
inequality constraints are handled as in PRSQP.
Equation-free model reduction is successfully coupled with PRSQP using a 2-step projection
scheme32. In order to include the decision variables in the low-dimensional subspace, an
extended orthonormal basis is defined:
Zext=( Z 00 I dof ) (10)
Here I dof ϵ R(N+dof )×dof is the identity matrix and dof is the number of decision (independent)
variables. A coordinate basis of the subspace of the independent variables can be computed as:
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Z r=(−H−1 ZT∇zGI ) (11)
while a basis, Υ , for the complement subspace is given by:
Υ=(I0) (12)
Hence, a projection basis, Z¿ , equivalent to the basis computed in rSQP5-8, can be calculated
as:
Z¿=Zext Z r=(−Z H−1 ZT ∇zGI )ϵ R(N+dof )×dof (13)
where only the inverse of the low-order matrix, H , is required. This projection is equivalent to
the one of the reduced Hessian method, but not equal as the latter is the null space of the
constraints and the former is essentially a double projection firstly on the low-order subspace of
the dominant modes and subsequently on the null space of the constraints. Even though, both
methods give the same basis size, rSQP requires the construction and inversion of a large matrix
(Jacobian of constrains), whilst our model reduction-based method computes and inverts only a
low-order matrix. The reduced Hessian, BR, is the restriction of the (unavailable) full system
Hessian, B , on Z¿27:
BR=Z¿ TB Z¿ (14)
Here B is the Hessian of the Lagrange function, L ( x) , defined as L ( x )=f ( x )+GT(x )λ+h¿T ( x) λ¿.
The reduced Hessian is efficiently computed taking advantage of the directional derivatives,
employing the same central finite difference-based scheme as the one used for the reduced
Jacobian (eq. 9). To accelerate this computation a BFGS17 approach can be followed preserving
positivity of the Hessian.
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It is important to mention that the dependent variables in the optimization problem satisfy the
constraints at every iteration, so the procedure is a feasible-point algorithm. Thus, the reduced
QP sub-problem is transformed as follows:
minpZ
(Z❑¿T∇ f❑)T pz+
12pz
T BR pzs . t .∇h ( x )Z¿ pZ ≤−h(x )xL−x≤ Z¿ pz≤x
U−x
(15)
Here pz ϵ Rdof is the component of the search direction onto the subspace of the decision variables.
The basis Z¿ and the reduced Hessian BR are given by eq. 13 and 14, respectively. The low-order
projections, φ, of the Lagrange multipliers, λ ϵ R(N+dof )×dof , of the equality constraints onto P are
computed as:
HT φ=ZT (Υ Τ∇ f +λ¿Τ∇h) (16)
where
φ=ZT λ (17)
It can be easily shown that the reduced optimization problem has the following KKT conditions:
∇ f ( x )T+[H ZT ZT ∇zG ( x ) ]T φ+∇h ( x )T λ¿=0PG ( x )=0
hi λi ni=0, for i=1…N ¿ (18)
3.1.1 Proposed optimization scheme
The proposed method, which handles large-scale systems and nonlinear inequality constraints, is
presented in Algorithm 1 (Table 1).
Table 1. Optimization scheme including inequality constraints (EF-PRSQP), Algorithm 1
1. Choose initial values for x
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2. Compute a feasible point using the black box simulator.
3. Model reduction: Compute basis Z and reduced Jacobian H .
4. Construct the basis Z¿ using eq. 13.
5. If iteration number >1 and ‖Z¿ pz‖>ε then compute the multipliers, φ , using values from
the previous iteration.
6. Compute the reduced Hessian, BR, in order to solve the reduced QP problem
minpZ
(Z k¿T∇ f k )
T pz+12pz
TBR pzs . t .∇h ( x )Z¿ pZ ≤−h(x )xL−x≤ Z¿ pz≤x
U−x
7. Update the values of the variables: x=x+Z¿ pZ, x previous=x
8. If ‖Z¿ pz‖<ε update the basis Z¿, and calculate the multipliers (eq. 16) based on the new
reduced Jacobian, H .
9. H previous=H
10. Check convergence (‖Z¿ pZ‖<ε); if the problem not converged then go to step 2
Implementation of Algorithm 1 produces satisfactory results as is illustrated in the case study
(see section 4), but has a major drawback: The number of inequality constraints may be large,
because the corresponding physical system is distributed-parameter. Also, inequality constraints
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may hold for the whole spatial domain. Consequently, calculating the gradient of the inequality
constraints with respect to all variables will be computationally expensive.
3.1.2 Equivalence of the computed optima
In this section, we show that an optimum of the full problem is an optimum of the reduced
problem (and vice-versa), if the reduced system is a good approximation of the full-scale one and
the reduced states can accurately reproduce the full states. The KKT conditions of the full
problem are the following:
∇ f +∇GT λ+∇hT λ¿=0
G ( x )=0 (19)
hi ( x ) λ i ni=0
for some λ¿>0, λ>0.
It can also be proven that our reduced optimization scheme has super-linear convergence
properties.
Theorem 1: For dissipative systems every optimum point satisfies eq. 19, iff satisfies eq. 18.
Proof: Eq. 19 can equivalently be written as:
∇ f T+ [ (P+Q )∇G (P ext+Qext ) ]T λ+∇hT λ¿=0
∇ f T+ [ (P )∇G (Pext )]T λ+ [P∇GQext+Q∇G Pext+Q∇GQext ]T λ+∇hT λ¿=0 (20)
Taking into account eq.17
∇ f T+Pext∇GT Z φ+[P∇GQext+Q∇G Pext+Q∇GQ ext ]
T λ+∇ hT λ¿=0 (21)
We can set
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[P∇GQext+Q∇GP ext+Q ∇GQext ]T λ=E (22)
E being the error associated with the model reduction operation. Therefore, eq. (21) can be
written as:
∇ f T+P ext∇GT Z φ+E+∇hT λ¿=0 (23)
This can be easily shown to be equivalent to:
∇ f T+ [H ZT ZT∇zG ( x ) ]Tφ+E+∇hT λ¿=0 (24)
As long as the basis, Z , approximates the maximal invariant subspace of ∇uG and the
dominant modes are captured by the model reduction with x≈ Z v30,31, ‖E‖ is small and bounded.
In addition, Z is udated at every iteration of the algorithm, to ensure that the full-scale model is
adequately captured by the dominant modes. Hence E→0 , throughout, and the following holds:
∇ f T+ [H ZT ZT∇zG ( x ) ]Tφ+E+∇hT λ¿=0 (25)
In addition, PG(x)=0 holds, because the (feasible point) algorithm uses a black-box
simulator, which solves the equality constraints. Also h ( x ) λ¿=0 holds, as the inequality
constraints are part of the solution of the QP problem at every iteration. Hence the equivalence of
eq. (19) with eq. (18) is proven. This of course does not guarantee that the reduced problem does
not exhibit additional stationary points which satisfy the KKT conditions.
The inverse can be shown accordingly. The KKT conditions corresponding to Algorithm 1 (eq.
18) can be written as:
∇ f T+Pext∇GT Z φ+∇hT λ¿=0∇ f T+P ext∇GT Pλ+∇hT λ¿=0 (26)
The term Pext∇GT Pλ from eq. (26) is equal to∇Gλ−E. As above we can assume that
E→0whenthe full-scale model is adequately captured by the dominant modes. Additionally,
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since the algorithm is feasible-path, G(x )=0, andPG ( x )=0. Also, h ( x ) λ¿=0 as explained above.
Then (eq. 26) becomes equal to (eq. 19) ∎
3.2. Equation-Free Reduced PRSQP with Aggregated Inequalities (EF-
PRSQP-KS)
Equation-Free Model Reduced PRSQP handles the inequality constraints effectively.
Nevertheless, the derivatives of all inequality constraints are needed, which means that the
computational efficiency may be jeopardised, since conditions, such as safety specifications or
economic restrictions, may be applied to the whole spatial domain.
An effective way to tackle this problem is to use an aggregation function, such as the KS
function in eq. 3, which can be combined with the Equation-Free PRSQP in order to produce an
efficient optimization algorithm suitable for a large number of both equality and inequality
constraints.
The main advantage of this KS aggregation function is the ability to substitute all the
inequality constraints with only one. It can easily be shown that if all the inequality constraints
are negative then the KS function is negative as well, and also if there is an active set of
inequality constraints then KS approximates zero as ρ→∞21. It is then easy to replace the
formulation of the nonlinear optimization problem of eq. 6 with the formulation of eq. 27
min f ( x )s . t .G ( x )=0 (27)KS ( x , ρ )≤0
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Yet, this formulation may produce sub-optimal results, and the solution gets closer to the
optimum for high values ofρ. This behaviour can be explained since the KS function creates a
smaller feasible region for the optimizer than the original one. Nevertheless, the parameter ρ
cannot be too large from the beginning of the algorithm, because numerical difficulties may
arise. As a result, an adaptive procedure should be implemented. Poon and Martins20 introduce
such an adaptive approach in order to avoid sub-optimal results. In this approach the aggregation
parameter, ρ , changes according to the sensitivity of the KS function. The aggregation parameter
is increased so that the derivative of KS, KS ', is less than (or equal to) a small number. This
number can be defined as a desired valueK Sd' . Hence, assuming that KS ' has a linear
dependence on the aggregation parameter20, a relationship between the current value, ρc, of the
aggregation parameter and the desired one ρd, can be found:
log K S1' −log K Sc
'
ρ1−ρc=
log K Sd' −log K Sc
'
ρd−ρc (28)
ρ1 being the value of the parameter at a small step ahead. Solving eq. 28 with respect to ρd the
following equation is derived:
ρd=exp ¿ (29)
In algorithm 2 (Table 2) an adaptive procedure is presented20,34, taking into account the constraint
functions, the current aggregation parameter ρc and the desired sensitivity.
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Table 2. Adaptation procedure of aggregation parameter Algorithm 2
Compute the derivative of KS(K S' ) at the current point
If the current value is smaller than the desired, return the current value
Otherwise compute ρd according to eq. 29
Compute KS using ρd
The aggregation function allows us to handle all inequality constraints with a single
constraint. Consequently, the computational time will decrease dramatically, as the case study
will show.
3.2.1 Proposed optimization scheme.
In Algorithm 3 (Table 3) a modification of Algorithm 1 (Table 1) is presented, including the
adaptation procedure for the aggregation function. This algorithm is implemented in section 4,
where all approaches are applied to a chemical engineering example and are evaluated and
compared.
Table 3. EF-PRSQP-KS Algorithm 3
1. Steps 1. -5. are the same with Algorithm 1
2. Compute the reduced Hessian27 in order to solve the reduced QP problem
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3. Compute the KS function and its derivative
4. Apply the Algorithm 2 (Table 2)
5. Solve the reduced QP problem and calculate the Lagrange multipliers for the inequality
constraints
minpZ
(Zk¿T∇ f k )
T pz+12pz
TBR pzs . t .KS ' ( x , ρ )Z¿ pZ ≤−KS ( x , ρ )
xL−x≤ Z¿ pz≤xU−x
6. Steps 7. -10. are the same with Algorithm 1
3.2.2 Equivalence of the computed optima
In this section it is shown that every optimization point of the full NLP problem is also an
optimum of the reduced problem when the aggregation function is applied. If the reduced space
is a good approximation of the real one, the reduced states can accurately reconstruct the full
states and also the adaptive procedure of the aggregation function produces a large enough ρ
when required. Then Theorem 1 can be applied to prove the equivalence of the computed
optima. An additional fair assumption has been posed here: The aggregation parameter ρ , is
large enough and can be produced by the adaptation procedure in Algorithm 2.
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3.3. Equation-Free Reduced PRSQP with Aggregated Inequalities and Slack
Variable (EF-PRSQP-KS-S)
Slack variables have been utilized to handle inequality constraints; however the
straightforward approach of introducing slack variables and treating inequality constraints as
equalities is not a reasonable approach when the methodology implemented, includes model
reduction or generally partitioning of the solution space. If the model reduction includes active
inequality constraints then 2 undesirable effects may arise:
The dimension of the basis would vary during run-time, depending on the number of
active inequality constraints. Then it will be difficult for the algorithm to give a good
initial guess for the basis for the dominant subspace. Hence the numerical efficiency will
be jeopardized.
If the eigenvalues of the active inequality constraints are aggregated with the eigen-
spectrum of the equality constraints then this will ruin the separation of scales due to the
addition of non-dissipative modes coming from the active inequality constraints.
These disadvantages are overcome using an aggregation function, as only one equality will be
added. The KS function, presented in section 2, can aggregate effectively all the inequality
constraints. If there is only one inequality in the problem then the eigen-spectrum will not change
significantly when slack variables are added. This is proven in Lemma 1.
Lemma 1:
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If only one inequality is added to the problem then only one eigenvalue equal to one will be
added in the eigenspectrum of the original problem.
Proof:
G ( x )=0h1 ( x )≤0 slack variables⇔
hnew=h1 ( x )+s=0 (30)
where s is the slack variable that is zero when the inequality constraints are active and positive
otherwise. The Jacobian of the augmented system is:
Jaug=( ∇xGT ∇sG
T
∇x hnew ∇sh )(31)
The derivative of G with respect to the slack variable is always zero and the derivative of hnew is
always one. Furthermore, the derivative of hnew with respect to the states is equal to the derivative
of the inequality constraints. As a result the augmented Jacobian can be written as
Jaug=(∇xGT ∅
∇x h 1 ) (32)
Every eigenvalue should be the solution of the following problem.
det (J aug−λeig IN+1 )=0 (33)
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❑⇒det ((∇xG
T ∅∇xh 1 )− λeig I N+1)=0❑
⇒det ((∇xG
T−λeig IN ∅∇x h 1− λeig))=0
❑⇒
(−1 )N+2 ∂h∂ x1
det((∂G1
∂x1− λeig ⋯ 0
⋮ ⋱ ⋮∂GN
∂x1⋯ 0))+…+ (1−λeig )det (∇xG
T−λeig I N) =0 (34)
In eq. 34 all the determinants are zero because all of them have (at least) one zero column, except
from the last one. So, eq. 34 can be rewritten as
(1− λeig )det (∇xGT− λeig IN)=0 (35)
The solution of eq. 35 produces the original eigenvalues of the system and/or one eigenvalue
equal to one. ∎
The lemma shows that a slack variable can be used without a significant disturbance in the eigen-
spectrum of the system. Specifically, to ensure that no critical eigenvalue will be overlooked, the
dominant space enlarged by one (m+1¿.
3.3.1 Proposed optimization scheme
In this optimization algorithm, model reduction proceeds as in Algorithm 1, the adaptive KS
function must be calculated in every step as in Algorithm 3 (Table 3). A slack variable is
introduced and the solver solves an additional equation. The algorithm is shown in Table 4.
Table 4. EF-slack variables-KS-S Algorithm 4
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1. Step 1. – 2. are the same with Algorithm 3
2. Use Algorithm 2 to find the adaptive KS
3. Find the slack variable
4. Step 3. – 5. are the same with Algorithm 1
5. Compute the reduced Hessian27 in order to solve the reduced QP problem
minpZ
(Z❑¿T ∇ f❑)T pz+
12pz
T BR pzxL−x≤ Z¿ pz≤xU−x
6. Steps 7. – 10. are the same with Algorithm 1
3.3.2 Equivalence of computed optima
The approach in this section aggregates all the inequality constraints into one KS function, and
then a slack variable is used to transform the resulting inequality constraint into an equality
constraint. According to Lemma 1, the eigenspectrum will not be disturbed when the KS
function is used in conjunction with a slack variable. Therefore, Theorem 1 can be used to show
that the computed optimum of the reduced problem is also an optimum problem of the full NLP.
4. Results
To illustrate the behaviour of the proposed optimization algorithms, a case study of a tubular
reactor is implemented. In the reactor an exothermic, first order, irreversible reaction takes place
(A→B ¿. The reactor has three heat exchangers on its jacket, the temperature in each heat
exchanger is considered to be constant and is used as a degree of freedom for the optimization
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problem. The model of the reactor consists of 2 PDEs42 (Note that in this case the system of
equations 36-37 is actually a set of ODEs; however we keep the term PDE as the methodology
does not change for problems with more partial derivatives, e.g. 2- or 3-dimensional systems):
1Pe1
∂2 x1
∂ y2 −∂x1
∂ y❑+Da (1−x1 ) exp( x2
1+x2
γ )=0(36)
1¿Pe2
∂2x2
∂ y2 −1¿∂ x2
∂ y❑+C¿ Da (1−x1 ) exp( x2
1+x2
γ )+ β¿ (x2w−x2)=0 (37)
where is the dimensionless concertation of the product, the dimensionless temperature
inside the reactor, the Damkohler number, the Lewis number, Pe1 and Pe2 the Peclet
numbers for mass and heat transfer respectively, the dimensionless heat transfer, C the
dimensionless adiabatic temperature rise, the dimensionless activation energy, ∈[0 , L] the
dimensionless longitude coordinate, L the length of the reactor and the dimensionless
adiabatic wall temperature. , whose expression is given as a function of the longitudinal
coordinate:
(38)
where is the Heaviside function, , , and the dimensionless
temperature at each of the 3 cooling zones. The boundary conditions are:
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(39)
(40)
The parameters of the physical model are
andL=1. The set of equations (36-37) are discretised with the central Finite Differences method
on a mesh of 250 nodes. This discretisation results in n = 500 dependent variables. The size of
the subspace, m, is chosen to be 10 so it can be large enough to capture the dominant dynamics
throughout the parameter space.
The optimization problem aims to maximize the concertation at the outlet of the reactor by
changing the 3 wall temperatures. The basic problem consists of only equality constraints the
steady state mass and energy balances27. In reactors that exothermic reactions take place, a
thermal runaway43 may produce uncontrollable situations. To control this phenomenon,
inequality constraints should be applied to meet some safety specifications. In this case study the
dimensionless reaction rate is selected to have an upper-bound in order to avoid thermal
explosions. The optimization problem is then set up as:
maxx2w
x1( y=1) (41)
s.t. G ( x )=0
h ( x )≤3.5
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0≤ x1≤1
0≤ x2≤8
0≤ x2w i≤4
i=1 ,…,3
where is the dimensionless reaction rate across the reactor, given by eq. 42, are the
PDEs of the physical model (eq. 36,37) in discretized form and x=[ x1T , x2
T , x❑2wiT ]T .
(42)
The inequality constraints, h(x), are applied to the whole length of the reactor to ensure the limit
is not surpassed at any point. Thus, not only the problem consists of a large number of equality
constraints but also of a large number of inequality constraints.
All three approaches produce the same optimal solution and in order to illustrate and compare
the 3 optimization algorithms, the methodologies are evaluated in terms of the number of
iterations for the same initial guess and CPU-time for different initial guesses. The initial guess
for the Equation Free-PRSQP-KS and the Equation Free -PRSQP should satisfy the inequality
constraints.
Firstly, a base case scenario is presented in order to observe the optimization path and the
general convergence behavior, where the initial guess for all wall temperatures is 0.2.
Convergence results are given in terms of ‖Z¿ pz‖, which is the norm of the solution update for
each iteration. As mentioned before, the optimum point computed for all three algorithms is the
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same. The results presented in Figures 2-3 depict the solution of the problem with inequality
constraints alongside with the solution of the problem taking in account only equality constraints
(Initial problem). As it can be seen in Figure 3, the reaction rate and the temperature of the
inequality-constrained problem are higher than those of the initial problem beyond y=0.2 to
allow the reactant concentration to reach the optimum point.
Figure 2. Dimensionless concentration at the optimum point with and without inequality
constraints.
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Figure 3. (a) Dimensionless reaction rate and (b) dimensionless temperatures at the optimum
point with and without inequality constraints.
Despite the fact that all 3 methods produce identical optimum point results, the convergence path
and computational time for each of the methods varies. Convergence paths for the three
approaches are depicted in Figure 4a showing that the norm of the solution follows almost the
same path for each method for the same initial guess ( dimensionless temperature all 3 cooling
zones being 0.2) but Equation Free-PRSQP-KS requires the minimum amount of iterations,
while Equation Free-PRSQP-KS-S requires the most. It is important to have a knowledge on
how an algorithm convergences for different initial guesses, because in most cases there is no a
priori knowledge of good guesses. In order to examine this behavior, 6 experiments have been
performed for every algorithm: Initially, all wall temperatures (degrees of freedom) start with the
value 0.00, then the code runs and the CPU-time is recorded. After that, wall temperatures are
given the previous initial guess value plus 0.075, and the algorithm starts again. These
experiments are conducted for all the proposed algorithms. The corresponding results are
depicted in Figure 4b.
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Figure 4. (a) Convergence behaviour for each method starting from the same initial guess. (b)
CPU- times for each method for a range of initial guesses for the degrees of freedom.
As it can be seen, EF-PRSQP-KS is the fastest with the minimum number of iterations, whilst
the slowest is the EF-PRSQP as expected. Figure 4 shows that EF-PRSQP-KS is faster in
comparison to the others, for almost all the different initial guesses tested. In conclusion, all three
algorithms seem to have the same trend in terms of iterations and computational time for all the
experiments tested, however the fastest algorithm regardless of the number of iterations is the
EF-PRSQP-KS, EF-PRSQP-KS-S comes next, and the slowest is EF-PRSQP. Thus, the
aggregation of the inequality constraints into one seems to have a significant impact into the
computational time as the CPU time of EF-PRSQP-KS is always around 80% faster than that of
EF-PRSQP.
To investigate how the fastest method, EF-PRSQP-KS, scales with problem size, we have tested
the solution of the same system (eq. 36-42) for n=1000 and 1500, respectively. In addition, we
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have compared its performance (in CPU s) against that of PRSQP and of a “standard” NAG
solver. As we can see in Table 1, EF-PRSQP-KS is approximately 8-10 times faster than PRSQP
for all problem sizes and 17- 58 times faster than standard NAG-based SQP which solves the full
model. All 3 methods converge to the same solution.
Table 1: Comparison of performance of EF_PRSQP-KS against PRSQP and NAG SQP for different problem sizes.
Problem size, n EF-PRSQP-KS, m=10(CPU, s)
PRSQP(CPU, s)
NAG SQP(CPU, s)
500 8.8 64 150
1000 39.1 509 1293
1500 75.8 796 4454
In addition, to investigate the effect of the size of m, we have tested the performance of
EF_PRSQP-KS for m ranging from 10 to 50 for system size n= 500 and 1000. As it can be seen
in Table 2, the method performs equally well for the whole range of m, which attests to its
robustness. It is worthwhile to note that the method is even faster for m = 20-50 than for m =10
as the system requires less iterations to converge as seen in Fig. 5, where the convergence
behavior is plotted for m = 10-30.
Table 2: Comparison of performance of EF_PRSQP-KS for different subspace, m, sizes. Problem size
(number of nodes, n)m=10(CPU, s)
m=20(CPU, s)
m=30(CPU, s)
m=40(CPU, s)
m=50(CPU, s)
500 8.8 5.5 5.6 5.8 6.1
1000 39.1 22 23.3 25.3 25.6
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Figure 5. Convergence behaviour of EF_PRSQP-KS from the same initial guess for different
subspace, m, sizes.
The Arnoldi iterations as well as the solution of each sub-QP and SQP-based solutions were
computed using Nag library Mark 25 in Nag Builder 6.1.
The times reported correspond to single threaded executions of FORTRAN source code on an
Intel core i7-6700 processor (3.40 GHz) and 16 GB of RAM, running 64-bit windows 7.
5. Conclusions
A model reduction-based deterministic optimisation method for large-scale nonlinear dissipative
PDE‐constrained problems, which involve a large number of nonlinear inequality constraints has
been presented. The work builds on previous research from the group presented earlier27. The
proposed algorithms are based on a 2 step-projection scheme. The first projection is onto the
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dominant space of the physical system, which is defined by the user in the beginning of the
algorithm, augmented in order to include the independent variables. Then the second projection
is onto the null space of the equality constraints. In this work all the methodologies make the
assumption that the system is dissipative and thus there exists a separation of scales. As a result,
there exists a low-order basis, which can be efficiently computed in order to approximate the full
large-scale system. Three approaches have been followed to handle (the large number of)
inequality constraints: the Equation-free-reduced-PRSQP method that reduces the number of
equality constraints and then includes the inequality constraints in the quadratic sub-problem,
This approach may exhibit numerical difficulties as the full derivatives of the inequality
constraints with respect to the states have to be computed. The second approach takes advantage
of the KS aggregation functions in order to handle only one inequality and combines it with the
former method. The last approach uses slack variables to convert the inequality constraints into
equalities. This is convenient when an aggregated function is used because the eigenvalue of the
one additional equality is known and is equal to one. The behaviour of the three approaches is
illustrated by applying them to the optimisation of a tubular reactor also testing the behaviour of
the three algorithms for different initial guesses. In conclusion, the EF-PRSQP-KS seems to have
the best performance, as it requires the least computational time and iterations compared to the
other two. This methodology, was applied to a static system assuming that there is an available
steady state simulator, however a dynamic model where G ( x )=xk+1−g (xk )=0 can be available
instead. In this case the method presented in Theodoropoulos and Luna-Ortiz33 can be extended
to include inequalities.
33
Acknowledgements
The EU program CAFE (KBBE-2008-212754) and the University of Manchester Presidential
Doctoral Scholarship Award to Panagiotis Petsagkourakis are gratefully acknowledged.
References
(1) COMSOL Multiphysics® v. 5.2. Www.comsol.com. COMSOL AB, Stockholm, Sweden.
(2) Guide, U. OpenFOAM, The Open Source CFD Toolbox. 2013, No. September, 211.
(3) Li, H. X.; Qi, C. Modeling of Distributed Parameter Systems for Applications - A
Synthesized Review from Time-Space Separation. J.ProcessControl 2010, 20 (8), 891.
(4) Biegler, L. T. New Nonlinear Programming Paradigms for the Future of Process
Optimization. AIChEJ. 2017, 63 (4), 1178.
(5) Drummond, R.; Howey, D. A.; Duncan, S. R. Low-Order Mathematical Modelling of
Electric Double Layer Supercapacitors Using Spectral Methods. J.PowerSources 2015,
277, 317.
(6) Balsa-Canto, E.; Alonso, A. A.; Banga, J. R. A Novel, Efficient and Reliable Method for
Thermal Process Design and Optimization. Part I: Theory. J.FoodEng. 2002, 52 (3), 227.
(7) Christofides, P. D. Nonlinear and Robust Control of PDE Systems: Methods and
Applications to Transport-Reaction Processes; Systems & Control: Foundations &
34
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Applications; Birkhäuser Boston, 2012.
(8) Hazra, S. B. Large-ScalePDE-ConstrainedOptimizationinApplications; Springer Berlin
Heidelberg, 2010; Vol. 54.
(9) Andersson, J.; Åkesson, J.; Diehl, M. {CasADI}: {A} Symbolic Package for Automatic
Differentiation and Optimal Control. In RecentAdvancesinAlgorithmicDifferentiation;
Forth, S., Hovland, P., Phipps, E., Utke, J., Walther, A., Eds.; Lecture Notes in
Computational Science and Engineering; Springer: Berlin, 2012; Vol. 87, pp 297–307.
(10) Bischof, C.; Corliss, C.; Green, L. L.; Griewank, A.; Haigler, K. J.; Newman, P. A.
AutomaticDifferentiationofAdvancedCFDCodesforMultidisciplinaryDesign; NASA
Langley Technical Report Server, 2003.
(11) Baydin, A. G.; Pearlmutter, B. A.; Radul, A. A. Automatic Differentiation in Machine
Learning: A Survey. CoRR 2015, abs/1502.0.
(12) Biegler, L. T.; Nocedal, J.; Schmid, C. A Reduced Hessian Method for Large-Scale
Constrained Optimization. SIAMJ.Optim. 1995, 5 (2), 314.
(13) Ternet, D. J.; Biegler, L. T. Recent Improvements to a Multiplier-Free Reduced Hessian
Successive Quadratic Programming Algorithm. Comput.Chem.Eng. 1998, 22 (7–8), 963.
(14) Wang, K.; Shao, Z.; Biegler, L. T.; Lang, Y.; Qian, J. Robust Extensions for Reduced-
Space Barrier NLP Algorithms. Comput.Chem.Eng. 2011, 35 (10), 1994.
(15) Bock, H. G.; Diehl, M.; Kühl, P.; Kostina, E.; Schiöder, J. P.; Wirsching, L. Numerical
Methods for Efficient and Fast Nonlinear Model Predictive Control. In Assessmentand
35
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
FutureDirectionsofNonlinearModelPredictiveControl; Findeisen, R., Allgöwer, F.,
Biegler, L. T., Eds.; Springer Berlin Heidelberg: Berlin, Heidelberg, 2007; pp 163–179.
(16) Theodoropoulos, C.; Qian, Y.-H.; Kevrekidis, I. G. “Coarse” stability and Bifurcation
Analysis Using Time-Steppers: A Reaction-Diffusion Example. Proc. Natl. Acad. Sci.
2000, 97 (18), 9840.
(17) Nocedal, J.; Wright, S. J. NumericalOptimization; 2006.
(18) Byrd, R. H.; Hriba, M. E.; Nocedal, J. An Interior Point Algorithm for Large Scale
Nonlinear Programming. SIAMJ.Opt. 2000, 9 (4), 877.
(19) Schulz, V. H. Reduced SQP Methods for Large-Scale Optimal Control Problems in DAE
with Application to Path Planning Problems for Satellite Mounted Robots. 1996, No.
November 1995.
(20) Poon, N.; Martins, J. Adaptive Constraint Aggregation for Structural Optimization Using
Adjoint Sensitivities. 2005Can.Aeronaut.Sp.Inst.Annu.Gen.Meet. 2005, 1.
(21) Raspanti, C. G.; Bandoni, J. a; Biegler, L. T. New Strategies for Flexibility Analysis and
Design under Uncertainty. Comput.Chem.Eng. 2000, 24 (9–10), 2193.
(22) Sun, W.; Yuan, Y. Optimization Theory and Methods: Nonlinear Programming. 2006.
(23) Bazaraa; Sherali; Shetty. NonlinearProgrammingTheoryandAlgorithms; 2006.
(24) Wills, A. G.; Heath, W. P. Interior-Point Algorithms for Nonlinear Model Predictive
Control. 2007, 207.
36
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
(25) Wang, Y.; Boyd, S. Fast Model Predictive Control Using Online Optimization. Control
Syst.Technol.IEEETrans. 2010, 18 (2), 267.
(26) Boyd, S. P.; Vandenberghe, L. ConvexOptimization; Berichte über verteilte messysteme;
Cambridge University Press, 2004.
(27) Bonis, I.; Theodoropoulos, C. Model Reduction-Based Optimization Using Large-Scale
Steady-State Simulators. Chem.Eng.Sci. 2012, 69 (1), 69.
(28) Theodoropoulos, C.; Qian, Y.; Kevrekidis, I. G. “ Coarse ” Stability and Bifurcation
Analysis Using Time-Steppers : A Reaction-Diffusion Example. 2000, 97 (18), 9840.
(29) Armaou, A.; Siettos, C. I.; Kevrekidis, I. G. Time-Steppers and “Coarse” Control of
Distributed Microscopic Processes. Int.J.RobustNonlinearControl 2004, 14 (2), 89.
(30) Shroff, G. M.; Keller, H. B. Stabilization of Unstable Procedures: The Recursive
Projection Method. SIAMJ.Numer.Anal. 1993, 30 (4), 1099.
(31) Theodoropoulos, C. Optimisation and Linear Control of Large Scale Nonlinear Systems:
A Review and a Suite of Model Reduction-Based Techniques. In Lecture Notes in
ComputationalScienceandEngineeringVol75; 2011; pp 37–61.
(32) Luna-Ortiz, E.; Theodoropoulos, C. An Input/output Model Reduction-Based
Optimization Scheme for Large-Scale Systems. MultiscaleModel.Simul. 2005, 4 (2), 691.
(33) Theodoropoulos, C.; Luna-ortiz, E. A Reduced Input / Output Dynamic Optimisation
Method. In Model Reduction and Coarse-Graining Approaches for Multiscale
Phenomena; 2006; pp 535–560.
37
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
(34) Martins, J. R. R. a; Poon, N. M. K. On Structural Optimization Using Constraint
Aggregation. Proc.6thWorldCongr.Struct.Multidiscip.Optim. 2005, No. June, 1.
(35) Saad, Y. Numerical Methods for Large Eigenvalue Problems. AlgorithmsArchit.Adv.Sci.
Comput. 1992, 346 p.
(36) Christofides, P. D.; Armaou, A. Control and Optimization of Multiscale Process Systems.
Comput.Chem.Eng. 2006, 30 (10–12), 1670.
(37) El-Farra, N. H.; Armaou, A.; Christofides, P. D. Analysis and Control of Parabolic PDE
Systems with Input Constraints. Automatica 2003, 39 (4), 715.
(38) Friedman, A. PartialDifferentialEquationsofParabolicType; Dover Publications INC:
Mineola, New York, 1964.
(39) Shroff, G. M.; Keller, H. B. Stabilization of Unstable Procedures: The Recursive
Projection Method. SIAMJ.Numer.Anal. 1993, 30 (4), 1099.
(40) Bonis, I.; Xie, W.; Theodoropoulos, C. A Linear Model Predictive Control Algorithm for
Nonlinear Large-Scale Distributed Parameter Systems. AIChEJournalAIChEJ. 2012, 58,
801.
(41) Biegler, L. T.; Nocedal, J.; Schmid, C. A Reduced Hessian Method for Large-Scale
Constrained Optimization. SIAMJ.Optim. 1995, 5 (2), 314.
(42) Jensen, K. F.; Harmon Ray, W. The Bifurcation Behaviour of Tubular Reactors. Chem.
EngheetinrhgSci. 1982, 37 (2), 199.
(43) Ni, L.; Mebarki, A.; Jiang, J.; Zhang, M.; Pensee, V.; Dou, Z. Thermal Risk in Batch
38
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Reactors: Theoretical Framework for Runaway and Accident. J.LossPrev.ProcessInd.
2016, 43, 75.
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Table of Contents (TOC) Graphical Abstract
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