new algorithms for mixed-integer dynamic optimization

New algorithms for mixed-integer dynamic optimization

Vikrant Bansal 1, Vassilis Sakizlis, Roderick Ross 2, John D. Perkins,Efstratios N. Pistikopoulos *

Centre for Process Systems Engineering, Department of Chemical Engineering, Imperial College, Prince Consort Road, London SW7 2BY, UK

Received 19 February 2001; received in revised form 21 October 2002; accepted 11 November 2002

Abstract

Mixed-integer dynamic optimization (MIDO) problems arise in chemical engineering whenever discrete and continuous decisions

are to be made for a system described by a transient model. Areas of application include integrated design and control, synthesis of

reactor networks, reduction of kinetic mechanisms and optimization of hybrid systems. This article presents new formulations and

algorithms for solving MIDO problems. The algorithms are based on decomposition into primal, dynamic optimization and master,

mixed-integer linear programming sub-problems. They do not depend on the use of a particular primal dynamic optimization

method and they do not require the solution of an intermediate adjoint problem for constructing the master problem, even when the

integer variables appear explicitly in the differential�/algebraic equation system. The practical potential of the algorithms is

demonstrated with two distillation design and control optimization examples.

# 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Mixed-integer optimization; Dynamic modelling; Process design; Process control; Distillation; gPROMS

1. Introduction

The last decade has seen a rapid increase in the use of

dynamic simulation as a tool for studying the transient

behaviour of process systems. In fact, common practice

is to use dynamic simulation as a means towards

‘optimizing’ process design and/or operation. This is

achieved by an engineer performing multiple, trial-and-

error simulations in order to choose a set of values for

the process parameters that leads to ‘good’ performance

(e.g. low cost or fast transition between steady-states)

while satisfying the constraints under which the process

operates. Although this approach is simple and intuitive,

it has the major drawback that the fraction of the

parameter space that can be searched is severely limited,

especially as the number of parameters increases. This

not only means that it is impossible to guarantee

optimality, but in many cases it can even be difficult

to find a set of parameter values that leads to a feasible

solution where all the constraints are satisfied (Pante-

lides, 1996).

The ad hoc approach described above can be avoided

by posing formal, dynamic optimization problems, where

the objective is to select the set of time-dependent and/or

time-invariant parameters that optimize some aspect of

the transient behaviour of the process while satisfying a

given set of constraints. A number of algorithms have

been developed in the literature for solving such

problems and their reliability has evolved to the extent

that realistic engineering problems involving thousands

of variables can now be readily solved with commercial

codes such as gPROMS/gOPT (Process Systems En-

terprise Ltd., 2000). Reported applications using

gPROMS include optimization of operating policies in

multi-component batch distillation and other processes

(Charalambides, Shah, & Pantelides, 1995; Furlonge,

Pantelides, & Sorensen, 1999; Ishikawa, Natori, Liberis,

& Pantelides, 1997); operation of catalytic reactors

(Lovik, Hillestad & Hertzberg, 1998); design and

operation of heat exchangers under fouling (Georgiadis,

Rotstein, & Macchietto, 1998); analysis of hazards in

* Corresponding author. Tel.: �/44-20-7594-6620; fax: �/44-20-

7594-6606.

E-mail address: [email protected] (E.N. Pistikopoulos).1 Present address: Orbis Investment Advisory Limited, Orbis

House, 5 Mansfield Street, London W1G 9NG, UK.2 Present address: United Technologies Research Center, 411 Silver

Lane, East Hartford, CT 06108, USA.

Computers and Chemical Engineering 27 (2003) 647�/668

www.elsevier.com/locate/compchemeng

0098-1354/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0098-1354(02)00261-2

mailto:[email protected]

batch processes (Viswanathan, Shah, & Venkatasubra-

manian, 2000); and the simultaneous optimization of the

process and control system design in complex distilla-

tion systems (Bansal, Ross, Perkins, & Pistikopoulos,2000; Ross, Bansal, Perkins, Pistikopoulos, Koot, & van

Schijndel, 1999). Other notable applications of dynamic

optimization to chemical engineering processes include

on-line model predictive control (Biegler & Sentoni,

2000; Nath & Alzein, 2000); fault diagnosis (Huang,

Reklaitis, & Venkatasubramanian, 2000); determination

of policies for switching between steady-states (Cer-

vantes, Tonelli, Brandolin, Bandoni & Biegler, 2000;Mcauley & Macgregor, 1992; White, Perkins, & Espie,

1996); and optimization of systems subject to excep-

tional events such as failures and sudden changes in

product demand (Abel & Marquardt, 2000).

Practical engineering problems generally involve dis-

crete as well as continuous decisions. For example, in

the optimal design of a distillation column, the number

of trays and feed location take discrete values, while thevalues of the column diameter and surface areas of the

reboiler and condenser are continuous. This gives rise to

mixed-integer dynamic optimization (MIDO) problems,

the solution of which is a formidable task. The devel-

opment of algorithms for solving such problems is still

in its infancy and thus few large-scale applications have

been reported. The objective of this article is to propose

new algorithms for solving MIDO problems that do notdepend on particular integration and dynamic optimiza-

tion solution methods, require significantly less inter-

mediate information than current algorithms, and as

such, are better suited to the solution of larger, more

realistic problems.

The remainder of the article is structured as follows.

After some mathematical preliminaries, the existing

solution approaches for MIDO are thoroughly andcritically reviewed. An alternative formulation and

algorithm based on generalised Benders decomposition

principles is then presented and its advantages are

discussed. An analogous algorithm based on outer-

approximation (OA) is also outlined and, finally, the

application of the two algorithms is illustrated with two

distillation design and control examples.

2. Preliminaries

A MIDO problem can be formulated as:

minu(t);d;y

J(xd(tf ); xd(tf ); xa(tf ); u(tf ); d; y; tf );

s:t: hd(xd(t); xd(t); xa(t); u(t); d; y; t)�0; � t � [t0; tf ];

ha(xd(t); xa(t); u(t); d; y; t)�0; � t � [t0; tf ];

h0(xd(t0); xd(t0); xa(t0); u(t0); d; y; t0)�0;

hp(xd(ti); xd(ti); xa(ti); u(ti); d; y; ti)�0; ti � [t0; tf ];

i�1; . . . ;N;

gp(xd(ti); xd(ti); xa(ti); u(ti); d; y; ti)00; ti � [t0; tf ];

i�1; . . . ;N;

hq(d; y)�0;

gq(d; y)00; (1)

where hd�/0 and ha�/0 represent the system of

differential�/algebraic equations (DAEs) with initial

conditions h0�/0; hp�/0 and gp0/0 represent the set

of point equalities and inequalities, respectively, that

must be satisfied at specific time instances (including the

end-point if required); hq�/0 and gp0/0 are the time-

invariant equality and inequality constraints; xd(t) andxa(t) are the vectors of differential state and algebraic

variables, respectively; u(t) is the vector of time-varying

control variables; d is the vector of time-invariant

continuous search variables (that may include the final

time, tf); and y is the vector of time-invariant integer

variables. In the sequel, we will consider y to be a vector

of binary (0�/1) variables since most process engineering

applications are modelled in this way and any integervariable can be expressed as a sum of binary variables

(Floudas, 1995). The case of time-varying binary vari-

ables will be considered later in the article.

Note that the formulation in Eq. (1) is quite general.

For example, integral terms in the objective function can

be eliminated by the addition of extra differential

equations. If there are path inequality constraints that

must be satisfied at all times during the time horizon,these can always be converted into end-point inequality

constraints (e.g. Bansal, 2000; Sullivan & Sargent, 1979),

which are a sub-set of gp0/0.

3. Review of existing solution approaches for MIDO

By definition, it is the presence of the binary variables

y that complicate the solution of a MIDO problem

compared to a continuous dynamic optimization pro-

blem (for which solution techniques are well-estab-

lished). In the context of optimally designing dynamicsystems for robustness, Samsatli, Papageorgiou and

Shah (1998) proposed the use of smooth approximations

for the binary variables in order to convert the discrete�/

continuous problem Eq. (1) into a purely continuous

one. For example, if a binary variable y is defined such

that it takes a value of 1 if another (continuous) variable

x �/0 and is zero otherwise, then the following smooth

approximation would be used:

y�1

2[tanh(bx)�1]; (2)

V. Bansal et al. / Computers and Chemical Engineering 27 (2003) 647�/668648

where b is a large, positive number. The major draw-

back of trying to solve general MIDO problems using

such an approach is that the smooth approximation

does not always lead to integral values of y (e.g., in Eq.(2), when x�/0, y�/0.5). Non-integral values of y will

often render the engineering problem physically mean-

ingless and lead to severe numerical difficulties.

Androulakis (2000) proposed a branch-and-bound

algorithm for solving MIDO problems that arise when

attempting to systematically reduce the complexity of

kinetic mechanisms. This also requires the solution of

dynamic optimization problems with relaxed binaries y.Furthermore, as acknowledged by the author, branch-

and-bound techniques are unsuitable for all but very

small MIDO problems since they will typically involve

the solution of a much larger number of dynamic

optimization problems (the computational rate-deter-

mining step) than the decomposition methods described

below.

The other MIDO solution approaches that haveappeared in the literature are all based on decomposi-

tion principles, but differ in their treatment of the DAE

system. One approach is to completely discretise the

system using a technique such as orthogonal collocation

on finite elements. This converts the MIDO problem Eq.

(1) into a finite-dimensional, mixed-integer non-linear

program (MINLP), where the continuous search vari-

ables for the optimization are the full set of discretisa-tion parameters for the differential, algebraic and

control variables, and the time-invariant variables d.

The MINLP can be solved using any conventional

method. Balakrishna and Biegler (1993) and Avraam,

Shah and Pantelides (1998, 1999) used the outer-

approximation/equality-relaxation/augmented-penalty

(OA/ER/AP) algorithm of Viswanathan and Gross-

mann (1990), while Dimitriadis and Pistikopoulos(1995) and Mohideen, Perkins and Pistikopoulos

(1996) used generalised Benders decomposition, or

GBD (Geoffrion, 1972), in the contexts of synthesising

reaction�/separation systems, optimizing hybrid pro-

cesses, analysing dynamic systems under uncertainty,

and simultaneous design and control, respectively. The

advantages of this approach are that the dynamic model

and optimizer constraints are simultaneously convergedand that general types of inequality constraints are

easily handled. There are two major drawbacks, how-

ever. Firstly, the MINLP resulting from the discretisa-

tion is very large even for relatively small problems. In

the context of solving realistic engineering problems the

size of the MINLP will often make the problem

intractable. Secondly, intermediate solutions during

the course of the optimization generally do not satisfythe original differential�/algebraic system equations

(hence the common use of the term ‘infeasible path’

methods). This effectively renders the approach an ‘all-

or-nothing’ strategy since if the optimizer fails to find an

optimum, the intermediate results are usually mean-

ingless and cannot be used, e.g., as new initial starting

points.

For continuous dynamic optimization problems, analternative approach to complete discretisation is to

only parameterise the control variables, u(t ), in terms of

time-invariant parameters (‘reduced space discretisation’

or ‘control vector parameterisation’). For given u(t) and

values of the other search variables, d, the DAE system

is integrated using standard algorithms. Gradient in-

formation with respect to the objective function and

constraints is obtained through finite difference pertur-bations, or more accurately through integration of the

sensitivity equations (e.g. Vassiliadis, Sargent & Pante-

lides, 1994a,b), or solution of the adjoint equations (e.g.

Sullivan & Sargent, 1979; Pytlak, 1998; Bloss, Biegler, &

Schiesser, 1999). An NLP solver then adjusts the values

of the controls and other search variables on the basis of

these gradients. Although this method has the disadvan-

tage that the solution of the DAEs and other equationscan be computationally expensive, it gives useful inter-

mediate (sub-optimal) solutions that are feasible with

respect to the DAE system; allows more accurate

control of the error associated with the solution of the

DAE system; and is better conditioned when handling

stiff models. As such, it is much more suited for large

dynamic systems than the complete discretisation ap-

proach, although work is ongoing aimed at redressingthe balance (e.g. Cervantes & Biegler, 1998).

The MIDO algorithms in the literature that utilise

reduced space methods (Allgor & Barton, 1999; Mo-

hideen, Pistikopoulos & Perkins, 1997; Narraway, 1992;

Schweiger & Floudas, 1997; Sharif, Shah & Pantelides,

1998) all decompose the problem into a series of primal

problems where the binary variables are fixed, and

master problems which determine a new binary config-uration for the next primal problem. In all cases the

primal problem corresponds to a dynamic optimization

problem that is solved using a reduced space approach,

and which gives an upper bound on the final solution.

The methods mainly differ in how they construct the

master problem, which gives a lower bound on the

solution. In the case of OA/ER/AP, the MILP master

problem is constructed by linearising the objectivefunction and constraints. This contrasts with GBD

where the master problem is constructed using the

solution of the primal problem and corresponding

dual information. The advantage of the former ap-

proach is that it requires the solution of fewer primal

problems since its (more constrained) master problem

always gives at least as tight a lower bound as the GBD

master problem (Duran & Grossmann, 1986). Thiscould be important for MIDO problems since the primal

problem (dynamic optimization) will tend to be much

more computationally expensive relative to the master

problem than for MINLPs. However, the OA methods

V. Bansal et al. / Computers and Chemical Engineering 27 (2003) 647�/668 649

are restricted to problems where the binary variables

appear linearly and separably in the objective function

and constraints, and are not designed to handle binary

variables that appear explicitly in the DAE system(pointed out by Schweiger & Floudas, 1997), which

would be the case, e.g., for the simultaneous design and

control optimization of a distillation column. Note that

the convexity conditions required for OA/ER/AP and

GBD to give the global optimum are rarely satisfied in

dynamic process engineering problems. Allgor and

Barton (1999) proposed a different master problem

approach where an MILP is solved based on a so-called‘screening model’. This approach may have the potential

for providing rigorous bounds on the global optimum;

its major drawback, however, is that the ‘screening

model’ is entirely case study-specific since it ‘is derived

from domain specific knowledge gathered from physical

laws and engineering insight’ .

Mohideen et al., 1997 and Schweiger and Floudas

(1997) have both developed MIDO algorithms based onvariant-2 of generalised Benders decomposition, v2-

GBD (Floudas, 1995), which are now explored in

more detail. In both cases, the k th primal problem

takes the form:

minu(t);d

J(xd(tf ); xd(tf ); xa(tf ); u(tf ); d; yk; tf );

s:t: hd(xd(t); xd(t); xa(t); u(t); d; yk; t)�0; � t � [t0; tf ];

ha(xd(t); xa(t); u(t); d; yk; t)�0; � t � [t0; tf ];

h0(xd(t0); xd(t0); xa(t0); u(t0); d; yk; t0)�0;

hp(xd(ti); xd(ti); xa(ti); u(ti); d; yk; ti)�0; ti � [t0; tf ];

i�1; . . . ;N;

gp(xd(ti); xd(ti); xa(ti); u(ti); d; yk; ti)00; ti � [t0; tf ];

i�1; . . . ;N;

hq(d; yk)�0;

gq(d; yk)00; (3)

where Eq. (3) corresponds to Eq. (1) with the binaryvariables fixed, y�/yk . In the approach of Mohideen et

al. (1997), the primal problem is solved using an implicit

Runge�/Kutta method to integrate the DAE system and

the NLP gradient information is obtained using adjoint-

based arguments. In the approach of Schweiger and

Floudas (1997), a backward difference formula algo-

rithm is used for the DAEs and the sensitivity equations

are integrated to obtain the required gradients. Thesolution of the k th primal problem is denoted as uk(t)

and dk; with resulting differential variables xk

d(t) and

algebraic variables xka(t):/

The master problem is formulated using dual infor-

mation and the solution of the primal problem in order

to construct supporting functions about uk(t) and dk: If

the primal problem is solved using adjoint-based argu-ments, as in Mohideen et al. (1997), then the dual

variables are available directly from the gradient evalua-

tion method used for the primal problem. If a sensitiv-

ity-based approach is used for the primal problem then

the dual variables are obtained through the solution of

the final-value problem (Bryson & Ho, 1975; Vassiliadis,

1993):

d

dt

��@hd

@xd

�T

nkd(t)

��

�@hd

@xd

�T

nkd(t)�

�@ha

@xd

�T

nka(t)�0;

��@hd

@xa

�T

nkd(t)�

�@ha

@xa

�T

nka(t)�0; (4)

with boundary conditions:�@J

@xd

�T

��@hd

@xd

�T jtf

nkd(tf )�

�@hd

@xd

�T jtf

jkd;f

��@ha

@xd

�T jtf

jka;f

��@hp

@xd

�T

mkp�

�@gp

@xd

�T

lkp�0; (5)

where nd, na, m and l are the dual multipliers associatedwith the differential equations, algebraic equations,

equality point constraints and inequality point con-

straints, respectively. jd, f and ja, f are multipliers that

are associated with the differential and algebraic equa-

tions at the final time; these are evaluated from the first

order optimality conditions of the primal problem

(Vassiliadis, 1993).

The dual multipliers mk and lk are obtained from thesolution of the primal problem, regardless of the method

used to solve it. The Lagrange function, Lk, for the case

of a feasible primal problem Eq. (3), is given by:

Lk(uk; dk; y)�J( ˆx

k

d(tf ); xkd(tf ); xk

a(tf ); uk(tf ); dk; y; tf )

�gtf

t0

nkT

d (t)hd( ˆxk

d(t); xkd(t); xk

a(t); uk(t); dk; y; t)dt

�gtf

t0

nkT

a (t)ha(xkd(t); xk


�jkT

d;fhd( ˆxk



�jkT

a;fha(xkd(tf ); xk


�jkT

d;0hd( ˆxk

d(t0); xkd(t0); xk

a(t0); uk(t0); dk; y; t0)


�jkT

a;0ha(xkd(t0); xk

a(t0); uk(t0); dk; y; t0)

�kkT

h0( ˆxk

d(t0); xkd(t0); xk

a(t0); uk(t0); dk; y; t0)

�mkT

p hp( ˆxk

d(ti); xkd(ti); xk

a(ti); uk(ti); dk; y; ti)

�lkT

p gp( ˆxk

d(ti); xkd(ti); xk


�mkT

q hq(dk; y)�lkT

q gq(dk; y); (6)

where k is the set of multipliers associated with the

initial conditions, and jd, 0 and ja, 0 are the sets of

multipliers associated with the initial values of the

DAEs. These are all evaluated in a similar manner tojd, f and ja, f.

The relaxed master problem is then:

miny;h

h;

s:t: Lk(uk; dk; y)0h; k � Kfeas; (7)

where Kfeas is the set of all feasible primal problems

solved up to the iteration under consideration. If the

variables y participate linearly in Eq. (1), then Eq. (7) is

an MILP that can be solved via conventional branch

and bound methods. Note that integer cuts of the form

(Floudas, 1995):Xi �Bk

yi�X

i �NBk

yi0 jBkj�1;

Bk�fi:yki �1g;

NBk�fi:yki �0g; (8)

where ½Bk ½ is the cardinality of Bk, are typically

introduced in Eq. (7) in order to exclude previous

integer solutions. Pure binary constraints on the original

system are also included in Eq. (7).

The MIDO algorithm terminates when the difference

between the least upper bound from the primal pro-

blems and the lower bound from the master problem isless than a specified tolerance, or if there is an infeasible

master problem, in which case all feasible integer

combinations have been considered. The solution to

the MIDO problem then corresponds to the solution of

the primal problem with the least upper bound.

4. An alternative approach

In this section an alternative MIDO approach is

proposed, still based on v2-GBD principles, but which

does not require the computation of the dual variablesvia the complex integration in Eq. (4), for example, and

which leads to the formulation of an equivalent but

much more compact master problem than Eqs. (6) and

(7). In order to achieve this, a modification of problem

Eq. (1) is considered where the original binary variables,

y, are relaxed, but are still forced to take integral values

by the addition of new, binary variables, y; andconstraints hy�y� y�0:/

4.1. Primal problem

At the k th iteration of the GBD algorithm, thefollowing dynamic optimization problem is solved:

minu(t);d;y

J(xd(tf ); xd(tf ); xa(tf ); u(tf ); d; y; tf );

hd(xd(t); xd(t); xa(t); u(t); d; y; t)�0; � t � [t0; tf ];


h0(xd(t0); xd(t0);xa(t0); u(t0); d; y; t0)�0;

hp(xd(ti); xd(ti); xa(ti); u(ti); d; y; ti)�0; ti � [t0; tf ];

i�1; . . . ;N;

gp(xd(ti); xd(ti); xa(ti); u(ti); d; y; ti)00; ti � [t0; tf ];

i�1; . . . ;N;

hq(d; y)�0;

gq(d; y)00;

hy�y� yk�0; (9)

where y is now a set of continuous, time-invariant searchvariables and y is the set of complicating (binary)

variables. If the same values are chosen for the

complicating variables in problems Eqs. (3) and (9),

then these two problems are clearly identical. Thus, their

solutions are both denoted by uk(t) and dk; with

resulting state variables xkd(t) and algebraic variables

xka(t): The vector y at the optimum solution to Eq. (9) is

given by yk .

4.2. Construction of the master problem

As before, the GBD relaxed master problem isformulated using dual information from the solution

of the primal problem. It is obvious that the dual

variables associated with hd, ha, h0, hp, gp, hq and gq are

identical for the ‘old’ and ‘new’ primal formulations

Eqs. (3) and (9). For a feasible primal problem Eq. (9),

the Lagrange function, Lk; is given by:


k


a(tf ); uk(tf ); dk; yk; tf )

�gtf

t0

nkT

d (t)hd( ˆxk

d(t); xkd(t); xk

a(t); uk(t); dk; yk; t)dt


�gtf

t0

nkT

a (t)ha(xkd(t); xk

a(t); uk(t); dk; yk; t)dt

�jkT

d;f hd( ˆxk

d(tf ); xkd(tf ); x

ka(tf ); uk(tf ); d

k; yk; tf )

�jkT

a;f ha(xkd(tf ); xk


�jkT

d;0hd( ˆxk

d(t0); xkd(t0); xk

a(t0); uk(t0); dk; yk; t0)

�jkT

a;0ha(xkd(t0); xk

a(t0); uk(t0); dk; yk; t0)

�kkT

h0(xkd(t0); xk

d(t0); xka(t0); uk(t0); d

k; yk; t0)

�mkT

p hp( ˆxk

d(ti); xkd(ti); xk

a(ti); uk(ti); d

k; yk; ti)

�lkT

p gp( ˆxk

d(ti); xkd(ti); xk

a(ti); uk(ti); dk; yk; ti)

�mkT

q hq(dk; yk)�lkT

q gq(dk; yk)

�vkT

(yk� y); (10)

where v is the set of dual multipliers for the equations

hy �/0. Since hd, ha, h0, hp and hq all equal 0 at the primal

solution and they are not functions of the complicating

variables y; their associated terms can be removed from

the Lagrangian in Eq. (10). Similarly, the complemen-

tarity optimality conditions for the NLP in the primal

problem ensure that lTp gp�lT

q gq�0; and since gp andgq are not functions of y either, their associated terms

can also be removed. Hence, the Lagrange function Lk

can be simplified to:


k



�vkT

(yk� y); (11)

and the master problem posed as:

miny;h

h;

s:t: Lk(uk; dk; y)0h; k � Kfeas: (12)

It is clear from the Lagrange function definition in

Eq. (11) that the master problem Eq. (12) does not

require any dual information with respect to the DAE

system and so no intermediate adjoint problem is

required for its construction. It should also be noted

that the master problem Eq. (12) is exactly equivalent to

the master problem Eq. (7). A proof of this can be found

in Appendix A.

5. New MIDO Algorithm I

Based on the theoretical developments presented so

far, a new, generalised Benders decomposition-based

algorithm for the solution of the general MIDO problem

Eq. (1) can be summarised as follows:

5.1. Step 1

Set the termination tolerance, o . Initialise the iteration

counter, k�/1; lower bound, LB�/�/�; and upperbound, UB�/�.

5.2. Step 2

For fixed values of the binary variables, y�/yk , solve

the k th primal problem Eq. (3) to obtain a solution Jk .

(For computational speed, it may be useful to omit the

search variables and constraints that are superfluous

due to the current choice of values for the binary

variables.) Set UB�/min(UB, Jk) and store the valuesof the continuous and binary variables corresponding to

the best primal solution so far.

5.3. Step 3

Solve the primal problem Eq. (9) at the solution found

in Step 2 with the full set of search variables and

constraints included. Convergence will be achieved in

one iteration. Obtain the multipliers vk needed to

construct the k th master problem Eq. (12)

5.4. Step 4

Solve the k th relaxed master problem Eq. (12) with

integer cuts Eq. (8) and the pure binary constraints of

the original system, to obtain a solution hk . Update the

lower bound, LB�/hk .

5.5. Step 5

If UB�/LB0/o , or the master problem is infeasible,

then stop. The optimal solution corresponds to UB andthe values stored in Step 2. Otherwise, set k�/k�/1 and

yk�1 equal to the integer solution of the k th master

problem, and return to Step 2.

Note that if in Step 2 a particular set of values yk

renders the primal problem Eq. (3) infeasible, then an

infeasibility minimisation problem is solved instead.

This can involve, for example, minimising the l1 or l�sum of constraint violations (Floudas, 1995). In Step 3,the infeasibility minimisation problem is then re-solved

at its optimal solution to obtain multipliers vk . The

formulation corresponding to minimising the l� sum of

violations is shown below:

mina;u(t);d;y

a;

s:t: hd(xd(t); xd(t); xa(t); u(t); d; y; t)�0; � t � [t0; tf ];



h0(xd(t0); xd(t0); xa(t0); u(t0); d; y; t0)�0;

hp(xd(ti); xd(ti); xa(ti); x(ti); d; y; ti)�0; ti � [t0; tf ];

i�1; . . . ;N;

gp(xd(ti); xd(ti); xa(ti); u(ti); d; y; ti)0ae; ti � [t0; tf ];

i�1; . . . ;N;

hq(d; y)�0; gq(d; y)00;

hy�y� yk�0;

aE0; (13)

where a is a positive scalar quantity and e is a vector

whose elements are all equal to one. In Step 4, the

master problem Eq. (12) is then augmented as:

miny;h

h;

s:t: J( ˆxk


a(tf ); uk(tf ); dk; yk; tf )�vkT

(yk� y)

0h;

k � Kfeas;

vkT

(yk� y)00; k � Kinfeas; (14)

where Kfeas denotes the set of iteration numbers for

which feasible primal problems are found and Kinfeas the

set for which infeasible primal problems are found.

6. Remarks on MIDO Algorithm I

(1) As already stated, one advantage of the algorithm

outlined above is that, even when the binary variables yappear explicitly (and non-linearly) in the DAE system,

the master problem Eq. (12) does not require any dual

information with respect to the DAE system and so no

intermediate adjoint problem is required for its con-

struction. This comes at the expense of adding more

equations, hy�/0, and search variables, y: However,

these are only used in Step 3 when the primal problem is

re-solved at its optimal solution, and convergence isachieved in just one iteration. The use of this novel

strategy also circumvents any numerical difficulties

(such as those suggested by Schweiger and Floudas,

1997) associated with reformulating the primal problem.

(2) The master problem in Step 4 of the new MIDO

algorithm has a much simpler form compared to

algorithms where adjoint variables are required. In the

latter case, the master problem defined by Eqs. (6) and(7) has the form:

miny;h

h;

s:t: J( ˆxk



�gtf

t0

nkT

d (t)hd( ˆxk

d(t); xkd(t); xk


�gtf

t0

nkT

a (t)ha(xkd(t); xk


�jkT

d;fhd( ˆxk



�jkT

a;fha(xkd(tf ); xk


�jkT

d;0hd( ˆxk

d(t0); xkd(t0); xk

a(t0); uk(t0); dk; y; t0)

�jkT

a;0ha(xkd(t0); xk

a(t0); uk(t0); dk; y; t0)

�kkT

h0( ˆxk

d(t0); xkd(t0); xk

a(t0); uk(t0); dk; y; t0)

�mkT

p hp( ˆxk

d(ti); xkd(ti); xk


�lkT

p gp( ˆxk

d(ti); xkd(ti); xk


�mkT

q hq(dk; y)�lkT

q gq(dk; y)

0h: (15)

Construction of this master problem thus requiresthat the objective function and all the DAEs, equality

and inequality constraints that involve the binary

variables y be considered together with their adjoint

and Lagrange multipliers. Their number may be con-

siderably larger than the number of binary variables y,

especially for complex models. In contrast, the master

problem defined by Eqs. (11) and (12) for the new

MIDO algorithm is:

miny;h

h;

s:t: J( ˆxk


d(tf ); uk(tf ); dk; yk; tf )�vkT

(yk� y)

0h;

(16)

which only requires the value of the primal objective

function together with extra terms whose complexity is

independent of the complexity of the process model. The

practical benefit of this will be seen in Example 2,

presented later in this article.

(3) Another important benefit of the new MIDO

algorithm is that it is independent of the type of method

used for solving the dynamic optimization primalproblem. The upshot of this is that with the formulation

presented, any existing dynamic optimization solver can

be used to solve MIDO problems since even the most

elementary NLP codes give the simple Lagrange multi-


plier information required for the master problem.

Note, however, that the advantages gained from the

new algorithm do depend on the method used for the

primal solution. The maximum benefits are obtained ifcontrol vector parameterisation is used since no evalua-

tion of the adjoint variables is then necessary. If a

multiple shooting approach is used, then the adjoints are

already computed at the primal solution; however, the

master problem formulated with Algorithm I is still

much simpler and more attractive from a computational

viewpoint than the master problem that would result

otherwise.(4) Since the new MIDO algorithm is based on v2-

GBD, like the MIDO approaches of Mohideen et al.

(1997) and Schweiger and Floudas (1997), it shares its

limitations. In particular, the algorithm is only guaran-

teed to converge to the global optimum under convexity

conditions for the primal problem with the binary

variables appearing linearly and separably (Floudas,

1995). In the much more common, non-convex case,there is a possibility that the GBD relaxed master

problems Eqs. (7) and (12) will exclude potentially

feasible choices of y from the solution set.

Note that for (MI)NLP problems, it has been shown

that GBD can lead to solutions which are not even local

optima if convexity criteria are not satisfied and a set of

continuous variables is chosen as the complicating

(fixed) variables for the primal problem (Floudas &Visweswaran, 1990; Sahinidis & Grossmann, 1991;

Bagajewicz & Manousiouthakis, 1991). Perhaps in

response to this, Barton, Allgor, Feehery and Galan

(1998) and Allgor and Barton (1999) incorrectly sug-

gested that the use of GBD approaches for MIDO ‘has

the property of convergence to an arbitrary point which is

not even guaranteed to be a local minimum .’ In fact, the

new MIDO algorithm (and the GBD algorithms ofMohideen et al., 1997 and Schweiger & Floudas, 1997)

will always converge to at least a local optimum, where

(Ross, Perkins, & Pistikopoulos, 1998):

Definition. A local optimum of an MINLP is one

which, for fixed binaries y, satisfies local optimality

conditions for the resulting (primal) problem.

The above definition is intuitively obvious since wheny is discrete there can be no feasible neighbourhood

around it which improves the objective. Therefore, the

(local optimal) solution of any of the feasible primal

problems will result in a local optimum of the MIDO

problem.

7. Extending Algorithm I to handle time-dependent

binary variables

Ross et al. (1998) showed that the GBD-based MIDO

algorithm of Mohideen et al. (1997) can be readily

extended to tackle MIDO problems which involve time-

dependent binary variables. Such problems can arise in

areas such as batch scheduling and adaptive control,

where in the latter, the control structure changes withtime. The new MIDO algorithm presented in this article

can also tackle problems involving time-dependent

binary variables, which will be denoted by uy(t). The

general formulation of such a problem is:

minu(t);d;y;uy(t)

J(xd(tf ); xd(tf ); xa(tf ); u(tf ); d; y; uy(tf ); tf );

s:t: hd(xd(t); xd(t); xa(t); u(t); d; y; uy(t); t)�0;

� t � [t0; tf ];

ha(xd(t); xa(t); u(t); d; y; uy(t); t)�0; � t � [t0; tf ];

h0(xd(t0); xd(t0);xa(t0); u(t0); d; y; uy(t0); t0)�0;

hp(xd(ti); xd(ti); xa(ti); u(ti); d; y; uy(ti); ti)�0; ti � [t0; tf ];

i�1; . . . ;N;

gp(xd(ti); xd(ti); xa(ti); u(ti); d; y; uy(ti); ti)00; ti � [t0; tf ];

i�1; . . . ;N;

hq(d; y)�0;

gq(d; y)00: (17)

uy(t) may be parameterised by means of a piecewise-

constant profile defined on Nuy sub-intervals over the

time horizon [t0, tf]. Hence:

uy(t)�uy(j); j�1; . . . ;Nuy: (18)

The values of uy(t) during the j th sub-interval are

given by a set of time-invariant binaries uy(j). With thesebinaries fixed, Eq. (17) becomes a standard dynamic

optimization problem. A similar approach to that

described earlier can thus be employed. The primal

problem in Step 3 of Algorithm I becomes:

minu(t);d;y;[uy(j) j�1;...;Nuy ]

J(xd(tf ); xd(tf ); xa(tf ); u(tf ); d; y; uy(tf ); tf );

s:t: hd(xd(t); xd(t); xa(t); u(t); d; y; uy(t); t)�0; � t � [t0; tf ];

ha(xd(t); xa(t); u(t); d; y; uy(t); t)�0; � t � [t0; tf ];

h0(xd(t0); xd(t0); xa(t0); u(t0); d; y; uy(t0); t0)�0;

hp(xd(ti); xd(ti); xa(ti); u(ti); d; y; uy(ti); ti)�0; ti � [t0; tf ];

i�1; . . . ;N;

gp(xd(ti); xd(ti); xa(ti); u(ti); d; y; uy(ti); ti)00; ti � [t0; tf ];

i�1; . . . ;N;

hq(d; y)�0;

gq(d; y)00;


hy�y�yk�0;

huy(j)�uy(j)�uk

y(j)�0; j�1; . . . ;Nuy: (19)

In Eq. (19) uy(j ), j�/1,. . ., Nuy , and y are continuous

search variables, while uy(j); j�1; . . . ;Nuy; and y are

the complicating, binary variables. The constraints

huy(j)�0; j�1; . . . ;Nuy; are interior-point constraints

that are handled in a similar fashion to the pointconstraints hp�/0. The multipliers associated with

huy(j) are denoted by vuy

(j). The values of the variables

at the optimum are denoted as before with the addition

of uky(j): The master problem is formulated in a similar

manner to Eq. (14);

miny;[uy ; j�1;...;Nuy];h

h;

Jk�vkT

(yk� y)�XNuy

j�1

[vkuy

(j)]T[uky(j)� uy(j)]0h;

k � Kfeas;

vkT

(yk� y)�XNuy

j�1

[vkuy

(j)]T[uky(j)� uy(j)]00; k � Kinfeas:

(20)

8. Using outer-approximation to construct the master

problem

As discussed in the review section earlier in this

article, current MIDO algorithms that use the principles

of OA/ER to construct the master problem are limited

in that they are not applicable to models where the

binary variables appear explicitly in the DAE system.This can be overcome using the reformulation strategy

that was applied in Algorithm I.

In an OA/ER-based MIDO algorithm, the primal

problems are solved in an identical manner to those in

Step 2 and Step 3 of the GBD-based MIDO Algorithm

I. However, in order to construct the master problems,

instead of obtaining Lagrange multiplier values in Step

3, gradient information is required of the objectivefunction and constraints with respect to the optimiza-

tion search variables. During solution of the primal

problem, the independent search variables are d, y and

the time-invariant variables resulting from the parame-

terisation of the controls, which are denoted by uj. The

k th master problem (for the case of feasible primal

problems) is thus:

miny;d;y;[uj ;�j ];h

h;

s:t: hEJk��@J

@d

�jk

(d� dk)�

�@J

@y

�jk

(y�yk)

�X�j

�@J

@uj

�jk

(uj� ukj );

0ETkp

�hk

p��@hp

@d

�jk

(d� dk)�

�@hp

@y

�jk

(y�yk)

�X�j

�@hp

@uj

�jk

(uj� ukj )

�

0Egkp�

�@gp

@d

�jk

(d� dk)�

�@gp

@y

�jk

(y�yk)

�X�j

�@gp

@uj

�jk

(uj� ukj );

0ETkp

�hk

q��@hq

@d

�jk

(d� dk)�

�@hp

@y

�jk

(y�yk)

�;

0Egkq�

�@gq

@d

�jk

(d� dk)�

�@hp

@y

�jk

(y�yk);

0�y� y;

k � Kfeas; (21)

together with the integer cuts Eq. (8) and pure binary

constraints of the original system. In Eq. (21), T�k , ��/

p, q, are diagonal matrices with elements defined as:

Tk�;ii�f�1 if lk

�;iB0;

�1 if lk�;i�0;

0 if lk�;i�0;

where lk�,i , ��/p, q are the Lagrange multipliers

pertaining to the equality constraints hp�/0 and hq�/

0. If control vector parameterisation is used to solve the

primal problem, then these multipliers and the gradients

in Eq. (21) are available directly from the solution of the

resulting NLP problem. If another type of dynamicoptimization technique is used, e.g. multiple shooting,

then the multipliers will still be directly available;

however, the gradients have to be computed by per-

forming an additional integration of the sensitivity or

adjoint-based DAE system at the optimal solution of the

primal problem (Rosen & Luus, 1991).

Note that Eq. (21) bears some resemblance to the

master problem proposed by Fletcher and Leyffer(1994). However, that work considers MINLP pro-

blems, not MIDO ones, where there are no equality

constraints and where linearisations of inactive con-

straints are not included in the master problem.


9. New MIDO Algorithm II

A new algorithm, based on OA/ER, for the solution

of the general MIDO problem Eq. (1) can now besummarised as follows:

9.1. Step 1

Set the termination tolerance, o . Initialise the iteration

counter, k�/1; lower bound, LB�/�/�; and upper

bound, UB�/�.

9.2. Step 2

For fixed values of the binary variables, y�/yk , solvethe k th primal problem Eq. (3) to obtain a solution Jk.

(For computational speed, it may be useful to omit the

search variables and constraints that are superfluous

due to the current choice of values for the binary

variables.) Set UB�/min(UB, Jk ) and store the values

of the continuous and binary variables corresponding to

the best primal solution so far.

9.3. Step 3

Solve the primal problem Eq. (9) at the solution found

in Step 2 with the full set of search variables and

constraints included. Convergence will be achieved in

one iteration. Obtain the gradients and Lagrange multi-

pliers needed to construct the k th master problem Eq.

(21).

9.4. Step 4

Solve the k th relaxed master problem Eq. (21), with

integer cuts Eq. (8) and the pure binary constraints of

the original system, to obtain a solution hk . Update the

lower bound, LB�/hk .

9.5. Step 5

If UB�/LB0/o , or the master problem is infeasible,

then stop. The optimal solution corresponds to the

values stored in Step 2. Otherwise, set k�/k�/1 and yk�1

equal to the integer solution of the k th master problem,

and return to Step 2.

As with the GBD-based MIDO Algorithm I, if in Step

2 a particular set of values yk renders the primal

problem infeasible, then an infeasibility minimisation

problem can be solved, which in Step 3 may correspond

to Eq. (13) (Viswanathan & Grossmann, 1990). The

matrices Tp and Tq and gradients @ hp/@ d, @ gp/@ d, @ hq/@ d, @ gq/@ d etc. are found from the solution of this

problem and then used to add extra linearisation

constraints to the master problem Eq. (21).

10. Remarks on MIDO Algorithm II

(1) As already stated, unlike current MIDO ap-

proaches that use OA principles to construct the masterproblems, the algorithm outlined above is applicable to

models where the binary variables appear explicitly in

the DAE system. Like Algorithm I, Algorithm II has the

advantage that it is independent of the type of method

that is used for solving the dynamic optimization primal

problem (although control vector parameterisation may

be preferred versus multiple shooting, for example, since

then no intermediate problem needs to be solved toobtain the gradients that are necessary for constructing

the master problem). Thus, again as with Algorithm I,

Algorithm II allows existing dynamic optimizers to be

used for solving MIDO problems. Algorithm II can also

be readily extended to tackle problems involving time-

dependent binary variables, in a similar fashion to that

described in Section 7 for Algorithm I.

(2) It should be noted that unlike the GBD-basedMIDO Algorithm I, where the master problem Eq. (16)

involves just one additional constraint per iteration, the

master problem Eq. (21) in Algorithm II involves the

addition of a large number of constraints at each

iteration. These constraints correspond to linearisations

about the optimal solution of the primal problem of the

objective function, equality and inequality constraints.

The MILP master problems in Algorithm II are there-fore computationally more expensive than those in

Algorithm I. (As noted in Section 2 of this paper,

however, for both GBD- and OA-based MIDO algo-

rithms, the master problems are usually much less

computationally demanding to solve than the primal,

dynamic optimization problems.) Because the master

problem in Algorithm II is more constrained than that

in Algorithm I, the lower bound will, in general, betighter, and so Algorithm II will often converge to a

locally optimal solution of the MIDO problem in fewer

iterations than Algorithm I. Note, however, that since

most MIDO problems involve non-convexities, the

additional constraints in the master problem of Algo-

rithm II may cause it to cut off larger parts of the

feasible region than Algorithm I. Thus, although Algo-

rithm II may converge to a local optimum faster thanAlgorithm I, there is a chance that the solution will be

worse.

11. Example 1

The application of the new algorithms for solving

MIDO problems is now demonstrated with an example

based on a binary distillation system from the MINOPTUser’s Guide (Schweiger, Rojnuckarin & Floudas,

1997). The column has a fixed number of trays and

the objective is to determine the optimal feed location


(discrete decision), vapour boil-up, V , and reflux flow

rate, R (continuous decisions), in order to minimise the

integral square error (ISE) between the bottoms and

distillate compositions and their respective set-points.The superstructure of the system is shown in Fig. 1. The

following modelling assumptions were used by Schwei-

ger et al.: (i) constant molar overflow; (ii) constant

relative volatility, a ; (iii) phase equilibrium; (iv) constant

liquid hold-ups, equal to m for each tray and 10m for

the reboiler and condenser; (v) no tray hydraulics; (vi)

negligible vapour hold-ups; and (vii) no pressure drops.

The system is initially at steady-state; at t�/0 there is astep change in the feed composition, zf; and the inequal-

ity constraints are that the distillate composition must

be greater than 0.98, and the bottoms composition must

be less than 0.02, at the end of the time horizon of 400

min. The problem can be stated mathematically as:

(1) Objective:

minfyf1;...;yfN ;V ;Rg

ISE(tf );

d(ISE)

dt� (xb�x+

b)2�(xN�1�x+N�1)2;

s.t.

(2) Component balances:

10mdxb

dt�L1x1�Vy0�Bxb;

mdxi

dt�Li�1xi�1�Lixi�V (yi�1�yi)�F �yfi �zf ;

i�1; . . . ;N�1;

mdxN

dt��LNxN �V (yN�1�yN)�F �yfN �zf �RxN�1;

10mdxN�1

dt�V (yN �xN�1);

(3) Overall balances:

0�L1�V�B;

0�Li�1�Li�Fyfi; i�1; . . . ;N�1;

0��LN �V�FyfN �R;

0�V�D�R;

(4) Vapour�/liquid equilibrium:

y0�axb

1 � (a� 1)xb

;

yi �axi

1 � (a� 1)xi

; i�1; . . . ;N;

(5) Initial conditions:

ISE(t�0)�0;

dxb

dt jt�0

�0;

dxi

dt jt�0

�0; i�1; ;N�1;

(6) Step disturbance (smoothly approximated):

zf �0:54�0:09e�10t;

(7) End-point inequality constraints:

g1�0:98�xN�1(tf )00;

g2�xb(tf )�0:0200;

(8) Time-horizon:

t � [0; 400];

(9) Pure binary constraints:

XN

i�1

yfi�1;

yfi�0; i�1; . . . ; 9;

yfi � f0; 1g; i�10; . . . ;N;

where yfi is a binary variable that takes a value of 1 if

tray i receives the feed, and is 0 otherwise. Only one tray

is allowed to receive the feed, which is constrained to

enter on tray 10 or above.

The MIDO Algorithms I and II were both imple-

mented using gPROMS/gOPT v1.7 to solve the dynamicoptimization primal problems, and GAMS/CPLEX

v2.50 (Brooke, Kendrick & Meeraus, 1992) for the

MILP master problems. All the primal solutions,Fig. 1. Superstructure for Example 1.


Lagrange multipliers and master solutions using Algo-

rithm I are summarised in Table 1, while the primal

solutions, gradients and master solutions using Algo-

rithm II are summarised in Table 2. Using a terminationtolerance of o�/0, both algorithms converged after four

iterations, giving the same optimal solution reported by

Schweiger et al. (1997). This corresponds to a feed

location at tray 25, vapour boil-up of 1.5426 kmol/min,

reflux flow rate of 1.0024 kmol/min, and ISE of 0.1817.

Note that integer cuts were added in each master

problem to prevent the re-occurrence of previous solu-

tions. For example, for the first master problem, the cut(yf20�ai"20yfi00); was added.

It is interesting to observe that, for this example, the

master problems of Algorithm II do not predict tighter

lower bounds than the master problems of Algorithm I

except on the fourth iteration when convergence of the

overall MIDO problem has already been achieved. This

behaviour can be understood in terms of the fact that,

qualitatively, the difference between a GBD-basedmaster problem and an OA-based one is that the former

only considers the inequality constraints that are active

at the primal solution, whereas the latter considers all

the inequality constraints in the model (Duran, 1984).

Thus, if the inequality constraints that are inactive at the

primal solution remain inactive at the solution of the

OA master problem, then GBD and OA will give

identical behaviour. This is demonstrated for thisexample in Appendix B.

12. Example 2

This example, adapted from Schweiger and Floudas

(1997), builds on Example 1, where the distillation

system is now subject to sinusoidal disturbances in the

feed flow-rate (9/20% variation, 3-h period) and com-position (9/20% variation, 1-h period). The objective is

to design the distillation column and its required control

system at minimum total annualised cost (comprising

capital cost of the column and operating cost of

utilities), capable of feasible operation over the whole

of a given time horizon (1440 min). Feasibility is defined

through the satisfaction of inequality path constraints

related to the product quality specifications and theminimum allowable column diameter required to avoid

flooding. It is assumed that proportional-integral (PI)

controllers are to be used, where the vapour boil-up

rate, V , controls the bottoms composition, xb, and the

reflux rate, R , controls the distillate composition, xN�1.

Solution of the problem thus requires the determination

of: (i) the optimal process design, in terms of the number

of trays (discrete decision), feed tray location, which isconstrained to lie at least five trays below the reflux tray

(discrete decision), and column diameter, Dcol (contin-

uous decision); and (ii) the optimal control design, in

terms of the gains (Kv and Kr), reset times (tv and tr)

and set-points (xb� and xN�1� ) of the two PI controllers.

The superstructure of this system, together with

relevant parameter values and definitions of variables,

is shown in Fig. 2. Using similar modelling assumptions

to those in Example 1, the MIDO problem can be stated

mathematically as:

(1) Objective:

minfyf ;yr;Dcol;Kv ;Kr;tvtr ;x

+b;x+

N�1g

Cost

�7756 �Vmean�1891:125�996:3 �D2col

Table 1

Progress of iterations of MIDO algorithm I for example 1

Iteration number 1 2 3 4

Primal solutions

Feed tray 20 26 24 25

V (kmol/min) 1.2231 1.7830 1.4111 1.5426

R (kmol/min) 0.6836 1.2426 0.8711 1.0024

ISE 0.1992 0.1827 0.1834 0.1817

UB 0.1992 0.1827 0.1827 0.1817

Lagrange multipliers

v1 0.4353 0.6147 0.6045 0.6204

v2 0.3319 0.3965 0.4077 0.4087

v3 0.2741 0.2864 0.3027 0.2984

v4 0.2414 0.2307 0.2463 0.2406

v5 0.2226 0.2024 0.2159 0.2101

v6 0.2114 0.1881 0.1992 0.1939

v7 0.2045 0.1808 0.1899 0.1853

v8 0.2001 0.1771 0.1847 0.1806

v9 0.1970 0.1753 0.1815 0.1780

v10 0.1948 0.1745 0.1796 0.1765

v11 0.1931 0.1742 0.1784 0.1757

v12 0.1916 0.1742 0.1775 0.1752

v13 0.1904 0.1744 0.1768 0.1749

v14 0.1893 0.1746 0.1763 0.1747

v15 0.1884 0.1749 0.1759 0.1746

v16 0.1878 0.1752 0.1755 0.1745

v17 0.1876 0.1755 0.1752 0.1745

v18 0.1879 0.1759 0.1749 0.1745

v19 0.1890 0.1763 0.1747 0.1745

v20 0.1909 0.1766 0.1746 0.1746

v21 0.1994 0.1770 0.1748 0.1747

v22 0.2085 0.1773 0.1753 0.1750

v23 0.2172 0.1777 0.1762 0.1754

v24 0.2246 0.1779 0.1776 0.1759

v25 0.2301 0.1777 0.1818 0.1763

v26 0.2325 0.1765 0.1846 0.1763

v27 0.2298 0.1698 0.1838 0.1721

v28 0.2189 0.1537 0.1763 0.1600

v29 0.1954 0.1204 0.1574 0.1339

v30 0.1535 0.0566 0.1209 0.0850

Master solutions

Feed tray 26 24 25 23

LB 0.1576 0.1813 0.1815 0.1848

UB�/LB0/0? No No No Yes: STOP


Table 2

Progress of iterations of MIDO algorithm II for example 1


Primal solutions

Feed tray 20 26 24 25

V (kmol/min) 1.2231 1.7830 1.4111 1.5426

R (kmol/min) 0.6836 1.2426 0.8711 1.0024

ISE 0.1992 0.1827 0.1834 0.1817

UB 0.1992 0.1827 0.1827 0.1817

Objective gradients

w.r.t. V , R 7.4445, �/6.7611 12.108, �/11.926 10.052, �/9.7077 10.962, �/10.701

yf1, yf2 �/4.9048, �/4.8013 �/7.3116, �/7.0977 �/6.2937, �/6.0972 �/6.7519, �/6.5413

yf3, yf4 �/4.7434, �/4.7107 �/6.9898, �/6.9352 �/5.9924, �/5.9361 �/6.4315, �/6.3740

yf5, yf6 �/4.6918, �/4.6806 �/6.9075, �/6.8934 �/5.9057, �/5.8891 �/6.3438, �/6.3277

yf7, yf8 �/4.6737, �/4.6692 �/6.8863, �/6.8827 �/5.8798, �/5.8746 �/6.3191, �/6.3144

yf9, yf10 �/4.6662, �/4.6639 �/6.8809, �/6.8802 �/5.8715, �/5.8695 �/6.3118, �/6.3103

yf11, yf12 �/4.6622, �/4.6608 �/6.8799, �/6.8798 �/5.8683, �/5.8674 �/6.3095, �/6.3090

yf13, yf14 �/4.6596, �/4.6587 �/6.8800, �/6.8802 �/5.8668, �/5.8663 �/6.3087, �/6.3085

yf15, yf16 �/4.6581, �/4.6580 �/6.8805, �/6.8808 �/5.8659, �/5.8656 �/6.3084, �/6.3084

yf17, yf18 �/4.6588, �/4.6611 �/6.8812, �/6.8817 �/5.8655, �/5.8655 �/6.3085, �/6.3087

yf19, yf20 �/4.6657, �/4.6743 �/6.8823, �/6.8831 �/5.8660, �/5.8673 �/6.3092, �/6.3101

yf21, yf22 �/4.7129, �/4.7640 �/6.8845, �/6.8869 �/5.8700, �/5.8754 �/6.3119, �/6.3153

yf23, yf24 �/4.8258, �/4.8989 �/6.8913, �/6.8998 �/5.8856, �/5.9049 �/6.3220, �/6.3349

yf25, yf26 �/4.9855, �/5.0899 �/6.9163, �/6.9487 �/5.9830, �/6.1007 �/6.3594, �/6.4534

yf27, yf28 �/5.2187, �/5.3816 �/7.0640, �/7.2642 �/6.2709, �/6.5180 �/6.6042, �/6.8391

yf29, yf30 �/5.5935, �/5.8772 �/7.6064, �/8.1968 �/6.8835, �/7.4364 �/7.2083, �/7.7990

Inequalities values

g1, g2 �/3.9013e�/3, 0 �/2.2065e�/3, 0 �/2.9734e�/3, 0 �/2.6192e�/3, 0

Gradients of g1

w.r.t. V , R 0.1819, �/0.3292 0.0423, �/0.0684 0.0821, �/0.1403 0.0615, �/0.1024

yf1, yf2 (�/1e6) �/385.63, �/415.24 �/1.0724, �/1.1254 �/11.278, �/12.010 �/3.7995, �/4.0215

yf3, yf4 (�/1e6) �/433.12, �/444.65 �/1.1541, �/1.1703 �/12.433, �/12.692 �/4.1467, �/4.2212

yf5, yf6 (�/1e6) �/452.71, �/458.81 �/1.1796, �/1.1838 �/12.862, �/12.976 �/4.2676, �/4.2958

yf7, yf8 (�/1e6) �/463.70, �/467.65 �/1.1825, �/1.1722 �/13.045, �/13.058 �/4.3067, �/4.2933

yf9, yf10 (�/1e6) �/470.56, �/471.93 �/1.1448, �/1.0837 �/12.981, �/12.740 �/4.2375, �/4.1012

yf11, yf12 (�/1e7) �/4706.6, �/4645.9 �/9.5487, �/6.9007 �/121.89, �/110.47 �/38.088, �/32.120

yf13, yf14 (�/1e7) �/4497.0, �/4185.2 �/1.5164, 9.3740 �/87.721, �/43.272 �/20.205, 3.3290

yf15, yf16 (�/1e6) �/357.15, �/239.88 3.1344, 7.5611 4.2762, 20.850 4.9574, 14.021

yf17, yf18 (�/1e5) �/1.9069, 39.354 1.6475, 3.4417 5.2700, 11.383 3.1759, 6.6453

yf19, yf20 (�/1e5) 116.14, 258.75 7.0530, 14.321 23.107, 45.587 13.429, 26.689

yf21, yf22 (�/1e4) 91.024, 181.41 2.8946, 5.8379 8.8680, 17.128 5.2608, 10.327

yf23, yf24 (�/1e3) 29.602, 43.746 1.1761, 2.3680 3.2961, 6.3307 2.0229, 3.9582

yf25, yf26 (�/1e3) 61.235, 83.232 4.7665, 9.5933 18.824, 38.274 7.7409, 22.509

yf27, yf28 (�/1e2) 11.157, 1.4905 2.7123, 5.8186 6.722, 11.031 4.6826, 8.5604

yf29, yf30 0.1998, 0.2700 0.1121, 0.2061 0.1754, 0.2751 0.1477, 0.2481

Gradients of g2

w.r.t. V , R �/1.8800, 1.7074 �/2.0744, 2.0432 �/1.9993, 1.9309 �/2.0286, 1.9803

yf1, yf2 1.1287, 1.1287 1.1473, 1.1481 1.1316, 1.1317 1.1347, 1.1349

yf3, yf4 1.1287, 1.1287 1.1484, 1.1486 1.1317, 1.1317 1.1350, 1.1351

yf5, yf6 1.1287, 1.1286 1.1487, 1.1488 1.1317, 1.1317 1.1350, 1.1351

yf7, yf8 1.1286, 1.1286 1.1488, 1.1488 1.1317, 1.1317 1.1350, 1.1351

yf9, yf10 1.1286, 1.1286 1.1488, 1.1488 1.1317, 1.1317 1.1350, 1.1351

yf11, yf12 1.1286, 1.1286 1.1488, 1.1488 1.1317, 1.1317 1.1351, 1.1351

yf13, yf14 1.1287, 1.1287 1.1488, 1.1488 1.1317, 1.1317 1.1351, 1.1351

yf15, yf16 1.1288, 1.1289 1.1488, 1.1488 1.1317, 1.1318 1.1351, 1.1351

yf17, yf18 1.1292, 1.1296 1.1488, 1.1489 1.1318, 1.1319 1.1351, 1.1352

yf19, yf20 1.1305, 1.1322 1.1489, 1.1489 1.1320, 1.1323 1.1353, 1.1354

yf21, yf22 1.1398, 1.1504 1.1491, 1.1495 1.1328, 1.1338 1.1357, 1.1363

yf23, yf24 1.1639, 1.1804 1.1502, 1.1516 1.1356, 1.1392 1.1375, 1.1398

yf25, yf26 1.2009, 1.2267 1.1544, 1.1602 1.1539, 1.1767 1.1442, 1.1617

yf27, yf28 1.2599, 1.3038 1.1811, 1.2182 1.2107, 1.2614 1.1903, 1.2360

yf29, yf30 1.3632, 1.4455 1.2825, 1.3946 1.3378, 1.4551 1.3092, 1.4276

Master solutions


�1494:45 �Dcol

�6�0:76 �

XN

i�1

i �yri

�

�61:25 �XN

i�1

i �yri(0:7�1:5 �D2col);

dVmean

dt�

V

tf

;

s.t.

(2) Component balances:

10m �dxb

dt�L1x1�Vy0�Bxb;

�XN

i0�i

yri?

�m �

dxi

dt�Li�1xi�1�Lixi�V (yi�1�yi)

�Fizf �RixN�1; i�1; . . . ;N;

Fi �F �yfi; i�1; . . . ;N;

Ri�R �yri; i�1; . . . ;N;

LN�1�0;

10m �dxN�1

dt�V (yN �xN�1);

m�7:538115 �D2col

��0:0014134

Dcol

�23�hweir

�;

(3) Overall balances:

0�L1�V�B;

0�Li�1�Li�Fi�Ri; i�1; . . . ;N;

0�V�D�R;

(4) Vapour�/liquid equilibrium:

y0�axb

1 � (a� 1)xb

;

yi �axi

1 � (a� 1)xi

; i�1; . . . ;N;

(5) Bottoms composition control loop:

V �Vss�Kv(ev�Iv);

ev�x+b�xmeas

b ;

dIv

dt�

ev

tv;

dVss

dt�0;

(6) Distillate composition control loop:

R�Rss�Kr(er�Ir);

er�x+N�1�xmeas

N�1;

dIr

dt�

er

tr

;

dRss

dt�0;

(7) Measurement lags:

xb�1:2dxmeas

b

dt�xmeas

b ;

Table 2 (Continued )


Feed tray 26 24 25 23

V , R (kmol/min) 1.7488, 1.2072 1.5179, 0.9777 1.6375, 1.0977 1.3856, 0.8440

LB 0.1576 0.1813 0.1815 0.2093

UB�/LB0/0? No No No Yes: STOP

Fig. 2. Superstructure for Example 2.


xN�1�1:2dxmeas

N�1

dt�xmeas

N�1;

(8) Initial conditions:

Vmean(t�0)�0;

dxb

dt jt�0

�0;

dxi

dt jt�0

�0; i�1; . . . ;N�1;

ev(t�0)�0;

Iv(t�0)�0;

er(t�0)�0;

Ir(t�0)�0;

dxmeasb

dt jt�0

�0;

dxmeasN�1

dt jt�0

�0;

(9) Sinusoidal disturbances:

F �1�0:2sin

�2pt

180

�;

zf �0:45�0:09sin

�2pt

60

�;

(10) Inequality path constraints:

g1�0:98�xN�1(tf )00;

g2�xb(tf )�0:0200;

g3�Dmincol �Dcol00;

Dmincol �0:6719

ffiffiffiffiV

p;

(11) Time horizon:

t � [0; 1440];

(12) Pure integer constraints:

XN

i�1

yfi�1;

XN

i�1

yri�1;

yfi�XN

i?�i�5

yri?00; i�1; . . . ;N;

yfi; yri � f0; 1g; i�1; . . . ;N;

where the new variables that have been defined com-

pared to Example 1 are Vmean, the average boil-up rate

over the time horizon; Dcolmin, the minimum allowable

column diameter due to flooding; and yri, a binary

variable that takes a value of 1 if tray i receives the

reflux, and is 0 otherwise. Note that the model above is

similar to that used by Schweiger and Floudas (1997),

although in that study the simpler case was consideredof a single step disturbance and only end-point con-

straints instead of path constraints. It should also be

noted that the binary variable term, (aNi?�1yri?); in the

component balances for the trays, is needed in order to

ensure that no separation is effected in the ‘shadow’

trays above the tray that receives the reflux. Without

this term, the vapour flow leaving the reflux tray would

be allowed to come into contact with the liquid hold-upon the ‘shadow’ trays. Omission of the term would thus

cause the properties of the vapour outlet from the top

tray N to be different from those of the vapour outlet

from the reflux tray, rendering such a superstructure

model invalid for determining the optimal number of

trays.

After conversion of the three path constraints into

end-point inequality constraints using the formulationdescribed by Bansal (2000), the MIDO problem was

solved using Algorithm I, again implemented within

gPROMS/gOPT v1.7 and GAMS v2.50/CPLEX. De-

spite the highly combinatorial nature of the problem

(465 discrete alternatives), the algorithm converged in

just four primal iterations. The primal and master design

and control solutions at each iteration are shown in

detail in Table 3. The optimal solution corresponds to adistillation column with 15 trays and the feed located on

tray 8. The profiles of the controlled distillate and

bottoms compositions for the optimal distillation system

are shown in Figs. 3 and 4, respectively. It can be seen

that the path constraints g1�/0.98�/xN�10/0 and g2�/

xb�/0.020/0 are satisfied at all times over one week’s

operation, even though the MIDO problem was only

solved using a time horizon of one day. Fig. 5 shows theprofile of the minimum allowable column diameter due

to flooding and compares it with the optimal column

diameter. Again, the associated path constraint, g3�Dmin

col �Dcol00; is always satisfied. The cyclical beha-

viour observed in Figs. 3�/5 and the fact that all three

path constraints are satisfied at all times suggests that

the solution found is not only economically optimal but

is also stable for the set of disturbances considered.Finally, as stated in Remark 2 of Section 6, earlier in

this article, one advantage of using Algorithm I com-

pared to current MIDO algorithms such as those of


Mohideen et al. (1997) and Schweiger and Floudas

(1997), is the far simpler form of the master problem.

This can be seen very clearly for this example. Using

Algorithm I, the k th master problem for this example

has the form:

minyf i(1;...;35);yri(1;...;35)

h;

s:t: hECostp�X35

i�1

wf ki (yf k

i �yf i)�X35

i�1

wrki (yrk

i �yri);

k � Kfeas;

(22)

where the pure binary constraints have not beenincluded for clarity of presentation; wfi are the Lagrange

multipliers for the constraints yfi�yfk

i �0; i�1; . . . ; 35; that are added when applying Step 3 of

Algorithm I; and wri are the Lagrange multipliers for

the constraints yri�yrki �0; i�1; . . . ; 35: These mul-

tipliers are all directly reported by the optimizer used for

solving the primal problems.

In sharp contrast, the master problem if Algorithm Iis not used is:

minyfi(1;...;35);yri(1;...;35)

h;

hE7756 �V pmean�1891:125�996:3(Dk

col)2

�1494:45 �Dcol

�6�0:76

X35

i�1

i �yri

�

�61:25X35

i�1

i �yri[0:7�1:5(Dkcol)

2]

�g1440

0

X35

i�1

�nk

1;i

��X35

i?�i

yri?

�m �

dxki

dt�Lk

i�1xki�1�Lk

i xki

�V k(yki�1�yk

i )�Fizf �Rki xk

N�1]gdt

�g1440

0

X35

i�1

[nk2;i(Fi�Fyfi)]dt;

Table 3

Progress of iterations of MIDO algorithm I for example 2

Number 1 2 3 4

Primal solutions

Discrete decisions

No. of trays 20 14 15 16

Feed tray 7 9 8 8

Process design

Dcol (m ) 0.8214 0.9138 0.8458 0.8323

Bottoms control loop

Kv �/55.92 �/43.32 �/54.64 �/43.83

tv 5.66 727.1 155.4 7.00

xb� 0.01823 0.01476 0.01564 0.01747

Distillate control loop

Kr 44.90 58.70 23.20 50.40

tr 654.2 27.2 2.01 211.5

xN�1� 0.9836 0.9824 0.9806 0.9830

Cost ($k yr�1) 40.303 39.633 36.729 36.901

UB 40.303 39.633 36.729 36.729

Master solutions

No. of trays 14 15 16 15

Feed tray 9 8 8 7

LB 32.989 34.502 36.349 36.879

UB�/LB0/0 No No No Yes: STOP

Fig. 3. Distillate composition profile for the optimal solution of Example 2.


�g1440

0

X35

i�1

[nk3;i(R

ki �Rkyri)]dt; k � Kfeas; (23)

where n1,i , n2,i and n3,i are the time-dependent dual

(adjoint) multipliers for the DAEs that contain the

binary variables, the values of which can only be

obtained by solving a complex, intermediate problem.

It is clear that Eq. (23) is much more complicated than

Eq. (22), since it involves integral terms and the storage

of a large number of primal solution values (e.g. the

values of dxki =dt; Lk

i�1, xki�1, yk

i�1 etc.).

13. Concluding remarks

This article has presented a new algorithm for solving

MIDO problems that is based on generalised Benders’

decomposition principles with the use of a reformulation

strategy. This algorithm has many advantages over

current approaches, namely that (i) it is independent

of the type of method used to solve the dynamic

optimization primal problems; (ii) no intermediate

adjoint problem needs to be solved in order to construct

the master problems, even when the binary variables

appear explicitly and non-linearly in the DAEs; and (iii)

the master problems are significantly simpler. The

algorithm can also be easily extended to handle the

case of time-dependent binary variables, as shown in the

article.

Fig. 4. Bottoms composition profile for the optimal solution of Example 2.

Fig. 5. Minimum column diameter profile for the optimal solution of Example 2.


With the reformulation strategy used in Algorithm I,

the article has also demonstrated how an algorithm can

be developed that uses OA/ER principles to construct

the master problems. This algorithm has similar proper-ties to Algorithm I and thus, in contrast to current OA-

based, MIDO algorithms, it is applicable to models

where the binary variables appear explicitly in the

DAEs.

The algorithms presented in this article are well-suited

for the practical solution of large engineering problems

using available tools. This offers the potential for much

progress to be made in the solution of more realisticproblems in areas such as simultaneous design and

control under uncertainty (Bansal, Pistikopoulos &

Perkins, 2002; Bansal, Ross, Pistikopoulos & Perkins,

2002).

Acknowledgements

The authors gratefully acknowledge financial supportfrom the EPSRC, the Centre for Process Systems

Engineering Industrial Consortium, DETR/ETSU and

Process Systems Enterprise Ltd. V. Bansal would also

like to thank ICI (Ph.D. Scholarship) and the Society of

the Chemical Industry (SCI Messel Scholarship).

Appendix A: Proof that master problems Eq. (12) and Eq.(7) are equivalent

In order to show that master problems Eq. (12) and

Eq. (7) are equivalent, equivalence needs to be estab-

lished between the Lagrange functions Lk in Eq. (11)

and Lk in Eq. (6). This can be achieved as follows.

Firstly, consider the optimality conditions that must be

satisfied by the continuous variables y in Eq. (9). Theseare determined by finding a stationary point for the

Lagrange function Lk in Eq. (10), i.e.:

0��@J

@y

�Tjyk

�gtf

t0

�@hd

@y

�Tjyk

�nkddt�g

tf

t0

�@ha

@y

�Tjyk

�nkadt

��@hd

@y

�Tjtf ;y

k

�jkd;f �

�@ha

@y

�Tjtf ;y

k

�jka;f

��@hd

@y

�Tjt0 ;y

k

�jkd;0�

�@ha

@y

�Tjt0;y

k

�jka;0�

�@h0

@y

�Tjyk

�kk

��@hp

@y

�Tjyk

�mkp�

�@gp

@y

�Tjyk

�lkp�

�@hq

@y

�Tjyk

�mkq

��@gq

@y

�Tjyk

�lkq�I �vk: (24)

Note that in Eq. (24), all of the Jacobian matrices

appear in transposed form. Thus, (@J/@y) is a (1�/ny)

vector, (@ hd/@y) is a (nxd�/ny) matrix, and so on, where

ny and nxdare the numbers of binary and differential

variables, respectively.

Substituting the expression for vk given by Eq. (24)

into Eq. (11) then gives:


k



��

@J

@y

�jyk

�gtf

t0

nkT

d ��@hd

@y

�jyk

dt�gtf

t0

nkT

a ��@ha

@y

�jyk

dt

�jkT

d;f ��@hd

@y

�jtf ;y

k

�jkT

a;f ��@ha

@y

�jtf ;y

k

�jkT

d;0 ��@hd

@y

�jt0;y

k

�jkT

a;0 ��@ha

@y

�jt0 ;y

k

�kkT ��@h0

@y

�jyk

�mkT

p ��@hp

@y

�jyk

�lkT

p ��@gp

@y

�jyk

�mkT

q ��@hq

@y

�jyk

�lkT

q ��@gq

@y

�jyk

�(y�yk): (25)

Now consider the original Lagrange function Lk , Eq.

(6), used in the master problems of the algorithms ofMohideen et al. (1997) and Schweiger and Floudas

(1997). For a system that is linear in y (as those

algorithms require), the objective function term in Eq.

(6) can be decomposed as:

J( ˆxk



�J1( ˆxk


a(tf ); uk(tf ); dk; tf )

�J2( ˆxk


a(tf ); uk(tf ); dk; tf ) �y: (26)

Similarly, the objective function evaluated at y�/yk is:

J( ˆxk



�J1( ˆxk


a(tf ); uk(tf ); dk; tf )

�J2( ˆxk


a(tf ); uk(tf ); dk; tf ) �y

k: (27)

Eliminating J1 from Eq. (26) and Eq. (27) gives:

J( ˆxk



�J( ˆxk



�J2( ˆxk


a(tf ); uk(tf ); dk; tf ) �(y�yk): (28)

Now, J2 is simply equal to (@J/@y) and so Eq. (28) canbe written as:

J( ˆxk


d(tf ); uk(tf ); dk; y; tf )


�J( ˆxk


a(tf ); uk(tf ); dk; yk; tf )�

�@J

@y

�jyk

�(y�yk): (29)

In an analogous manner to that above, the terms

within the Lagrange function Eq. (6) that relate to the

equality constraints can be written as:

h�(. . . ; y . . .)�h�(. . . ; yk . . .)��@h�

@y

�jyk

�(y�yk);

��d; a; 0; p; q;

(30)

with the added simplification that h�(. . .,yk . . .)�/0 at

the primal solution, i.e.

h�(. . . ; y . . .)��@h�

@y

�jyk

�(y�yk);

��d; a; 0; p; q:

(31)

Similarly, the terms in Eq. (6) that pertain to the

inequality constraints become:

lkT

� �g�(. . . ; y . . .)

�lkT

� �g�(. . . ; yk . . .)�lkT

� ��@g�

@y

�jyk

�(y�yk);

��p; q;

(32)

with the simplification that l�kT �/g�(. . .,yk . . .)�/0 due to

the complementarity optimality conditions of the primal

problem. Hence:

lkT

� �g�(. . . ; y . . .)�lkT

� ��@g�

@y

�jyk

�(y�yk);

��p; q:

(33)

Substituting Eqs. (29) and (31) and Eq. (33) in Eq. (6)

gives:


k



��

@J

@y

�jyk

�gtf

t0

nkT

d ��@hd

@y

�jyk

dt�gtf

t0

nkT

a ��@ha

@y

�jyk

dt

�jkT

d;f ��@hd

@y

�jtf ;y

k

�jkT

a;f ��@ha

@y

�jtf ;y

k

�jkT

d;0 ��@hd

@y

�jt0 ;y

k

�jkT

a;0 ��@ha

@y

�jt0;y

k

�kkT ��@h0

@y

�jyk

�mkT

p ��@hp

@y

�jyk

�lkT

p ��@gp

@y

�jyk

�mkT

q ��@hq

@y

�jyk

�lkT

q ��@gq

@y

�jyk

��(y�yk); (34)

which is identical to Eq. (25) as required.

Appendix B: Demonstrating the equivalence of GBD and

OA for Example 1

In Step 3 of Algorithm I/II, a dynamic optimizationproblem is solved. During the course of this, the

optimizer ‘sees’ a NLP formed from the elimination of

the differential and algebraic variables during an

integration phase. For Example 1, this NLP takes the

form:

minV ;R;yfi(i�1;...;30)

ISE(V ;R; yf);

s:t: g1(V ;R; yf)00;

g2(V ;R; yf)00;

yf�yfk�0; (35)

where yf is a vector consisting of thirty elements yfi , i�/

1,. . ., 30, and the solution of Eq. (35) is denoted byV

k; R

kand yfk: When applying Algorithm I, the k th

master problem Eq. (16) for this example is:

minyf ;h

h;

s:t: hEISE(Vk; R

k; yfk)�vkT �(yfk�yf); (36)

where the Lagrange multipliers vk satisfy the Karush�/

Kuhn�/Tucker (KKT) optimality conditions of Eq. (35):

@(ISE)

@V jk

�lk1 �@g1

@V jk

�lk2 �@g2

@V jk

�0; (37)

@(ISE)

@R jk

�lk1 �@g1

@R jk

�lk2 �@g2

@Rjk

�0; (38)

@(ISE)

@(yf) jk

�lk1 �

@g1

@(yf)jk�lk2 �

@g2

@(yf)jk�vk�0: (39)

Now assume that at the solution of the primal

problem Eq. (35), inequality constraint g10/0 is inactive

but g20/0 is active. In this case, l1k �/0; eliminating l2

k

from Eq. (37) and Eq. (39) then gives:

vk��@(ISE)

@(yf) jk

�

@(ISE)

@V�@g2

@(yf)

@g2

@V

jk: (40)

Substituting Eq. (40) in Eq. (36) thus gives the

following master problem for Algorithm I:

minyf;h

h;


s:t: hEISE(Vk; R

k; yfk)

�@(ISE)

@(yf) jk

�(yf� yfk)

�

@(ISE)

@V�@g2

@(yf)

@g2

@V

jk �(yf� yfk): (41)

When applying the OA-based, Algorithm II, the k th

master problem Eq. (21) for this example is:

minyf;V ;R;yf

h;

s:t: hEISE(Vk; R

k; yfk)

�@(ISE)

@V jk

�(V�Vk)�

@(ISE)

@R jk

�(R�Rk)

�@(ISE)

@(yf) jk

�(yf� yfk);

0Eg1(Vk; R

k; yfk)

�@g1

@V jk

�(V�Vk)

�@g1

@Rjk

�(R�Rk)�

@g1

@(yf)jk �(yf� yfk);

0Eg2(Vk; R

k; yfk)

�@g2

@V jk

�(V�Vk)

�@g2

@Rjk

�(R�Rk)�

@g2

@(yf)jk �(yf� yfk);

0�yf�yf: (42)

Now consider the case where, at the solution of the

master problem above, the linearised constraint asso-

ciated with g2 is active but the constraint associated with

g1 remains inactive. In this case, the latter constraint canbe ignored, while the equality of the former constraint

implies that:

(V�Vk)

��

@g2

@(yf) jk � (yf � yfk) �@g2

@R jk

� (R � Rk)

@g2

@V jk

: (43)

Utilising Eq. (43) in Eq. (42) thus gives the following

master problem for Algorithm II:

minyf

h;

s:t: hEISE(Vk; R

k; yfk)

�@(ISE)

@(yf) jk

�(yf� yfk)

�

@(ISE)

@V�@g2

@(yf)

@g2

@V

jk �(yf� yfk)

�

�@(ISE)

@R jk

�

@(ISE)

@V�@g2

@R

@g2

@V

jk

��(R�R

k): (44)

However, the KKT optimality conditions Eq. (37)

and Eq. (38) with l1k �/0 imply that:

@(ISE)

@R jk

�

@(ISE)

@V�@g2

@R

@g2

@V

jk�0: (45)

Thus, the master problem for Algorithm II becomes:

minyf

k;h

h;

s:t: hEISE(Vk; R

k; yfk)

�@(ISE)

@(yf) jk

�(yf� yfk)

�

@(ISE)

@V�@g2

@(yf)

@g2

@V

jk �(yf� yfk); (46)

which is identical to the master problem for Algorithm I,

Eq. (41). Hence, if g2 is active at the primal solution and

is still the only active inequality constraint at the

solution of the master problem, Algorithm II will give

an identical solution to the master problem of Algo-

rithm I. However, if any inequalities become active at

the master solution that were inactive at the primalsolution, then the master problem of Algorithm II will

give a tighter lower bound than that of Algorithm I.

This behaviour is indeed observed for Example 1, as

shown in Table 4.


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Table 4

Comparison of the performance of the master problems of algorithms

I and II for example 1


Feed tray

Algorithm I 26 24 25 23

Algorithm II 26 24 26 23

Lower bound, LB

Algorithm I 0.1576 0.1813 0.1815 0.1848

Algorithm II 0.1576 0.1813 0.1815 0.2093

Active Inequalities, gjk

Algorithm I g11 g1

2 g12 g1

3

Algorithm II g11 g1

2 g12 g1

1, g12

LB from Algorithm I Yes Yes Yes No

�/LB from Algorithm II?


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