a v a l A r c h it e c tu r e & O c e a n E n g in e e r in g
Computer Aided Ship Design Part I. Optimization Method
Ch. 2 Problem Statement of Optimal Design
Computer Aided Ship Design Lecture Note
September, 2013 Prof. Myung-Il Roh
Department of Naval Architecture and Ocean Engineering,
Seoul National University of College of Engineering
N a v a l A r c h it e c tu r e & O c e a n E n g in e e r in g
Computer Aided Ship Design, I-2. Problem Statement of Optimal Design, Fall 2013, Myung-Il Roh
Ch. 2 Problem Statement of Optimal Design
2.1 Components of Optimal Design Problem
2.2 Formulation of Optimal Design Problem
2.3 Classification of Optimization Problems
2.4 Classification of Optimization Methods
2.1 Components of Optimization Problem (1/3)
þ Design variable
n A set of variables that describes the system such as size and position, etc.
n It is also called ‘Free variable’ or ’Independent variable’.
n Dependent Variable
: A variable that is dependent on the design variable(independent variable)
þ Constraint
n A certain set of specified requirements and restrictions placed on a design
n Inequality Constraint, Equality Constraint
2.1 Components of Optimization Problem (2/3)
þ Objective function
n A criteria to compare the different design and
determine the proper design such as cost, profit, weight, etc.
n It is a function of the design variables.
Constraint
Objective Function (Minimization)
Design variable
Design variable
2.1 Components of Optimization Problem (3/3)
Local optimum
= Global optimum
The region satisfying the
constraint
Optimal design can be changed according to the
constraints.
Determination of the optimal design considering
the objective function(maximization) and constraints
2.2 Formulation of Optimization Problem
Objective Function Objective Function
Minimize:
Constraints Constraints
Subject to: g j ( x ) £ 0 , j = 1 , L , m
: Inequality constraint
p k
h k ( x ) = 0 , = 1 , L ,
: Equality constraint
u
l x x
x £ £
: Constraint(limit for the design variable) 2
1 5
4 x x f = - -
Minimize:
1 2
0 £ x x ,
2 2
4 4
6 6
x 1 x 2
x 1 + x 2 = 6
(f * = -29 ) Optimal solution
5x 1 +x 2 =10
1 2
1 2
4 6 x x x x - + £
+ £
Subject to 1 2
1 2
4 0 6 0 x x
x x
- + - £ + - £
1 2
5 x + x = 10 5 x
1+ x
2- 10 = 0
feasible region
( ) x
f
2.3 Classification of Optimization Problems (1/4)
þ Existence of constraints
n Unconstrained optimization problem
l Minimize the objective function f(x) without any constraints on the design variables x.
n Constrained optimization problem
l Minimize the objective function f(x) with some constraints on the design variables x.
Minimize f(x)
Minimize f(x) Subject to h(x)=0
g(x)≤0
2.3 Classification of Optimization Problems (2/4)
þ Number of objective functions
n Single-objective optimization problem
n Multi-objective optimization problem
l Weighting Method, Constraint Method
Minimize f(x) Subject to h(x)=0
g(x)≤0
Minimize f 1 (x), f 2 (x), f 3 (x) Subject to h(x)=0
g(x)≤0
2.3 Classification of Optimization Problems (3/4)
þ Linearity of objective function and constraints
n Linear optimization problem
l The objective function(f(x)) and constraints(h(x), g(x)) are linear functions of the design variables x.
n Nonlinear optimization problem
l The objective function(f(x)) or constraints(h(x), g(x)) are nonlinear functions of the design variables x.
1 2
( ) 2
f x = x + x
1 2
1
( ) 5 0
( ) 0
h x x
g x
= + =
= - £ x
x
2 1 2
2 2
1 3
)
( x x x x
f x = + - f ( x ) = x 1 2 + x 2 2 - 3 x 1 x 2
Minimize
Subject to
Minimize Minimize
2.3 Classification of Optimization Problems (4/4)
þ Type of design variables
n Continuous optimization problem
l Design variables are continuous in the optimization problem.
n Discrete optimization problem
l Design variables are discrete in the optimization problem.
l It is also called a ‘combinatorial optimization problem’.
l Example) Integer programming problem
2.4 Classification of Optimization Method
¾ Global Optimization Method
n Advantage
l It is useful for solving the global optimization problem which has many local optima.
n Disadvantage
l It needs many iterations(much time) to obtain the optimum.
n Genetic Algorithms(GA), Simulated Annealing, etc.
n Local Optimization Method
n Advantage
l It needs relatively few iterations(less time) to obtain the optimum.
n Disadvantage
l It is only able to find the local optimum which is near to the starting point.
n Sequential Quadratic Programming(SQP), Method of Feasible
Directions(MFD), Multi-Start Optimization Method, etc.
