4460 Lecture 8 2011 - Optimization Dr.MarioRichardEden...

Optimization
CHEN 4460 – Process Synthesis,
Simulation and Optimization
Dr. Mario Richard Eden
Department of Chemical Engineering
Auburn University
Lecture No. 8 – Mathematical Optimization
October 25, 2011
Contains Material Developed by Dr. Daniel R. Lewin, Technion, Israel

Lecture 10 – Objectives

Understand  the  different  types  of  optimization
problems and their formulation

Be  able  to  formulate  and  solve  a  variety  of
optimization problems in LINGO
On completion of this part, you should:

Optimization Basics
•
What is Optimization?
–
The  purpose  of  optimization  is  to maximize  (or
minimize) the  value  of  a  function  (called objective
function )  subject  to  a  number  of  restrictions  (called
constraints ).
•
Examples
1.Maximize reactor conversion
Subject to reactor modeling equations
kinetic equations
limitations on T, P and x

Optimization Basics
•
Examples (Continued)
2.Minimize cost of plant
Subject to mass & energy balance equations
equipment modeling equations
environmental, technical and logical
constraints

Optimization Basics
•
Examples (Continued)
3.Maximize your grade in this course
Subject to extracurricular activities
full-time-job requirements
constant  demand  by  other  courses
and/or your advisor/boss

Optimization Basics
•
Formulation of Optimization Problems
min (or max) f(x
1
,x
2
,……,x
N
)
subject to g
1
(x
1
,x
2
,……,x
N
) ≤ 0
g
2
(x
1
,x
2
,……,x
N
) ≤ 0
g
m
(x
1
,x
2
,……,x
N
) ≤ 0
h
1
(x
1
,x
2
,……,x
N
)=0
h
2
(x
1
,x
2
,……,x
N
)=0
h
E
(x
1
,x
2
,……,x
N
)=0
Inequality
Constraints
Equality
Constraints
Feasibility
Any vector (or point) which
satisfies all the constraints
of the optimization program
is called a feasible vector
(or a feasible point)
The set of all feasible points
is called feasibility region
or feasibility domain
Any optimal solution
must lie within the
feasibility region!

Optimization Basics
•
Classification of Optimization Problems
–
Linear Programs (LP’s)
•
A mathematical program is linear if
f(x
1
,x
2
,……,x
N
) and g
i
(x
1
,x
2
,……,x
N
) ≤ 0 are linear in each of their
arguments:
f(x
1
,x
2
,……,x
N
) = c
1
x
1
+ c
2
x
2
+ …. c
N
x
N
g
i
(x
1
,x
2
,……,x
N
) = a
i1
x
1
+ a
i2
x
2
+ …. a
iN
x
N
where c
i
and a
ij
are known constants.
Linear Programs (LP’s) can be solved to yield a
global optimum. Solver routines can guarantee a
truly optimal solution.

Optimization Basics
•
Classification of Optimization Problems
–
Non-Linear Programs (NLP’s)
•
A mathematical program is non-linear if any of the arguments
are non-linear. For example:
min 3x + 6y
2
s.t. 5x + x ⋅ y ≥ 0
–
Integer Programming
•
Optimization  programs  in  which ALL the  variables  must