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Optimization under Uncertainty -
Advances in
Multi-Parametric Mixed Integer Linear
Programming and its Applications
Martina Wittmann-Hohlbein
Efstratios N. Pistikopoulos
QUADS Seminar, DoC Imperial College, 10.11.2011
Outline
I. Parametric programming
II. Multi-parametric mixed integer linear programming
i. Global optimization of mp-MILP problems
ii. Two-stage method for the approximate
solution of mp-MILP problems
III. An application: Pro-active scheduling of short-term
batch processes under uncertainty
IV. Conclusions
Motivation - MPC
Model Predictive Control (MPC):
• Mathematical model to predict effect of control actions
• Aim is to regulate the system to a reference value
Solver
System
states controller
Online MPC:
Repetitive solution of optimization
problems with identical system and
MPC-related data,
but with varying states
• Based on current measurements an optimization
problem is solved
• First optimal control applied, refined MPC-model
at next time step
Motivation - MPC
Offline MPC1:
Solver
System
Look-up Table
System
Parametric Solver
states states controller controller
parametric
profiles
• Future control actions as optimization variables
• Initial states as parameters
Offline MPC Formulation for Time Invariant Systems:
At the decision stage, a single optimization
problem with uncertain data is solved using
multi-parametric programming techniques.
1 Pistikopoulos et al. (2000)
Parametric Programming
1) Exact solution method: as a function of ,
characterization of critical regions
If Then
Optimization under Uncertainty:
Multi-Parametric Programming:
2) Strategies for the efficient exploration of the parameter space
Parametric Programming
Algorithms and Applications:
Linear Discrete
Systems (MPC)
Pistikopoulos et al. (2000); Bemporad et al. (2002); Tondel et al. (2003);
Sakizlis et al. (2005)
Robust Control Sakizlis et al. (2004a/b); Bemporad et al. (2003); Kouramas et al.
(2009,2011)
Hybrid Control Sakizlis et al. (2002); Borelli et al. (2005); Morari et al. (2006);
Scheduling Li et al. (2007,2008), Ryu et al. (2007a/b)
mp-LP/mp-QP Gal et al. (1972, 1975); Dua et al. (2002); Tondel et al. (2003); Borelli
(2003)
mp-NLP Bemporad et al. (2006); Acevedo (2003); Dominguez et al. (2010)
mp-MI Acevedo et al. (1997) ; Pertsinidis et al. (1998) ; Dua et al. (2002) ; Li et
al. (2007,2008); Faisca et al. (2009) ; Mitsos et al. (2009); Wittmann-
Hohlbein et al. (2011)
mp-GO Benson (1982); Dua et al. (2004)
mp-DO Faisca et al. (2008)
Outline
I. Parametric programming
II. Multi-parametric mixed integer linear programming
i. Global optimization of mp-MILP problems
ii. Two-stage method for the approximate
solution of mp-MILP problems
III. An application: Pro-active scheduling of short-term
batch processes under uncertainty
IV. Conclusions
Applications:
• Pro-active Scheduling under price, demand and processing time
uncertainty (Ryu et al. (2006,2007), Li et al.(2007,2008), Lin et al. (2004))
• Explicit Model Predictive Control of Hybrid Systems:
Control actions as optimization variables, states as parameters,
input and model disturbances as parameters (Pistikopoulos et al. (2006) Vol 2.)
The general mp-MILP Problem:
• and analogously for
State-of-the-Art Algorithms
mp-MILP
(mp-MIQP)
Dua et al. (2002) (RHS)
Pertsinidis et al. (1998) (RHS, single parametric)
Faisca et al. (2009) (RIM=RHS+OFC)
Acevedo et al. (1997) (RHS)
Li et al. (2008)* (LHS)
Mitsos et al. (2009) (LHS, single parametric)
* LHS-mp-MILP using enumeration of parameter-space
Research Objective:
Efficient approaches to the solution of the general mp-MILP problem
(in particular with LHS-uncertainty)
Algorithms vary in:
• Addressing the integer nodes
• Exploring the parameter space
Solution of mp-MILP Problems
• General mp-MILP problem non-convex mp-MINLP problem
• Bottleneck is the solution of LHS-mp-LP sub-problem:
• Optimal solution is piecewise fractional polynomial,
possibly discontinuous
• Critical regions need not be convex
Dinkelbach (1969), altered Li et al.
(2008)
Global Solution of mp-MILP Problems
Adapt strategies from deterministic global optimization
to multi-parametric framework. (Benson (1982), Fiacco (1990), Dua et al. (2004))
Suitable under- and
overestimating problems
Branch-and-Bound
procedure
Question:
Can we make use of the special structure of the problem ?
Motivation:
Global Solution of mp-MILP Problems
• McCormick-type relaxations of bilinear terms are employed
• Transformation of LHS-uncertainty into RHS-uncertainty
Approximation of optimal solution with piecewise affine functions;
Approximation of optimal objective value with piecewise affine
underestimating (overestimating) functions
• Variables and parameters participate in non-linear terms
• Bivariate multi-parametric branch-and-bound procedure is
propagated in the overall global optimization routine
• Modelling OFC-uncertainty as LHS-uncertainty A)
B)
C)
Aim:
Relaxations of Bilinear Terms
Underestimating bilinear terms in a LHS-mp-LP problem by
1. Classic McCormick relaxations RHS-mp-LP
McCormick
relaxation
Dinkelbach (1969), altered
• Optimal solution is piecewise affine and continuous
• Critical Regions and union of critical regions are
polyhedral convex
RHS-mp-LP:
Relaxations of Bilinear Terms
Underestimating bilinear terms in a LHS-mp-LP problem by
1. Classic McCormick relaxations RHS-mp-LP
• Optimal solution is piecewise affine and continuous
• Critical Regions and union of critical regions are
polyhedral convex
RHS-mp-LP:
Relaxations of Bilinear Terms
Underestimating bilinear terms in a LHS-mp-LP problem by
1. Classic McCormick relaxations RHS-mp-LP
2. Piecewise affine relaxations w.r.t. RHS-mp-MILP
nf4r
(Gounaris et al. (2009))
• Partition factor
• Number of binary
variables scales
linearly with
• additional
parameters
=1
=3
Relaxations of Bilinear Terms
Underestimating bilinear terms in a LHS-mp-LP problem by
1. Classic McCormick relaxations RHS-mp-LP
2. Piecewise affine relaxations w.r.t. RHS-mp-MILP
Dinkelbach (1969)
nf4r
(Gounaris et al. (2009))
• Partition factor
• Number of binary
variables scales
linearly with
nf4r
(Gounaris et al. (2009))
• Partition factor
• Number of binary
variables scales
linearly with
Relaxations of Bilinear Terms
Underestimating bilinear terms in a LHS-mp-LP problem by
1. Classic McCormick relaxations RHS-mp-LP
2. Piecewise affine relaxations w.r.t. RHS-mp-MILP
Dinkelbach (1969)
=15
# CR‟s: 24
Relaxations of Bilinear Terms
Underestimating bilinear terms in a LHS-mp-LP problem by
1. Classic McCormick relaxations RHS-mp-LP
2. Piecewise affine relaxations w.r.t. RHS-mp-MILP
• Optimal solution is piecewise affine,
possibly discontinuous
• Critical Regions are polyhedral
convex
• Union of critical regions not
necessarily convex
RHS-mp-MILP: nf4r
(Gounaris et al. (2009))
• Partition factor
• Number of binary
variables scales
linearly with
2 Partitions:
Multi-Parametric B&B Procedure
1 Partition
Tighter Relaxation:
• Partitioning of the
parameter space
• Underestimating problem
in each partition
• Related to piecewise
affine relaxation of bilinear
terms w.r.t. parameters RHS-mp-MILP problem
Multi-Parametric B&B Procedure
1 Partition
15 Partitions; 21 CR’s
Tighter Relaxation:
• Partitioning of the
parameter space
• Underestimating problem
in each partition
• Related to piecewise
affine relaxation of bilinear
terms w.r.t. parameters
RHS-mp-MILP problem
Branch on continuous variables Refine bounds:
1. Solve auxiliary RHS-mp-LP problem with
2. Identify region where optimal objective value is worse
than current best upper bound
3. Fathoming confine in this region
Multi-Parametric B&B Procedure
Branch on continuous variables Refine bounds:
1. Solve auxiliary RHS-mp-LP problem with
2. Identify region where optimal objective value is worse
than current best upper bound
3. Fathoming confine in this region
Multi-Parametric B&B Procedure
Set the tolerance . Set partitioning factor for each
- component in bilinear terms. Initialize a set of w.r.t. .
In each :
Step 1 Obtain current best lower bound:
Step 2 Obtain current best upper bound:
Step 3
Compare bounds: Stop in regions where
Exclude , store ;
Update as union of convex regions
Step 4
Branch on variables: Fathom a partition in regions where
Update as union of convex regions
Step 5 Partition each region w.r.t and goto Step 1 with new .
Algorithm for LHS-mp-LP Problems
Candidates
for Global
Optimum
Algorithm for LHS-mp-LP Problems
Number of critical regions of globally optimal solution is
influenced by:
• Computational complexity of the RHS-mp-(MI)LP algorithm for
the solution of under-, overestimating and auxiliary problems
• Tolerance criteria, Step 3, and the success of fathoming, Step 4
• Partitioning Factor w.r.t. parameters
Trade-off between quality of approximation and computational
requirements:
• Classic McCormick relaxation vs. piecewise affine relaxation
• Emphasizing different aspects within multi-parametric B&B
procedure: univariate vs. bivariate partitioning scheme
Initialize as set of feasible parameters; Solve initial MINLP
problem treating as variable; Obtain optimal integer node,
Step 1
(mp-LP)
Fix , solve LHS-mp-LP in to global optimality;
Obtain and add corresponding solution to the envelope
of parametric profiles;
Update by incorporating the newly identified regions
Step 2
(MINLP)
In each solve a MINLP problem with as variable, and
integer and parametric cuts w.r.t. already identified profiles;
Goto Step 1 if a new is found
Step 3 Termination in if MINLP problem is infeasible
Algorithm for LHS-mp-MILP Problems
Decomposition Algorithm for the global solution of the
general mp-MILP problem:
Example 2 - LHS-mp-MILP Algorithm
Step 0 – MINLP master problem solved to global optimality:
(BARON, Sahinidis & Tawarmalani (2010))
Li et al. (2008)
Step 1 – LHS-mp-LP sub-problem:
Iteration 1 Iteration 2 (CPU 9s)
Iteration 3
• 3330 CR‟s
• CPU 100s
•
Step 2 –
MINLP master problem:
• In each of 3330 CR‟s
infeasible
Global Optimization of mp-MILP„s1
Challenges in Global Optimization of mp-MILP Problems:
• Comparison of parametric profiles, not scalar values
• High computational requirements
• Price for global solution is large number of critical regions
Multi-Parametric Global Optimization:
• Globally optimal solution is a piecewise affine function
• Polyhedral convex critical regions
Motivation:
Can we find “good solutions” of an mp-MILP problem with less effort?
1 Wittmann-Hohlbein, Pistikopoulos; submitted
Hybrid Approach: Two-Stage Method for
mp-MILP„s1
1 Wittmann-Hohlbein, Pistikopoulos (2011)
General mp-MILP problem
Stage 1 – Reformulation
Partially robust RIM-mp-MILP* model
Stage 2 – Solution
Decomposition Algorithm (Faisca et al. (2009))
Optimal partially robust solution; Upper bound
on optimal objective function value
*objective function coefficient and
right hand side vector uncertainty
Two-Stage Method for mp-MILP„s
Stage 1:
• Immunize model against uncertainty in constraint matrix
• Worst-case scenario (Soyster (1973), Ben-Tal et al. (2000), Lin et al. (2004)):
• Partially robust RIM-mp-MILP model (RIM=OFC+RHS)
where
• Combined robust optimization / multi-parametric programming
approach for the approximate solution of mp-MILP problems
Stage 2: Decomposition Algorithm (Faisca et al. (2009))
Two-Stage Method for mp-MILP„s
• Iteration between MINLP master- and mp-LP sub-problems
• Envelope of parametric profiles stores solutions of sub-problems
(optimal objective values of sub-problems need not be convex)
Key Features of Two-Stage Method:
• No need (for global optimization) to solve LHS-mp-LP
problems due to partial robustification step
• Solution of a single RIM-mp-MILP problem
Properties of Partially Robust Profiles:
• Piecewise affine functions
• Polyhedral convex critical regions
Example 2 - Two-Stage Method
Special Case:
Partially robust model is
deterministic problem
• Optimal partially robust solution:
• Two-stage method requires
solution of the initial MILP
master problem only
Optimal solution:
Outline
I. Parametric programming
II. Multi-parametric mixed integer linear programming
i. Global optimization of mp-MILP problems
ii. Two-stage method for the approximate
solution of mp-MILP problems
III. An application: Pro-active scheduling of short-term
batch processes under uncertainty
IV. Conclusions
A Scheduling Model
Short-term Scheduling of Batch Processes:
Unit-specific event based approach (Ierapetritrou et al. (1998))
• Continuous time formulation
• Set of time related instances at which tasks start in a unit
• Binary variables
- activation status of task i in unit j at event point n
• MILP model
Objective: Maximization of profit, minimization of makespan, etc.
Subject to:
• Allocation and sequence constraints
• Storage and capacity constraints
• Material balances
• Time and duration constraints
• Market demands
MILP
Scheduling
Formulation
Pro-active Scheduling
Real Life Scenario:
• Unsteady prices
• Varying demands
• Uncertainty in processing times and conversion rates
Uncertainty
Pro-active Scheduling:
• Finding the scheduling policy that performs best in
the face of disturbances
Pro-active Scheduling
Cost Uncertainty
Demand Uncertainty
Processing Time
Uncertainty
Multi-Parametric
Programming (Li et al. (2007,2008),
Ryu et al. (2007a/b))
Robust Optimization (Li et al. (2008,2011),
Lin et al. (2004))
• Deterministic model
• Solutions are feasible
for all data variations
Two-Stage Method
• Combines optimization with multi-parametric
programming techniques
• Flexible towards incorporation of data once its actual
value is known
• Efficient treatment of all types of uncertainty
STN - Representation of an example batch process:
S1 S2
S3 S4
mixing reaction
purification
Results for the Two-Stage Method
• Parameters:
• Price of S4:
• Demand of S4:
• Mean processing time for mixing: with 33% variability
worst case scenario for variable proc. time
Optimal Partially Robust Schedule:
• 3 integer nodes identified
• Obtained after 9 MINLP and
3 mp-LP problems have been
solved
Results for the Two-Stage Method
• Consider
• Exact profit at :
• Robust Model is infeasible
Optimal Partially Robust Profit:
Results for the Two-Stage Method
In Scheduling under Uncertainty:
• Contributes to pro-active scheduling strategies
• Efficient use of multi-parametric programming techniques:
Lower bound on the overall profit
Results for the Two-Stage Method
Motivation:
Can we embed different uncertainty set-induced robust models1
into the two-stage method?
Do they have applications in pro-active scheduling of batch processes?
In Comparison with Rigorous Robust Optimization Approach:
• Partially robust scheduling policy is less conservative
1 Li et al. (2011)
Alternative Robust Model
• Worst-case oriented approach is restrictive
• In practice not all uncertain coefficients are prone to vary
from an estimated nominal value at the same time
Robust MILP model with an adjustable degree of
conservatism1:
• Budget parameter is introduced into the model
• Controlling trade-off between conservatism of the solution
and robustness of the method
Optimal solution is immune against variation of
up to coefficients from their nominal values
1 Bertsimas et al. (2003, 2004)
- Number of supported deviations from nominal value in the coefficients in i-th row of A/E
Alternative Robust Model
Partially robust RIM-mp-MILP model with an adjustable degree
of conservatism:
- Index set of all uncertain coefficients in i-th row of A/E
Alternative Robust Model
• Robust model with an adjustable degree of conservatism is
induced by a combined interval + polyhedral uncertainty set
S1
Example 4
S2
S8
S5
S3
S6
S7
S4
S9
Heat
Sep
R3 R1
R2
• Parameters:
• Price of S8:
• Demand of S8:
• Production rate of S8:
• Production rate of S7:
40%
60%
50%
50% 20%
80%
35%
60%
10%
90%
• Price of S9:
• Demand of S9:
90%
Four available units:
Two units U1 and U2
suitable for R1, R2 and R3;
Plus one each for heating
and separation
Alternative Robust Model
• Scheduling with uncertain conversion rates: Storage constraints
• Uncertain coefficients not independent Sensible choice of
Partially robust constraints with adjust. degree of conservatism:
a) between types of constraints
b) related to event points Consistency
Example 4
Same profit for
CR3 – CR4
• We assume that of the production rates for S7, resp. S8, in U1
and U2 at most one is likely to change from the nominal value:
• Optimal partially robust schedule is independent of
• Optimal robust profit (lower bound):
• We assume that of the production rates for S7, resp. S8, in U1
and U2 at most one is likely to change from the nominal value:
• : all are likely to change; worst-case scenario
• : no deviation supported; nominal model
Conclusions I
Two-Stage Method in Scheduling under Uncertainty1:
• Pro-active scheduling approach
• Scheduling model with various types of uncertainty is address
Key Advances:
A) Computational complexity
• Partially robust schedules are efficiently obtained
B) Quality of scheduling policy
• Partially robust model according to individual requirements:
worst-case oriented vs. adjustable degree of conservatism
• Flexibility to “robustify” model against all complicating, non-
measurable, less relevant uncertainty
C) Valuable apriori insight into the production process
1 Wittmann-Hohlbein, Pistikopoulos; accepted
Conclusions II
Solution of general mp-MILP Problems
A) Two-Stage Method B) Global Solution
• Combined robust optimization/
multi-parametric programming
approach
• Multi-parametric global optimization
Key Features of Multi-Parametric Programming
• Problem contaminated with uncertainty is solved offline
• For a given parameter realization the optimal solution is
retrieved via function evaluation
Ongoing research:
• Theoretical/algorithmical
developments for mp-MIQP‟s
Ongoing research:
• Uncertainty set induced partially
robust models
• Conceptual importance
Thank you.
Comments and Questions?
Optimality Conditions:
• Describe subset of parameter space where optimal basis
remains valid
Foundation: LP optimality conditions
Optimal Solution:
If solvable for
Optimal basis
Standard
mp-LP problem:
Solution of mp-MILP Problems
Gantt-Chart for the optimal partially robust schedule
at using 5 event points:
Results for the Two-Stage Method
overall profit: 128.6
Example 1 - LHS-mp-LP Algorithm
Partitioning Factor: 2
Iteration Depth: 3
• Branching on
• # UE‟s : 37
• # CR‟s: 51 (6 exact)
Partitioning Factor: 2
Iteration Depth: 3
• No branching on
• # UE‟s : 20
• # CR‟s: 18 (6 exact)
Example 1 - LHS-mp-LP Algorithm
Partitioning Factor: 2
Iteration Depth: 3
• Branching on
• # UE‟s : 37
• # CR‟s: 51 (6 exact)
Partitioning Factor: 2
Iteration Depth: 3
• No branching on
• # UE‟s : 20
• # CR‟s: 18 (6 exact)
Partitioning Factors:
=2, =1
Iteration Depth: 3
• No branching on
• # UE‟s (mp-LP): 20
• # CR‟s: 18 (6 exact)
Partitioning Factors:
=2, =3
Iteration Depth: 3
• No branching on
• # UE‟s (mp-MILP): 50
• # CR‟s: 47 (6 exact)
Example 1 - LHS-mp-LP Algorithm