-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathSolver.py
358 lines (278 loc) · 17.3 KB
/
Solver.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
"""
Solver.py
Python function template to solve the discounted stochastic
shortest path problem.
Dynamic Programming and Optimal Control
Fall 2023
Programming Exercise
Contact: Antonio Terpin aterpin@ethz.ch
Authors: Abhiram Shenoi, Philip Pawlowsky, Antonio Terpin
--
ETH Zurich
Institute for Dynamic Systems and Control
--
"""
import numpy as np
def solution(P, G, alpha):
"""Computes the optimal cost and the optimal control input for each
state of the state space solving the discounted stochastic shortest
path problem by:
- Value Iteration;
- Policy Iteration;
- Linear Programming;
- or a combination of these.
Args:
P (np.array): A (K x K x L)-matrix containing the transition probabilities
between all states in the state space for all control inputs.
The entry P(i, j, l) represents the transition probability
from state i to state j if control input l is applied
G (np.array): A (K x L)-matrix containing the stage costs of all states in
the state space for all control inputs. The entry G(i, l)
represents the cost if we are in state i and apply control
input l
alpha (float): The discount factor for the problem
Returns:
np.array: The optimal cost to go for the discounted stochastic SPP
np.array: The optimal control policy for the discounted stochastic SPP
"""
from scipy.sparse import csr_matrix
K, L = G.shape
P_csr = [csr_matrix(P[:, :, action]) for action in range(L)]
J_opt = np.full(K, 1e8)
u_opt = np.zeros(K)
epsilon = 1e-08
delta_v = float('inf')
while delta_v > epsilon:
J_prev = J_opt.copy()
for action in range(L):
J_opt_col_vector = J_opt.reshape(-1, 1)
total_cost_action = G[:, action] + alpha * (P_csr[action].dot(J_opt_col_vector)).flatten()
better_cost = total_cost_action < J_opt
J_opt[better_cost] = total_cost_action[better_cost]
u_opt[better_cost] = action
delta_v = np.max(np.abs(J_opt - J_prev))
return J_opt, u_opt
def freestyle_solution(Constants):
"""Computes the optimal cost and the optimal control input for each
state of the state space solving the discounted stochastic shortest
path problem with a 200 MiB memory cap.
Args:
Constants: The constants describing the problem instance.
Returns:
np.array: The optimal cost to go for the discounted stochastic SPP
np.array: The optimal control policy for the discounted stochastic SPP
"""
from ComputeStageCosts import compute_stage_cost
import itertools
from scipy.sparse import coo_matrix
def compute_transition_probabilities_sparse():
"""Computes the transition probability matrix P.
It is of size (K,K,L) where:
- K is the size of the state space;
- L is the size of the input space; and
- P[i,j,l] corresponds to the probability of transitioning
from the state i to the state j when input l is applied.
Args:
- None
Returns:
- scipy.sparse: Transition probability matrix of shape
"""
t = np.arange(0, Constants.T)
z = np.arange(0, Constants.D)
y = np.arange(0, Constants.N)
x = np.arange(0, Constants.M)
state_space = np.array(list(itertools.product(t, z, y, x)))
K = Constants.T * Constants.D * Constants.N * Constants.M
input_space = np.array([Constants.V_DOWN, Constants.V_STAY, Constants.V_UP])
L = len(input_space)
# Initialize lists for COO format
rows = []
cols = []
data = []
matrix_dict={}
def add_to_sparse_matrix(i, j, l, value):
"""
Adds a non-zero transition probability to the sparse matrix.
Adding to COO matrix.
Args:
- matrix_dict (dict): Dictionary for helping construct the coo matrix
- i (int): Current state index.
- j (int): Next state index.
- l (int): Action index.
- value (float): Probability value to be added.
- L (int): Total number of actions.
"""
if value != 0.0:
row_index= i*L+l
matrix_dict[(row_index,j)]= value
for i in range (K):
t_i=state_space[i][0]
z_i=state_space[i][1]
y_i=state_space[i][2]
x_i=state_space[i][3]
if(t_i<(Constants.T-1)):
t_j=t_i+1
else:
t_j=0
if(z_i<(Constants.D-1)):
z_up_j=z_i+1
else:
z_up_j=z_i
if(z_i>0):
z_down_j=z_i-1
else:
z_down_j=0
if(y_i<(Constants.N-1)):
y_north_j= y_i+1
else:
y_north_j=y_i
if(y_i>0):
y_south_j=y_i-1
else:
y_south_j= 0
if(x_i<(Constants.M-1)):
x_east_j=x_i+1
else:
x_east_j=0
if(x_i>0):
x_west_j=x_i-1
else:
x_west_j=Constants.M-1
j_up = t_j*(Constants.D*Constants.N*Constants.M) + z_up_j*(Constants.N*Constants.M) + y_i*Constants.M + x_i
j_stay = t_j*(Constants.D*Constants.N*Constants.M) + z_i*(Constants.N*Constants.M) + y_i*Constants.M + x_i
j_down = t_j*(Constants.D*Constants.N*Constants.M) + z_down_j*(Constants.N*Constants.M) + y_i*Constants.M + x_i
j_up_east = t_j*(Constants.D*Constants.N*Constants.M) + z_up_j*(Constants.N*Constants.M) + y_i*Constants.M + x_east_j
j_up_west = t_j*(Constants.D*Constants.N*Constants.M) + z_up_j*(Constants.N*Constants.M) + y_i*Constants.M + x_west_j
j_stay_east = t_j*(Constants.D*Constants.N*Constants.M) + z_i*(Constants.N*Constants.M) + y_i*Constants.M + x_east_j
j_stay_west = t_j*(Constants.D*Constants.N*Constants.M) + z_i*(Constants.N*Constants.M) + y_i*Constants.M + x_west_j
j_down_east = t_j*(Constants.D*Constants.N*Constants.M) + z_down_j*(Constants.N*Constants.M) + y_i*Constants.M + x_east_j
j_down_west = t_j*(Constants.D*Constants.N*Constants.M) + z_down_j*(Constants.N*Constants.M) + y_i*Constants.M + x_west_j
j_up_north = t_j*(Constants.D*Constants.N*Constants.M) + z_up_j*(Constants.N*Constants.M) + y_north_j*Constants.M + x_i
j_up_south = t_j*(Constants.D*Constants.N*Constants.M) + z_up_j*(Constants.N*Constants.M) + y_south_j*Constants.M + x_i
j_stay_north = t_j*(Constants.D*Constants.N*Constants.M) + z_i*(Constants.N*Constants.M) + y_north_j*Constants.M + x_i
j_stay_south = t_j*(Constants.D*Constants.N*Constants.M) + z_i*(Constants.N*Constants.M) + y_south_j*Constants.M + x_i
j_down_north = t_j*(Constants.D*Constants.N*Constants.M) + z_down_j*(Constants.N*Constants.M) + y_north_j*Constants.M + x_i
j_down_south = t_j*(Constants.D*Constants.N*Constants.M) + z_down_j*(Constants.N*Constants.M) + y_south_j*Constants.M + x_i
y_north_limit = 0
y_south_limit = 0
if y_i == Constants.N - 1:
y_north_limit = 1
if y_i == 0:
y_south_limit = 1
# ----- Constants.V_DOWN -----
if z_i > 0:
p_value_j_stay = Constants.P_V_TRANSITION[0]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_STAY]
p_value_j_down = Constants.P_V_TRANSITION[1]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_STAY]
add_to_sparse_matrix(i, j_stay, Constants.V_DOWN, p_value_j_stay)
add_to_sparse_matrix(i, j_down, Constants.V_DOWN, p_value_j_down)
p_value_j_stay_east = Constants.P_V_TRANSITION[0]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_EAST]
p_value_j_stay_west = Constants.P_V_TRANSITION[0]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_WEST]
add_to_sparse_matrix(i, j_stay_east, Constants.V_DOWN, p_value_j_stay_east)
add_to_sparse_matrix(i, j_stay_west, Constants.V_DOWN, p_value_j_stay_west)
p_value_j_down_east = Constants.P_V_TRANSITION[1]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_EAST]
p_value_j_down_west = Constants.P_V_TRANSITION[1]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_WEST]
add_to_sparse_matrix(i, j_down_east, Constants.V_DOWN, p_value_j_down_east)
add_to_sparse_matrix(i, j_down_west, Constants.V_DOWN, p_value_j_down_west)
p_value_j_stay_north = Constants.P_V_TRANSITION[0]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_NORTH]
p_value_j_stay_south = Constants.P_V_TRANSITION[0]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_SOUTH]
add_to_sparse_matrix(i, j_stay_north, Constants.V_DOWN, p_value_j_stay_north)
add_to_sparse_matrix(i, j_stay_south, Constants.V_DOWN, p_value_j_stay_south)
p_value_j_down_north = Constants.P_V_TRANSITION[1]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_NORTH]
p_value_j_down_south = Constants.P_V_TRANSITION[1]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_SOUTH]
add_to_sparse_matrix(i, j_down_north, Constants.V_DOWN, p_value_j_down_north)
add_to_sparse_matrix(i, j_down_south, Constants.V_DOWN, p_value_j_down_south)
if y_north_limit: # If north limit, akin to staying
p_value_j_stay = (Constants.P_V_TRANSITION[0]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_STAY] +
Constants.P_V_TRANSITION[0]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_NORTH])
p_value_j_down = (Constants.P_V_TRANSITION[1]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_STAY]+
Constants.P_V_TRANSITION[1]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_NORTH])
add_to_sparse_matrix(i, j_stay, Constants.V_DOWN, p_value_j_stay)
add_to_sparse_matrix(i, j_down, Constants.V_DOWN, p_value_j_down)
elif y_south_limit: # If north limit, akin to staying
p_value_j_stay = (Constants.P_V_TRANSITION[0]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_STAY] +
Constants.P_V_TRANSITION[0]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_SOUTH])
p_value_j_down = (Constants.P_V_TRANSITION[1]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_STAY]+
Constants.P_V_TRANSITION[1]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_SOUTH])
add_to_sparse_matrix(i, j_stay, Constants.V_DOWN, p_value_j_stay)
add_to_sparse_matrix(i, j_down, Constants.V_DOWN, p_value_j_down)
# ----- Constants.V_STAY (always possible) -----
p_value_j_stay = Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_STAY]
add_to_sparse_matrix(i, j_stay, Constants.V_STAY, p_value_j_stay)
p_value_j_stay_east = Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_EAST]
p_value_j_stay_west = Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_WEST]
add_to_sparse_matrix(i, j_stay_east, Constants.V_STAY, p_value_j_stay_east)
add_to_sparse_matrix(i, j_stay_west, Constants.V_STAY, p_value_j_stay_west)
p_value_j_stay_north = Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_NORTH]
p_value_j_stay_south = Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_SOUTH]
add_to_sparse_matrix(i, j_stay_north, Constants.V_STAY, p_value_j_stay_north)
add_to_sparse_matrix(i, j_stay_south, Constants.V_STAY, p_value_j_stay_south)
if y_north_limit: # If north limit, akin to staying
p_value_j_stay = (Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_STAY] +
Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_NORTH])
add_to_sparse_matrix(i, j_stay, Constants.V_STAY, p_value_j_stay)
elif y_south_limit: # If north limit, akin to staying
p_value_j_stay = (Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_STAY] +
Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_SOUTH])
add_to_sparse_matrix(i, j_stay, Constants.V_STAY, p_value_j_stay)
# ----- Constants.V_UP -----
if (z_i < (Constants.D - 1)):
p_value_j_up = Constants.P_V_TRANSITION[1]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_STAY]
p_value_j_stay = Constants.P_V_TRANSITION[0]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_STAY]
add_to_sparse_matrix(i, j_up, Constants.V_UP, p_value_j_up)
add_to_sparse_matrix(i, j_stay, Constants.V_UP, p_value_j_stay)
p_value_j_up_east = Constants.P_V_TRANSITION[1]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_EAST]
p_value_j_up_west = Constants.P_V_TRANSITION[1]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_WEST]
add_to_sparse_matrix(i, j_up_east, Constants.V_UP, p_value_j_up_east)
add_to_sparse_matrix(i, j_up_west, Constants.V_UP, p_value_j_up_west)
p_value_j_stay_east = Constants.P_V_TRANSITION[0]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_EAST]
p_value_j_stay_west = Constants.P_V_TRANSITION[0]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_WEST]
add_to_sparse_matrix(i, j_stay_east, Constants.V_UP, p_value_j_stay_east)
add_to_sparse_matrix(i, j_stay_west, Constants.V_UP, p_value_j_stay_west)
p_value_j_up_north = Constants.P_V_TRANSITION[1]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_NORTH]
p_value_j_up_south = Constants.P_V_TRANSITION[1]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_SOUTH]
add_to_sparse_matrix(i, j_up_north, Constants.V_UP, p_value_j_up_north)
add_to_sparse_matrix(i, j_up_south, Constants.V_UP, p_value_j_up_south)
p_value_j_stay_north = Constants.P_V_TRANSITION[0]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_NORTH]
p_value_j_stay_south = Constants.P_V_TRANSITION[0]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_SOUTH]
add_to_sparse_matrix(i, j_stay_north, Constants.V_UP, p_value_j_stay_north)
add_to_sparse_matrix(i, j_stay_south, Constants.V_UP, p_value_j_stay_south)
if y_north_limit: # If north limit, akin to staying
p_value_j_stay = (Constants.P_V_TRANSITION[0]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_STAY] +
Constants.P_V_TRANSITION[0]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_NORTH])
p_value_j_up = (Constants.P_V_TRANSITION[1]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_STAY]+
Constants.P_V_TRANSITION[1]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_NORTH])
add_to_sparse_matrix(i, j_stay, Constants.V_UP, p_value_j_stay)
add_to_sparse_matrix(i, j_up, Constants.V_UP, p_value_j_up)
elif y_south_limit: # If north limit, akin to staying
p_value_j_stay = (Constants.P_V_TRANSITION[0]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_STAY] +
Constants.P_V_TRANSITION[0]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_SOUTH])
p_value_j_up = (Constants.P_V_TRANSITION[1]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_STAY]+
Constants.P_V_TRANSITION[1]*Constants.P_H_TRANSITION[z_i].P_WIND[Constants.H_SOUTH])
add_to_sparse_matrix(i, j_stay, Constants.V_UP, p_value_j_stay)
add_to_sparse_matrix(i, j_up, Constants.V_UP, p_value_j_up)
rows, cols, data = zip(*[(key[0], key[1], val) for key, val in matrix_dict.items()])
P_sparse = coo_matrix((data, (rows, cols)), shape=(K*L, K))
return P_sparse
K = Constants.T * Constants.D * Constants.N * Constants.M
J_opt = np.zeros(K)
u_opt = np.zeros(K)
P=compute_transition_probabilities_sparse()
G=compute_stage_cost(Constants)
K, L = G.shape
P = P.tocsr()
J_opt = np.full(K, 1e5) # Based on testing performance
u_opt = np.zeros(K)
# Convergence parameters
epsilon = 1e-08
delta_v = float('inf')
while delta_v > epsilon:
J_prev = J_opt.copy()
for action in range(L):
action_indices = np.arange(action, K * L, L)
P_action = P[action_indices, :]
total_cost_action = G[:, action] + Constants.ALPHA * P_action.dot(J_opt)
better_cost = total_cost_action < J_opt
J_opt[better_cost] = total_cost_action[better_cost]
u_opt[better_cost] = action
delta_v = np.max(np.abs(J_opt - J_prev))
return J_opt, u_opt