optimization under uncertainty - advances in multi-parametric mixed integer linear ... ·...

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








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:




• Optimal solution is piecewise affine and continuous

• Critical Regions and union of critical regions are

polyhedral convex

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





Dinkelbach (1969)

nf4r




variables scales

linearly with

nf4r




variables scales

linearly with





Dinkelbach (1969)

=15

# CR‟s: 24





• Optimal solution is piecewise affine,

possibly discontinuous

• Critical Regions are polyhedral

convex

• Union of critical regions not

necessarily convex

RHS-mp-MILP: nf4r




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


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


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








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


• Consider

• Exact profit at :

• Robust Model is infeasible

Optimal Partially Robust Profit:


In Scheduling under Uncertainty:

• Contributes to pro-active scheduling strategies

• Efficient use of multi-parametric programming techniques:

Lower bound on the overall profit


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


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


• 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


• 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:


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)

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

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