1Computer Aided Ship Design, I-2. Problem Statement of Optimal Design, Fall 2013, Myung-Il Roh

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Computer Aided Ship Design, I-2. Problem Statement of Optimal Design, Fall 2013, Myung-Il Roh

Computer Aided Ship Design

Part I. Optimization MethodCh. 2 Problem Statement of Optimal Design

Computer Aided Ship Design Lecture Note

September, 2013Prof. Myung-Il Roh

Department of Naval Architecture and Ocean Engineering,Seoul National University of College of Engineering

2Computer Aided Ship Design, I-2. Problem Statement of Optimal Design, Fall 2013, Myung-Il Roh

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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 Problem2.2 Formulation of Optimal Design Problem2.3 Classification of Optimization Problems2.4 Classification of Optimization Methods

3Computer Aided Ship Design, I-2. Problem Statement of Optimal Design, Fall 2013, Myung-Il Roh

2.1 Components of Optimization Problem (1/3)

þDesign variablen 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)

þConstraintn A certain set of specified requirements and restrictions

placed on a designn Inequality Constraint, Equality Constraint

4Computer Aided Ship Design, I-2. Problem Statement of Optimal Design, Fall 2013, Myung-Il Roh

2.1 Components of Optimization Problem (2/3)

þObjective functionnA 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

5Computer Aided Ship Design, I-2. Problem Statement of Optimal Design, Fall 2013, Myung-Il Roh

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 consideringthe objective function(maximization) and constraints

6Computer Aided Ship Design, I-2. Problem Statement of Optimal Design, Fall 2013, Myung-Il Roh

2.2 Formulation of Optimization Problem

Objective FunctionObjective FunctionMinimize:

ConstraintsConstraintsSubject to: mjg j ,,1,0)( L=£x: Inequality constraint

pkhk ,,1,0)( L==x: Equality constraint

ul xxx ££: Constraint(limit for the design variable)

21 54 xxf --=Minimize:

1 20 ,x x£

2

2

4

4

6

6

x1

x2

x1 + x2 = 6

(f *= -29 )Optimal solution

5x1+x2=10

1 2

1 2

46

x xx x- + £+ £

Subject to 1 2

1 2

4 06 0

x xx x- + - £+ - £

1 25 10x x+ = 1 25 10 0x x+ - =

feasible region

( )xf

7Computer Aided Ship Design, I-2. Problem Statement of Optimal Design, Fall 2013, Myung-Il Roh

2.3 Classification of Optimization Problems (1/4)

þExistence of constraintsn Unconstrained optimization problem

l Minimize the objective function f(x) without any constraints on the design variables x.

n Constrained optimization probleml 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

8Computer Aided Ship Design, I-2. Problem Statement of Optimal Design, Fall 2013, Myung-Il Roh

2.3 Classification of Optimization Problems (2/4)

þNumber of objective functionsn Single-objective optimization problem

n Multi-objective optimization probleml Weighting Method, Constraint Method

Minimize f(x)Subject to h(x)=0

g(x)≤0

Minimize f1(x), f2(x), f3(x)Subject to h(x)=0

g(x)≤0

9Computer Aided Ship Design, I-2. Problem Statement of Optimal Design, Fall 2013, Myung-Il Roh

2.3 Classification of Optimization Problems (3/4)

þLinearity of objective function and constraintsn 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 probleml The objective function(f(x)) or constraints(h(x), g(x)) are nonlinear

functions of the design variables x.

1 2( ) 2f x x= +x

1 2

1

( ) 5 0( ) 0

h x xg x

= + == - £

xx

2122

21 3)( xxxxf -+=x

1 2

1

( ) 5 0( ) 0

h x xg x

= + == - £

xx

2 21 1 2

2 1

1 1( ) 1.0 06 6

( ) 0

g x x

g x

= + - £

= - £

x

x

2122

21 3)( xxxxf -+=x

Minimize

Subject to

Minimize

Subject to

Minimize

Subject to

10Computer Aided Ship Design, I-2. Problem Statement of Optimal Design, Fall 2013, Myung-Il Roh

2.3 Classification of Optimization Problems (4/4)

þType of design variablesn Continuous optimization problem

l Design variables are continuous in the optimization problem.

n Discrete optimization probleml Design variables are discrete in the optimization problem.l It is also called a ‘combinatorial optimization problem’.l Example) Integer programming problem

11Computer Aided Ship Design, I-2. Problem Statement of Optimal Design, Fall 2013, Myung-Il Roh

2.4 Classification of Optimization Method

¾ Global Optimization Methodn Advantage

l It is useful for solving the global optimization problem which has many local optima.

n Disadvantagel It needs many iterations(much time) to obtain the

optimum.

n Genetic Algorithms(GA), Simulated Annealing, etc.

n Local Optimization Methodn Advantage

l It needs relatively few iterations(less time) to obtain the optimum.

n Disadvantagel 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.