1. MK. OPTMIMASI BAGIAN 2
Kuliah 1: 12 NOVEMBER 2017

2. TUJUAN
Meningkatkan kemampuan mahasiswa dalam berfikir secara logis analitis dalam penyelesaian masalah optimasi yang bersifat non linear.
Setelah mengikuti mata kuliah ini mahasiswa diharapkan mempunyai kompetensi menyelesaian masalah optimasi non linear dan dapat merapkanya pada masalah nyata

3. DESKRIPSI ISI
Masalah pengambilan keputusan selalu melibatkan penentuan pilihan terbaik dari semua alternative pilihan yang mungkin.
Sehingga dalam pemilihan keputusan terbaik (optimal) diperlukan teknik2 khusus.
Dalam perkuliahan ini dibahas metoda optimasi non linear dengan dan tanpa batasan.

4. Optimization Terminology
An optimization problem or a mathematical program or a mathematical programming problem.
Minimize f(x)
subject to
h(x)=c
g(x)> b
f(x): Objective function
x: Decision variables
h(x)=c: Equality constraint
g(x)> b: Inequality constraint
x*: solution

5. Classification of Optimization Problems
Continuous Optimization
Unconstrained Optimization
Bound Constrained Optimization
Derivative-Free Optimization
Global Optimization
Linear Programming
Network Flow Problems
Nondifferentiable Optimization
Nonlinear Programming
Optimization of Dynamic Systems
Quadratic Constrained Quadratic Programming
Quadratic Programming
Second Order Cone Programming
Semidefinite Programming
Semiinfinite Programming
Discrete and Integer Optimization
Combinatorial Optimization
Traveling Salesman Problem
Integer Programming
Mixed Integer Linear Programming
Mixed Integer Nonlinear Programming
Optimization Under Uncertainty Robust Optimization
Stochastic Programming
Simulation/Noisy Optimization
Stochastic Algorithms
Complementarity Constraints and Variational Inequalities
Complementarity Constraints
Game Theory
Linear Complementarity Problems
Mathematical Programs with Complementarity Constraints
Nonlinear Complementarity Problems
Systems of Equations
Data Fitting/Robust Estimation
Nonlinear Equations
Nonlinear Least Squares
Systems of Inequalities
Multiobjective Optimization

6. Outline Unconstrained Optimization Functions of One Variable
General Ideas of Optimization
First and Second Order Conditions
Local v.s. Global Extremum
Functions of Several Variables
Constrained Optimization
Kuhn-Tucker Conditions
Sensitivity Analysis
Second Order Conditions
Covex Programming 6

7. Unconstrained Optimization
An unconstrained optimization problem is one where you only have to be concerned with the objective function you are trying to optimize.
An objective function is a function that you are trying to optimize.
None of the variables in the objective function are constrained.

8. General Ideas of Optimization
There are two ways of examining optimization.
Maximization (example: maximize profit)
In this case you are looking for the highest point on the function.
Minimization (example: minimize cost)
In this case you are looking for the lowest point on the function.

9. Fall 2010 Olin Business School
Maximization f(x) is equivalent to minimization –f(x)
Fin500J Topic 4
Fall Olin Business School

10. Graphical Representation of a Maximumy 16
y = f(x) = -x2 + 8x x 4 8

11. Questions Regarding the Maximum
What is the sign of f '(x) when x < x*?
Note: x* denotes the point where the function is at a maximum.
What is the sign of f '(x) when x > x*?
What is f '(x) when x = x*?
Definition:
A point x* on a function is said to be a critical point if f ' (x*) = 0.
This is the first order condition for x* to be a maximum/minimum.

12. Second Order Conditions
If x* is a critical point of function f(x), can we decide whether it is a max, a min or neither?
Yes! Examine the second derivative of f(x) at x*, f ' '(x*);
x* is a maximum of f(x) if f ' '(x*) < 0;
x* is a minimum of f(x) if f ' '(x*) > 0;
x* can be a maximum, a minimum or neither if f ' '(x*) = 0;

13. An Example of f''(x*)=0
Suppose y = f(x) = x3, then f '(x) = 3x2 and f ''(x) =6x,
This implies that x* = 0 and f ''(x*=0) = 0.
y=f(x)=x3 y x
x*=0 is a saddle point where the point is neither a maximum nor a minimum

14. Example of Using First and Second Order Conditions
Suppose you have the following function:
f(x) = x3 – 6x2 + 9x
Then the first order condition to find the critical points is:
f’(x) = 3x2 - 12x + 9 = 0
This implies that the critical points are at x = 1 and x = 3.

15. Example of Using First and Second Order Conditions (Cont.)
The next step is to determine whether the critical points are maximums or minimums.
These can be found by using the second order condition.
f ' '(x) = 6x – 12 = 6(x-2)
Testing x = 1 implies:
f ' '(1) = 6(1-2) = -6 < 0.
Hence at x =1, we have a maximum.
Testing x = 3 implies:
f ' '(3) = 6(3-2) = 6 > 0.
Hence at x =3, we have a minimum.