Biological process simulation and optimization

생물공정모사 및 최적화

Biological process simulation and optimization

Major: Interdisciplinary program of the integrated biotechnology

Graduate school of bio- & information technology Youngil Lim (N110), Lab. FACS

Youngil Lim (N110), Lab. FACS

phone: +82 31 670 5200 (secretary), +82 31 670 5207 (direct) phone: +82 31 670 5200 (secretary), +82 31 670 5207 (direct)

Fax: +82 31 670 5445, mobile phone: +82 10 7665 5207 Fax: +82 31 670 5445, mobile phone: +82 10 7665 5207

Email:

Email: [email protected][email protected], homepage: , homepage:  http://http://

hknu.ac.kr/~limyi/index.htm hknu.ac.kr/~limyi/index.htm

Part I. Problem Formulation

Mathematical form :

• Objective function (economic criteria): profit,

variables.

• Process model (constrains): interrelationship

equations).

Part II. Optimization theory and methods

Computational methods and algorithms:

• Ch. 4: mathematical concepts

• Ch. 5: one-dimensional search

• Ch. 6: unconstrained multivariable optimization

• Ch. 7: Linear programming (LP)

• Ch. 8: Nonlinear programming (NLP)

• Ch. 9: mixed-integer Nonlinear programming (MINLP)

• Ch. 10: Global optimization

Ch 4. Basic concepts of optimization

4.1 Continuity functions (So-Jeong)

- 0^th-order continuous, 1^st-order continuous function - derivative-based optimization

4.2 NLP problem statement (Hae-Jeong) - linear constraints, nonlinear constraints 4.3 Convexity and its applications (Thanh)

- convex function

- convex programming problem - role of convexity

- determination of convexity and concavity

4.4 Interpretation of the objective function in terms of its quadratic approximation - quadratic functions

- eigenvectors

4.5 Necessity and sufficient conditions for an extremum of an unconstrained function - necessity conditions

- sufficient conditions

4.1 Continuity functions (So-Jeong)

A. continuity

+ 0^th-order continuous function + 1^st-order continuous function + 2^nd-order continuous function

B. derivative-based optimization

4.2 NLP problem statement (Hae-Jeong)

A. Quadratic program: linearly constrained problem with a quadratic objective  local extremum = global extremum

B. Feasible/infeasible region

C. Local extremum, global extremum

4.3 Convexity and its applications (Thanh)

A. Convex/concave function

B. convex programming problem

If the objective function and inequality constraints are convex,

 Local minimum of f(x) is also the global minimum.

) (

) 1

( ) ( ]

) 1

(

[ x

x

f x

f x

f         

4.3 Convexity and its applications (Thanh)

C. role of convexity

If any of h(x) is nonlinear, the NLP may not be a convex problem.

Although convexity is desirable, many real-world problems turn out to be nonconvex !

] ...,

, [

,...

1 ,

0 )

(

,....

1 ,

0 )

( .

. ) ( min

1 n

k i x

x x

x where

n r

k x

h

m i

x g t s

x f













4.3 Convexity and its applications (Thanh)

D. Determination of convexity and concavity + one variable function, f(x)

1) F(x) is strictly convex, if and only if its H(x) is positive-definite.

2) F(x) is convex, if and only if its H(x) is positive-semidefinite 3) positive definite: x^THx > 0 for all x  0

4) positive semidefinite: x^THx ≥ 0 for all x  0 + multi-variable function, f(x)

1) positive definite: eigenvalues of H(x) > 0

2) positive semidefinite: eigenvalues of H(x) ≥ 0

* What is the eigenvalue ?

 0

 I

A 

4.4 Interpretation of the objective function in terms of its quadratic approximation

A. Quadratic function or approximated quadratic function of two variables

+ maximum + minimum + saddle point + contour map

B. well-posed optimization problems = convex functions for minimization C. Eigenvector (v₁ and v₂) and orthogonal (v₁^Tv₂=0):

The eigenvectors correspond to the directions of the principle axes of the contours of f(x).

2 1 2 12

2 2 22

1 11 2

2 1

1

) 0

( x b b x b x b x b x b x x

f      

vector column

a is V where

V I A

,

0 )

(   

Ex. Determine the geometric shape, contour characteristics and

eigenvectors of the following Hessian matrix of a quadratic function.





 



 

10 0

0

H 1

4.5 Necessity and sufficient conditions for an extremum of an unconstrained function

A. Necessity and sufficient conditions for optimality + if and only if =necessity and sufficient

B. Talyor expansion

What is SQP for the NLP solver?

- Quadratic approximation  to find f(x) = 0

- line search using eigenvectors  to find the steepest way for local minimum

extremum presumed

x and x

x x

where

x O

x x

f x

x x

f x

f x

f

: ,

) ( )

2 ( ) 1

( )

( )

(

*

*

* 3 2

*

*



























definite positive

is x

H x

f condiiton

y sufficienc

x f b

continuous order

is x f a condiiton necessity

x f

nd x











 ) ( )

( :

) 2

0 ) ( )

2 )

( ) : )

1

) ( min

*

* 2

*

Example 4.10 min f(x)=2x

See p. 138 - f(x)=x⁴

- examine the necessity and sufficiency conditions!

definite positive

is x

H condiiton

y sufficienc

x x

x f b

continuous order

is x

f a condiiton necessity

x x

f

nd x



















6 )

( :

) 2

0 0

6 )

( )

2 )

( )

: )

1

3 )

( min

*

* 2

Example 4.11 calculation of extrema

2 2 4 1

1 2

2 1 2 2

1 2

1

4 2 2 2

5 . 4 4

)

( x x x x x x x x x x

f

Min

       

How do you solve this problem?

Example 4.12 Production rate for fermentation process

] [

) (

) (

] / [

) ( ,

) ) (

( max

kg throughput

cumulative the

is t

P

h time cleaning

the is

t

h kg rate

production overall

the is

t R where

t t

t t P

R

c





How do you solve this problem?

1.6 General procedure for solving optimization problems

1. Analyze the process itself so that the process variables and specific characteristics of interest are defined  make a list of the variables/parameters

2. Determine the criterion for optimization, and specify the objective function w.r.t variables and parameters  performance model

3. Using mathematical expressions, develop a valid process or equipment model that relates the input/output variables. Include both equality and inequality constraints. Use first-principle models (mass/energy balances, equilibrium equations), empirical equations, implicit concepts and external restrictions. Identify the number of degree of freedom.  equality/inequality constraints

4. If the problem formulation is too large in scope,  reduced model development

1. Break it up into manageable parts or

2. Simplify the objective function and model

5. Apply a suitable optimization technique (SQP, GA, GCMC, etc. or Matlab, GAMS, etc.) to the mathematical statement of the problem.

6. Check the answers, and examine the sensitivity of the result to change in the parameters  parameter sensitivity analysis.

Rep eat

Exercise and homework 4

• Select two problems of ch. 4 and solve them using Matlab or others.

• Each student should select a different problem each other.

• If there is no specific value to be needed, please set the

values by yourself.