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OPTMASS.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME OPTMASS
* Problem :
* *********
* A constrained optimal control problem
* adapted from Gawande and Dunn
* The problem is that of a particle of unit mass moving on a
* frictionless plane under the action of a controlling force whose
* magnitude may not exceed unity. At time=0, the particle moves through
* the origin of the plane in the direction of the positive x-axis with
* speed SPEED. The cost function incorporates two conflicting control
* objectives, namely: maximization of the particle's final (at time=1)
* distance from the origin and minimization of its final speed. By
* increasing the value of the penalty constant PEN, more stress can be
* placed on the latter objective.
* Gawande and Dunn originally use a starting point (in the control
* only) that is much closer to the solution than the one chosen
* here.
* Source:
* M. Gawande and J. Dunn,
* "A Projected Newton Method in a Cartesian Product of Balls",
* JOTA 59(1): 59-69, 1988.
* SIF input: Ph. Toint, June 1990.
* classification QQR2-AN-V-V
* Number of discretization steps in the time interval
* The number of variables is 6 * (N + 2) -2 , 4 of which are fixed.
*IE N 10 $-PARAMETER n = 70 original value
*IE N 100 $-PARAMETER n = 610
*IE N 200 $-PARAMETER n = 1210
IE N 500 $-PARAMETER n = 3010
* Initial speed (as in reference)
RE SPEED 0.01
* Penalty parameter (as in reference)
RE PEN 0.335
* Constants
IE 0 0
IE 1 1
IE 2 2
* Derived parameters
IA N+1 N 1
RI RN N
RD 1/N RN 1.0
RM -1/N 1/N -1.0
R* 1/N2 1/N 1/N
RM -1/2N2 1/N2 -0.5
VARIABLES
DO I 0 N
DO J 1 2
X X(J,I)
X V(J,I)
X F(J,I)
ND
DO J 1 2
X X(J,N+1)
X V(J,N+1)
ND
GROUPS
* Objective
N F
* State equations
DO I 1 N+1
IA I-1 I -1
DO J 1 2
XE A(J,I) X(J,I) 1.0 X(J,I-1) -1.0
ZE A(J,I) V(J,I-1) -1/N
ZE A(J,I) F(J,I-1) -1/2N2
XE B(J,I) V(J,I) 1.0 V(J,I-1) -1.0
ZE B(J,I) F(J,I-1) -1/N
ND
* Limit on the force
DO I 0 N
XL C(I)
ND
CONSTANTS
DO I 0 N
X OPTMASS C(I) 1.0
ND
BOUNDS
FR OPTMASS 'DEFAULT'
XX OPTMASS X(1,0) 0.0
XX OPTMASS X(2,0) 0.0
ZX OPTMASS V(1,0) SPEED
XX OPTMASS V(2,0) 0.0
START POINT
XV OPTMASS 'DEFAULT' 0.0
Z OPTMASS V(1,0) SPEED
ELEMENT TYPE
EV SQ X
ELEMENT USES
* Objective
T O1 SQ
ZV O1 X X(1,N+1)
T O2 SQ
ZV O2 X X(2,N+1)
T O3 SQ
ZV O3 X V(1,N+1)
T O4 SQ
ZV O4 X V(2,N+1)
* Limit on the force
DO I 0 N
DO J 1 2
XT D(J,I) SQ
ZV D(J,I) X F(J,I)
ND
GROUP USES
* Objective
E F O1 -1.0 O2 -1.0
ZE F O3 PEN
ZE F O4 PEN
* Limit on the force
DO I 0 N
XE C(I) D(1,I) D(2,I)
ND
OBJECT BOUND
* Solution
*LO SOLTN(10) -0.04647
*LO SOLTN(100) ???
*LO SOLTN(200) ???
*LO SOLTN(500) ???
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS OPTMASS
INDIVIDUALS
T SQ
F X * X
G X X + X
H X X 2.0
ENDATA