2. Construction
1. Introduction
2. Overview of Genetic Algorithm
3. Nonlinear Stochastic Optimal Control
4. Application of NSOC for GA
5. Numerical Experiments
6. Conclusion
2
3. NSOC
GA
Application
3
Introduction
Background
u Nonlinear Stochastic Optimal Control
Foundation problem for control theory
u Genetic Algorithm
Derived from the principles of Darwinian natural
selection and evolution.
Previous research
uMarkov Chain Analysis on Simple Genetic Algorithms
Suzuki Joe (1995)
uLinearly-Solvable Markov Decision Problems
E. Todorov (2006)
5. Algorithm of GA
5
1. Obtain two parents by the selection function.
2. Produce their child by the crossover function
3. Mutate it by the mutation function.
4. Put the result into the next generation.
5. less than population members, go to Step 1.
l Optimization
l Automatic Programming
l Economic Models
l Social Systems Models
0
1
1
0
1
0
0
1
6. Nonlinear Stochastic Optimal
Control(NSCO)
Continuous Model
𝑥⃗ are given at an initial time t , the SCO problem is to find the
control path 𝑢 ' can minimized. the cost to go function is
continuous-time stochastic dynamics
d) = 𝑓 𝑋, 𝑢 𝑑/ + 𝐹 𝑥, 𝑢 𝑑2
optimal control law is
𝑢∗ = −𝜎6 𝐵 𝑥 8 𝜐)(𝑋)
6
𝐽(𝑥⃗,𝑡) = min
A
→
𝐶(𝑥⃗, 𝑡, 𝑢 ('))
𝑥 ∈ ℝFGstate vector
𝑢 ∈ ℝFAcontrol vector
𝜔 ∈ ℝFPBrown motion
7. Discrete Model
7
when we let 𝑝 𝑥T
𝑥, 𝑢 = 𝑢 𝑥T
𝑥
The bellman function of υ 𝑥 can write as
υ 𝑥 = min
V∈𝒰 X
{ℓ 𝑖, 𝑢 + EX]~_ ' 𝑥, 𝑢 𝜐(𝑥T
) }
optimal action will be
𝑢∗ 𝑥T 𝑥 =
𝑝 𝑥T 𝑥 𝓏 𝑥T
𝒢 𝓏 𝑥
𝓏 𝑥 = exp −𝑞 𝑥 𝒢 𝓏 𝑧 𝓏 = 𝑄𝑃𝓏
action cost of
current state
the value function
of next state
11. Numerical Experiments
β: Constant
ϵ =0.3
𝜆 =10
𝜒 = 0.4
𝜇j = ϵ𝑒𝑥𝑝 −𝜆𝛽
Iteration no. 800
fitness function is to compute the number of 1,
obtained by individuals.
Initial conditions
12. 12
we set 3 patterns to GA, all of which are
Constant function, Linear Function and
Exponential function.
13. where we list the result of transition probabilities
from state0 into other states. the result shows,
after the iteration of Numerical experiment,
individual is prefer to go to the state 7.
0
2
3
4
5 6
7
1
14. Introduce theory of GA
Introduce theory of NSOC
Conclusion
5
Overview of previous works
Across the discrete model of NSOC, we apply
it into GA , showed in numerical results.
Future Works
fitness function should
be change by complex
Compute speed should
be improved.
Apply by NSOC