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DTOC2.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME DTOC2
* Problem :
* *********
* This is a discrete time optimal control (DTOC) problem.
* The system has N time periods, 2 control variables and 4 state variables.
* The problem is not convex.
* Sources: problem 2 in
* T.F. Coleman and A. Liao,
* "An Efficient Trust Region Method for Unconstrained Discret-Time Optimal
* Control Problems",
* Tech. Report, ctc93tr144, Advanced Computing Research Institute,
* Cornell University, 1992.
* L.Z. Liao and C.A. Shoemaker,
* "Advantages of differential dynamic programming over Newton's method for
* discrete-time optimal control problems",
* Tech. Report ctc92tr97, Advanced Computing Research Institute,
* Cornell University, 1992.
* SIF input: Ph. Toint, August 1993
* classification OOR2-AN-V-V
* Problem variants: they are identified by the value of the parameter N.
* The problem has (N-1)*NX+N*NY variables (of which NY are fixed),
* and (N-1)*NY constraints
*IE N 10 $-PARAMETER # periods } original value
*IE NX 2 $-PARAMETER # controls } n= 58, m= 36
*IE NY 4 $-PARAMETER # states }
*IE N 50 $-PARAMETER # periods }
*IE NX 2 $-PARAMETER # controls } n= 298, m= 196
*IE NY 4 $-PARAMETER # states }
*IE N 100 $-PARAMETER # periods }
*IE NX 2 $-PARAMETER # controls } n= 598, m= 396
*IE NY 4 $-PARAMETER # states }
*IE N 500 $-PARAMETER # periods }
*IE NX 2 $-PARAMETER # controls } n= 2998, m=1996
*IE NY 4 $-PARAMETER # states }
IE N 1000 $-PARAMETER # periods }
IE NX 2 $-PARAMETER # controls } n= 5998, m=3996
IE NY 4 $-PARAMETER # states }
*IE N 10 $-PARAMETER # periods }
*IE NX 5 $-PARAMETER # controls } n= 145, m= 90
*IE NY 10 $-PARAMETER # states }
*IE N 50 $-PARAMETER # periods }
*IE NX 5 $-PARAMETER # controls } n= 745, m= 490
*IE NY 10 $-PARAMETER # states }
*IE N 100 $-PARAMETER # periods }
*IE NX 5 $-PARAMETER # controls } n= 1495, m= 990
*IE NY 10 $-PARAMETER # states }
*IE N 500 $-PARAMETER # periods }
*IE NX 5 $-PARAMETER # controls } n= 7495, m=4990
*IE NY 10 $-PARAMETER # states }
*IE N 1000 $-PARAMETER # periods }
*IE NX 5 $-PARAMETER # controls } n=14995, m=9990
*IE NY 10 $-PARAMETER # states }
IA N-1 N -1
IE 1 1
IE 2 2
IE 3 3
IE 4 4
IA NY-1 NY -1
I+ 2NY NY NY
RI R2NY 2NY
RD 1/2NY R2NY 1.0
* Build the transition matrix C
DO J 1 NX
DO I 1 NY
I+ I+J I J
RI RI+J I+J
A* C(I,J) RI+J 1/2NY
OD J
OD I
VARIABLES
DO T 1 N-1
DO I 1 NX
X X(T,I)
OD I
OD T
DO T 1 N
DO I 1 NY
X Y(T,I)
OD I
OD T
GROUPS
* Objective function
DO T 1 N
XN O(T)
OD T
* Transition constraints
DO T 1 N-1
IA T+1 T 1
DO J 1 NY
XE TT(T,J) Y(T+1,J) -1.0
OD J
OD T
BOUNDS
FR DTOC2 'DEFAULT'
DO I 1 NY
RI RI I
R* TMP RI 1/2NY
ZX DTOC2 Y(1,I) TMP
OD I
START POINT
DO I 1 NY
RI RI I
R* TMP RI 1/2NY
ZV DTOC2 Y(1,I) TMP
OD I
ELEMENT TYPE
EV OEL YY1 YY2
EV OEL YY3 YY4
EV OEL XX1 XX2
EV SQ YY
EV SINE ZZ
ELEMENT USES
DO T 1 N-1
XT EO(T) OEL
ZV EO(T) YY1 Y(T,1)
ZV EO(T) YY2 Y(T,2)
ZV EO(T) YY3 Y(T,3)
ZV EO(T) YY4 Y(T,4)
ZV EO(T) XX1 X(T,1)
ZV EO(T) XX2 X(T,2)
DO J 1 NY
XT SY(T,J) SINE
ZV SY(T,J) ZZ Y(T,J)
OD J
DO I 1 NX
XT SX(T,I) SINE
ZV SX(T,I) ZZ X(T,I)
OD J
OD T
DO J 1 NY
XT YNSQ(J) SQ
ZV YNSQ(J) YY Y(N,J)
OD J
GROUP USES
DO T 1 N-1
XE O(T) EO(T)
OD T
DO J 1 NY
XE O(N) YNSQ(J)
OD J
DO T 1 N-1
DO J 1 NY
XE TT(T,J) SY(T,J)
DO I 1 NX
ZE TT(T,J) SX(T,I) C(J,I)
OD I
OD J
OD T
OBJECT BOUND
LO DTOC2 0.0
*LO SOLUTION( 10) 0.485983918948
*LO SOLUTION( 20) 0.486212213803
*LO SOLUTION( 30) 0.486383392574
*LO SOLUTION( 40) 0.486572686778
*LO SOLUTION( 50) 0.486884900389
*LO SOLUTION( 100) 0.487532342563
*LO SOLUTION( 500) 0.490996540460
*LO SOLUTION(1000) 0.490200910983
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS DTOC2
TEMPORARIES
R XN2
R YN2
R SZ
R CZ
R SZ2
R SC
R CCSS
M SIN
M COS
INDIVIDUALS
T SQ
F YY * YY
G YY YY + YY
H YY YY 2.0
T SINE
A SZ SIN( ZZ )
F SZ
G ZZ COS( ZZ )
H ZZ ZZ - SZ
T OEL
A XN2 XX1 * XX1 + XX2 * XX2
A YN2 YY1 * YY1 + YY2 * YY2 + YY3 * YY3
A+ + YY4 * YY4
A SZ SIN( 0.5 * XN2 )
A CZ COS( 0.5 * XN2 )
A SZ2 SZ * SZ + 1.0
A SC SZ * CZ
A CCSS CZ * CZ - SZ * SZ
F YN2 * SZ2
G XX1 2.0 * YN2 * SC * XX1
G XX2 2.0 * YN2 * SC * XX2
G YY1 2.0 * YY1 * SZ2
G YY2 2.0 * YY2 * SZ2
G YY3 2.0 * YY3 * SZ2
G YY4 2.0 * YY4 * SZ2
H XX1 XX1 2.0 * YN2 * ( SC + XX1 * XX1 * CCSS )
H XX1 XX2 2.0 * YN2 * XX1 * XX2 * CCSS
H XX1 YY1 4.0 * YY1 * SC * XX1
H XX1 YY2 4.0 * YY2 * SC * XX1
H XX1 YY3 4.0 * YY3 * SC * XX1
H XX1 YY4 4.0 * YY4 * SC * XX1
H XX2 XX2 2.0 * YN2 * ( SC + XX2 * XX2 * CCSS )
H XX2 YY1 4.0 * YY1 * SC * XX2
H XX2 YY2 4.0 * YY2 * SC * XX2
H XX2 YY3 4.0 * YY3 * SC * XX2
H XX2 YY4 4.0 * YY4 * SC * XX2
H YY1 YY1 2.0 * SZ2
H YY2 YY2 2.0 * SZ2
H YY3 YY3 2.0 * SZ2
H YY4 YY4 2.0 * SZ2
ENDATA