Piecewise-Linear Approximation: Multidimensional
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1 OptIntro 1 / 38 Piecewise-Linear Approximation: Multidimensional Eduardo Camponogara Department of Automation and Systems Engineering Federal University of Santa Catarina October 2016
2 OptIntro 2 / 38 Summary Motivation CC Model DCC Model SOS2 Model
3 OptIntro 3 / 38 Motivation Summary Motivation CC Model DCC Model SOS2 Model
4 OptIntro 4 / 38 Motivation Well Production (Lift-gas, Well-head Pressure).
5 OptIntro 5 / 38 Motivation Pressure Loss in Flowlines
6 OptIntro 5 / 38 Motivation Pressure Loss in Flowlines
7 OptIntro 6 / 38 Motivation Multidimensional Models Let us consider functions with multidimensional domains: f : D R, in which D R n is the domain. An issue that comes up is how to partition the domain. Let P be a set of polyhedrons defining a partitioning of D: P P P = D, int(p) P =, P, P P The piecewise-linear models can be applied depending on the structure of the partitioning P (geometry/topology).
8 OptIntro 6 / 38 Motivation Multidimensional Models Let us consider functions with multidimensional domains: f : D R, in which D R n is the domain. An issue that comes up is how to partition the domain. Let P be a set of polyhedrons defining a partitioning of D: P P P = D, int(p) P =, P, P P The piecewise-linear models can be applied depending on the structure of the partitioning P (geometry/topology).
9 OptIntro 7 / 38 Motivation Models and Partitioning These models can be applied to any convex partitioning of the domain: CC, DCC, DLog e Multiple Choice. Can be applied only to a J-1 triangulation of the domain: Log. Can be applied only to regular lattices: SOS2.
10 OptIntro 7 / 38 Motivation Models and Partitioning These models can be applied to any convex partitioning of the domain: CC, DCC, DLog e Multiple Choice. Can be applied only to a J-1 triangulation of the domain: Log. Can be applied only to regular lattices: SOS2.
11 OptIntro 7 / 38 Motivation Models and Partitioning These models can be applied to any convex partitioning of the domain: CC, DCC, DLog e Multiple Choice. Can be applied only to a J-1 triangulation of the domain: Log. Can be applied only to regular lattices: SOS2.
12 OptIntro 8 / 38 Motivation Models and Partitioning
13 OptIntro 9 / 38 CC Model Summary Motivation CC Model DCC Model SOS2 Model
14 OptIntro 10 / 38 CC Model CC Model Definitions: P is the set of polyhedra defining the domain. V is the set of breakpoints. V (P) V is the set of vertices of polyhedron P P. Variables: λ v [0, 1] is a weighting variable associated with vertex v V. y P {0, 1} assumes value 1 if x P, P P, otherwise assumes value 0.
15 OptIntro 10 / 38 CC Model CC Model Definitions: P is the set of polyhedra defining the domain. V is the set of breakpoints. V (P) V is the set of vertices of polyhedron P P. Variables: λ v [0, 1] is a weighting variable associated with vertex v V. y P {0, 1} assumes value 1 if x P, P P, otherwise assumes value 0.
16 OptIntro 11 / 38 CC Model CC Model f = f (v)λ v, x = vλ v, v V v V 1 = λ v, 1 = y P, v V P P λ v y P, P P(v) λ v 0, v V, y P {0, 1}, P P in which: P(v) = {P P : v V (P)} is the set of polyhedra that contain breakpoint v.
17 OptIntro 12 / 38 CC Model Sample Function Consider the nonlinear function: f (x, y) = e y + e x2 x 2 + xy 2y y x 1 2
18 OptIntro 12 / 38 CC Model Sample Function Consider the nonlinear function: f (x, y) = e y + e x2 x 2 + xy 2y y x 1 2
19 OptIntro 13 / 38 CC Model Example: CC Model Vertex set: V = { 2, 1, 0, 1, 2} { 2, 1, 0, 1, 2}} Polyhedra set: P = { 2, 1, 0, 1} { 2, 1, 0, 1}} in which, the index (i, j) P corresponds to the polyhedron P i,j = {(x, y) R 2 : i x i + 1, j y j + 1}
20 OptIntro 13 / 38 CC Model Example: CC Model Vertex set: V = { 2, 1, 0, 1, 2} { 2, 1, 0, 1, 2}} Polyhedra set: P = { 2, 1, 0, 1} { 2, 1, 0, 1}} in which, the index (i, j) P corresponds to the polyhedron P i,j = {(x, y) R 2 : i x i + 1, j y j + 1}
21 OptIntro 14 / 38 CC Model Example: CC Model f = v V f (v)λ v = f ( 2, 2)λ ( 2, 2) + f ( 2, 1)λ ( 2, 1) + f ( 2, 0)λ ( 2,0). + f ( 2, 1)λ ( 2,1) + f ( 2, 2)λ ( 2,2) + f (2, 2)λ (2, 2) + f (2, 1)λ (2, 1) + f (2, 0)λ (2,0) + f (2, 1)λ (2,1) + f (2, 2)λ (2,2)
22 OptIntro 15 / 38 CC Model Example: CC Model x = v V v xλ v = ( 2)λ ( 2, 2) + ( 2)λ ( 2, 1) + ( 2)λ ( 2,0) + ( 2)λ ( 2,1) + ( 2)λ ( 2,2). + (2)λ (2, 2) + (2)λ (2, 1) + (2)λ (2,0) + (2)λ (2,1) + (2)λ (2,2)
23 OptIntro 16 / 38 CC Model Example: CC Model y = v V v y λ v = ( 2)λ ( 2, 2) + ( 1)λ ( 2, 1) + (0)λ ( 2,0) + (1)λ ( 2,1) + (2)λ ( 2,2). + ( 2)λ (2, 2) + ( 1)λ (2, 1) + (0)λ (2,0) + (1)λ (2,1) + (2)λ (2,2)
24 OptIntro 17 / 38 CC Model Example: CC Model 1 = v V λ v = λ ( 2, 2) + λ ( 2, 1) + λ ( 2,0) + λ ( 2,1) + λ ( 2,2) + λ ( 1, 2) + λ ( 1, 1) + λ ( 1,0) + λ ( 1,1) + λ ( 1,2). + λ (2, 2) + λ (2, 1) + λ (2,0) + λ (2,1) + λ (2,2)
25 OptIntro 18 / 38 CC Model Example: CC Model 1 = P P y P = y ( 2, 2) + y ( 2, 1) + y ( 2,0) + y ( 2,1) + y ( 1, 2) + y ( 1, 1) + y ( 1,0) + y ( 1,1). + y (2, 2) + y (2, 1) + y (2,0) + y (2,1)
26 OptIntro 19 / 38 CC Model Example: CC Model The constraint family λ v y P, P P(v) becomes λ ( 2, 2) y ( 2, 2) λ ( 2, 1) y ( 2, 2) + y ( 2, 1) λ ( 2,0) y ( 2, 1) + y ( 2,0) λ ( 2,1) y ( 2,0) + y ( 2,1) λ ( 2,2) y ( 2,1).. λ ( 1, 2) y ( 2, 2) + y ( 2, 1) λ ( 1, 1) y ( 2, 2) + y ( 2, 1) + y 1, 2 + y ( 1, 1) λ ( 1,0) y ( 2, 1) + y ( 1, 1) + y ( 2,0) + y ( 1,0) λ ( 1,1) y ( 2,0) + y ( 1,0) + y ( 2,1) + y ( 1,1) λ ( 1,2) y ( 2,1) + y ( 1,1)..
27 OptIntro 20 / 38 DCC Model Summary Motivation CC Model DCC Model SOS2 Model
28 OptIntro 21 / 38 DCC Model Disaggregated Convex Combination (DCC) Model f = f (v)λ P,v, P P v V(P) y P = λ P,v, P P, v V(P) 1 = P P y P, x = P P vλ P,v, v V(P) λ P,v 0, P P, v V(P), y P {0, 1}, P P where: λ P,v is associated to vertex v of polyhedron P (breakpoint).
29 OptIntro 22 / 38 DCC Model Example: Function
30 OptIntro 23 / 38 DCC Model Example: Domain y 2 P( 2,1) P( 1,1) P(0,1) P(1,1) x 1 P( 2, 2) P( 1, 2) P( 0, 2) P(1, 2) 2
31 OptIntro 24 / 38 DCC Model Example: DCC Model f = P P v V(P) f (v)λ P,v, = f ( 2, 2)λ P( 2, 2) ( 2, 2) +f ( 1, 2)λ P( 2, 2) ( 1, 2) +f ( 1, 2)λ P( 1, 2) ( 1, 2) +f (0, 2)λ P( 1, 2) (0, 2). +f (1, 1)λ P(1,1) (1,1) + f (1, 2)λ P(1,1) (1,2) + f ( 2, 1)λ P( 2, 2) ( 2, 1) + f ( 1, 1)λ P( 2, 2) ( 1, 1) + f ( 1, 1)λ P( 1, 2) ( 1, 1) + f (0, 1)λ P( 1, 2) (0, 1) +f (2, 1)λ P(1,1) (2,1) + f (2, 2)λ P(1,1) (2,2)
32 OptIntro 25 / 38 DCC Model Example: DCC Model (x, y) = P P v V(P) vλ P,v, = ( 2, 2)λ P( 2, 2) ( 2, 2) +( 1, 2)λ P( 2, 2) ( 1, 2) +( 1, 2)λ P( 1, 2) ( 1, 2) +(0, 2)λ P( 1, 2) (0, 2). +(1, 1)λ P(1,1) (1,1) + (1, 2)λ P(1,1) (1,2) + ( 2, 1)λ P( 2, 2) ( 2, 1) + ( 1, 1)λ P( 2, 2) ( 1, 1) + ( 1, 1)λ P( 1, 2) ( 1, 1) + (0, 1)λ P( 1, 2) (0, 1) +(2, 1)λ P(1,1) (2,1) + (2, 2)λ P(1,1) (2,2)
33 OptIntro 26 / 38 DCC Model Example: DCC Model The constraint family y P = λ P,v, P P, v V(P) becomes y P( 2, 2) = λ P( 2, 2) ( 2, 2) y P( 1, 2) = λ P( 1, 2) ( 1, 2).. + λ P( 2, 2) ( 2, 1) + λ P( 1, 2) ( 1, 1) + λ P( 2, 2) ( 1, 2) + λ P( 1, 2) (0, 2) y P(1,1) = λ P(1,1) (1,1) + λ P(1,1) (1,2) + λ P(1,1) (2,1) + λ P(1,1) (2,2) + λ P( 2, 2) ( 1, 1) + λ P( 1, 2) (0, 1)
34 OptIntro 27 / 38 DCC Model Example: DCC Model The constraint family 1 = P P y P, becomes 1 = y P( 2, 2) + y P( 1, 2) + y P(0, 2) y P(1,1)
35 OptIntro 28 / 38 SOS2 Model Summary Motivation CC Model DCC Model SOS2 Model
36 OptIntro 29 / 38 SOS2 Model SOS2 Model Objectives: Piecewise-linear approximation of the function f : R 2 R. Definitions: Breakpoint set in x : X = {x 1, x 2,..., x n}. Breakpoint set in y : Y = {y 1, y 2,..., y n}. Given the values of the function f for (x, y) X Y. Polyhedron set given by: P = {[x i, x i+1 ] [y j, y j+1 ] : i = 1,..., n 1, j = 1,..., n 1}.
37 OptIntro 29 / 38 SOS2 Model SOS2 Model Objectives: Piecewise-linear approximation of the function f : R 2 R. Definitions: Breakpoint set in x : X = {x 1, x 2,..., x n}. Breakpoint set in y : Y = {y 1, y 2,..., y n}. Given the values of the function f for (x, y) X Y. Polyhedron set given by: P = {[x i, x i+1 ] [y j, y j+1 ] : i = 1,..., n 1, j = 1,..., n 1}.
38 OptIntro 30 / 38 SOS2 Model SOS2 Model f = f (x, y)λ x,y, x X y Y x = xλ x,y, x X y Y y = yλ x,y, x X y Y 1 = λ x,y, λ x,y 0, x X, y Y x X y Y η x = λ x,y, x X, y Y η y = λ x,y, y Y, x X (η x) x X is SOS2 (η y) y Y is SOS2
39 OptIntro 31 / 38 SOS2 Model Example: Function
40 OptIntro 32 / 38 SOS2 Model Example: Domain y x 1 2
41 OptIntro 33 / 38 SOS2 Model Example: SOS2 Model Function approximation becomes f = f (x, y)λ x,y x X y Y = +f ( 2, 2)λ ( 2, 2) + f ( 2, 1)λ ( 2, 1) + f ( 2, 0)λ ( 2,0) +f ( 2, 1)λ ( 2,1) + f ( 2, 2)λ ( 2,2). +f (2, 2)λ (2, 2) + f (2, 1)λ (2, 1) + f (2, 0)λ (2,0) +f (2, 1)λ (2,1) + f (2, 2)λ (2,2)
42 OptIntro 34 / 38 SOS2 Model Example: SOS2 Model Domain variables (x, y) = x X (x, y)λ x,y y Y = +( 2, 2)λ ( 2, 2) + ( 2, 1)λ ( 2, 1) + ( 2, 0)λ ( 2,0) +( 2, 1)λ ( 2,1) + ( 2, 2)λ ( 2,2). +(2, 2)λ (2, 2) + (2, 1)λ (2, 1) + (2, 0)λ (2,0) +(2, 1)λ (2,1) + (2, 2)λ (2,2)
43 OptIntro 35 / 38 SOS2 Model Example: SOS2 Model 1 = x X y Y λ x,y = λ ( 2, 2) + λ ( 2, 1) + λ ( 2,0) + λ ( 2,1) + λ ( 2,2). +λ (2, 2) + λ (2, 1) + λ (2,0) + λ (2,1) + λ (2,2)
44 OptIntro 36 / 38 SOS2 Model SOS2 Model The constraint family η x = y Y λ x,y, x X becomes η x 2 = λ ( 2, 2) + λ ( 2, 1) + λ ( 2,0) + λ ( 2,1) + λ ( 2,2) η x 1 = λ ( 1, 2) + λ ( 1, 1) + λ ( 1,0) + λ ( 1,1) + λ ( 1,2). =. η x 2 = λ (2, 2) + λ (2, 1) + λ (2,0) + λ (2,1) + λ (2,2)
45 OptIntro 37 / 38 SOS2 Model SOS2 Model The constraint family η y = x X λ x,y, y Y becomes η y 2 = λ ( 2, 2) + λ ( 1, 2) + λ (0, 2) + λ (1, 2) + λ (2, 2) η y 1 = λ ( 2, 1) + λ ( 1, 1) + λ (0, 1) + λ (1, 1) + λ (2, 1). =. η y 2 = λ ( 2,2) + λ ( 1,2) + λ (0,2) + λ (1,2) + λ (2,2)
46 OptIntro 38 / 38 SOS2 Model Piecewise-Linear Approximation: Multidimensional End! Thank you for your attention.
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