Deterministic Global Optimization Algorithm and Nonlinear Dynamics - PowerPoint PPT Presentation

Deterministic Global Optimization Algorithm and Nonlinear Dynamics C. S. Adjiman and I. Papamichail Centre for Process Systems Engineering Department of Chemical Engineering and Chemical Technology Imperial College London Global Optimization Theory Institute, Argonne National Laboratory September 8-10, 2003 Financial support : Engineering and Physical Sciences Research Council 1

Outline • Motivation • Dynamic Optimization: Problem and Methods • Convex Relaxation of the Dynamic Information • Deterministic Global Optimization Algorithm • Convergence of the Algorithm • Case Studies • Conclusions and Perspectives 2

Applications of Dynamic Optimization • Chemical systems − Determination of Kinetic Constants from Time Series Data (Parameter Estimation) − Optimal Control of Batch and semi-Batch Chemical Reactors − Safety Analysis of Industrial Processes • Biological and ecological systems • Economic and other dynamic systems 3

Dynamic Optimization Problem Solution approaches • Simultaneous approach: full discretization min J ( x ( t i , p ) , p ; i = 0 , 1 , ..., NP ) p • Sequential approach: subject to: p p 0 opt NLP Algorithm g i ( x ( t i , p ) , p ) ≤ 0 , i = 0 , 1 , ..., NP p L ≤ p ≤ p U p x = f ( t, x, p ) , ∀ t ∈ [ t 0 , t NP ] ˙ x,x p x ( t 0 , p ) = x 0 ( p ) Integrator 4

Usually nonconvex NLP Problems • Multiple optimum solutions exist • Commercially available numerical solvers guarantee local optimum solutions • Poor economic performance Algorithm classes (combined with simultaneous or sequential approach) Stochastic algorithms Deterministic algorithms 5

Global optimization methods Simultaneous approaches • Application of global optimization algorithms for NLPs • Issues: problem size; quality of discretization. • Smith and Pantelides (1996): spatial BB + reformulation • Esposito and Floudas (2000): α BB algorithm 6

Global optimization methods: Sequential approaches • Stochastic algorithms – Luus et al. (1990): direct search procedure. – Banga and Seider (1996), Banga et al. (1997): randomly directed search. • Deterministic algorithms – New techniques for convex relaxation of time-dependent parts of problem – Lack of analytical forms for the constraints / objective function – Esposito and Floudas (2000): extension of α BB to handle nonlinear dynamics. – Singer and Barton (2002): convex relaxation of integral objective function with linear dynamics – Papamichail and Adjiman (2002): convex relaxations of nonlinear dynamics. 7

Outline • Motivation • Dynamic Optimization: Problem and Methods • Convex Relaxation of the Dynamic Information • Deterministic Global Optimization Algorithm • Convergence of the Algorithm • Case Studies • Conclusions and Perspectives 8

Reformulated NLP Problem min x,p J (ˆ x i , p ; i = 0 , 1 , ..., NP ) ˆ subject to: g i (ˆ x i , p ) ≤ 0 , i = 0 , 1 , ..., NP ˆ x i = x ( t i , p ) , i = 0 , 1 , ..., NP p ∈ [ p L , p U ] x = f ( t, x, p ) , ∀ t ∈ [ t 0 , t NP ] ˙ x ( t 0 , p ) = x 0 ( p ) 9

Convex Relaxation of J and g i (1) f ( z ) = f CT ( z ) + � bt i =1 b i z B i , 1 z B i , 2 + � ut i =1 f UT,i ( z i ) + � nt i =1 f NT,i ( z ) Underestimating Bilinear Terms (McCormick, 1976) [ z L 1 , z U 1 ] × [ z L 2 , z U w = z 1 z 2 over 2 ] w ≥ z L 1 z 2 + z L 2 z 1 − z L 1 z L 2 w ≥ z U 1 z 2 + z U 2 z 1 − z U 1 z U 2 w ≤ z L 1 z 2 + z U 2 z 1 − z L 1 z U 2 w ≤ z U 1 z 2 + z L 2 z 1 − z U 1 z L 2 Underestimating Univariate Concave Terms f UT ( z ) = f UT ( z L ) + f UT ( z U ) − f UT ( z L ) ( z − z L ) [ z L , z U ] ⊂ ℜ ˘ over z U − z L 10

Convex Relaxation of J and g i (2) Underestimating General Nonconvex Terms in C 2 (Maranas and Floudas, 1994; Androulakis et al. , 1995) m α i ( z L i − z i )( z U [ z L , z U ] ⊂ ℜ m ˘ � f NT ( z ) = f NT ( z ) + i − z i ) over i =1 ˘ • f NT ( z ) is always less than f NT ˘ f NT ( z ) is convex if α i is big enough • f NT ( z ) = H f NT ( z ) + 2 diag( α i ) H ˘ Rigorous α calculations using the scaled Gerschgorin method (Adjiman et al. , 1998) ∀ z ∈ [ z L , z U ] H f NT ( z ) ∈ [ H f NT ] = H f NT ([ z L , z U ]) 11

Convex Relaxation of ˆ x i = x ( t i , p ) ˆ x i − x ( t i , p ) ≤ 0 x ( t i , p ) − ˆ x i ≤ 0 Constant bounds : x ( t i ) ≤ ˆ x i ≤ x ( t i ) Affine bounds : M ( t i ) p + N ( t i ) ≤ ˆ x i ≤ M ( t i ) p + N ( t i ) α -based bounds (Esposito and Floudas, 2000): r kij ( p L j − p j )( p U � x ik − x k ( t i , p ) + ˆ α − j − p j ) ≤ 0 j =1 r α + kij ( p L j − p j )( p U � x k ( t i , p ) + j − p j ) − ˆ x ik ≤ 0 j =1 12

Illustrative example x ( t ) = − x ( t ) 2 + p , ∀ t ∈ [0 , 1] ˙ x (0) = 9 p ∈ [ − 5 , 5] 10 p L =−5 8 p U =5 6 How do we find 4 bounding x p=5 2 p=2 p=0 trajectories? 0 p=−2 −2 p=−5 −4 0 0.2 0.4 0.6 0.8 1 t 13

Differential Inequalities (1) Consider the following parameter dependent ODE: x = f ( t, x, p ) , ∀ t ∈ [ t 0 , t NP ] ˙ x ( t 0 , p ) = x 0 ( p ) p ∈ [ p L , p U ] ⊂ ℜ r x ∈ ℜ n , f : ( t 0 , t NP ] × ℜ n × [ p L , p U ] �→ ℜ n and where x and ˙ x 0 : [ p L , p U ] �→ ℜ n . Let x = ( x 1 , x 2 , ..., x n ) T and x k − = ( x 1 , x 2 , ..., x k − 1 , x k +1 , ..., x n ) T . The notation f ( t, x, p ) = f ( t, x k , x k − , p ) is used. Theory on diff. inequalities (Walter, 1970) has been extended. 14

Differential Inequalities (2) Bounds on the solutions of the parameter dependent ODE: x k = inf f k ( t, x k , [ x k − , x k − ] , [ p L , p U ]) ˙ x k = sup f k ( t, x k , [ x k − , x k − ] , [ p L , p U ]) ˙ ∀ t ∈ [ t 0 , t NP ] and k = 1 , ..., n x ( t 0 ) = inf x 0 ([ p L , p U ]) x ( t 0 ) = sup x 0 ([ p L , p U ]) x ( t ) is a subfunction and x ( t ) is a superfunction for the solution of the ODE, i.e., x ( t ) ≤ x ( t, p ) ≤ x ( t ) , ∀ p ∈ [ p L , p U ] , ∀ t ∈ [ t 0 , t NP ] 15

Quasi-monotonicity Definition 1: Let g(x) be a mapping g : D �→ ℜ with D ⊆ ℜ n . Again the notation g ( x ) = g ( x k , x k − ) is used. The function g is called unconditionally partially isotone (antitone) on D with respect to the variable x k if g ( x k , x k − ) ≤ g (˜ x k , x k − ) for x k ≤ ˜ x k ( x k ≥ ˜ x k ) and for all ( x k , x k − ) , (˜ x k , x k − ) ∈ D . Definition 2: Let f ( t, x ) = ( f 1 ( t, x ) , ..., f 2 ( t, x )) T and each f k ( t, x k , x k − ) be unconditionally partially isotone on I 0 × ℜ × ℜ n − 1 with respect to any component of x k − , but not necessarily with respect to x k . Then f is quasi-monotone increasing on I 0 × ℜ n with respect to x (Walter, 1970) 16

Example: Constant bounds x ( t ) = − x ( t ) 2 + 5 x ( t ) = − x ( t ) 2 − 5 ˙ ˙ x (0) = 9 x (0) = 9 10 p L =−5 8 p U =5 6 4 x x 2 0 x −2 −4 0 0.2 0.4 0.6 0.8 1 t 17

Parameter Dependent Bounds Let f ( t, x, p ) ≤ f ( t, x, p ) ∀ x ∈ [ x ( t ) , x ( t )] , ∀ p ∈ [ p L , p U ] , ∀ t ∈ [ t 0 , t NP ] and x 0 ( p ) ≤ x 0 ( p ) ∀ p ∈ [ p L , p U ] , where f : [ t 0 , t NP ] × ℜ n × [ p L , p U ] �→ ℜ n and x 0 : [ p L , p U ] �→ ℜ n . If f is quasi-monotone increasing w.r.t. x and x ( t, p ) is the solution of the ODE: x = f ( t, x, p ) , ∀ t ∈ [ t 0 , t NP ] ˙ x ( t 0 , p ) = x 0 ( p ) p ∈ [ p L , p U ] x ( t, p ) ≤ x ( t, p ) , ∀ p ∈ [ p L , p U ] , ∀ t ∈ [ t 0 , t NP ] . then 18

Affine Bounds Let f ( t, x, p ) = A ( t ) x + B ( t ) p + C ( t ) and x 0 ( p ) = Dp + E , where A ( t ), B ( t ) and C ( t ) are continuous on [ t 0 , t NP ]. Then the analytical solution is (Zadeh and Desoer, 1963): � t � � x ( t, p ) = Φ( t, t 0 ) D + Φ( t, τ ) B ( τ ) dτ p t 0 � t +Φ( t, t 0 ) E + Φ( t, τ ) C ( τ ) dτ, t 0 where Φ( t, t 0 ) is the transition matrix: ˙ Φ( t, t 0 ) = A ( t )Φ( t, t 0 ) ∀ t ∈ [ t 0 , t NP ] Φ( t 0 , t 0 ) = I and I is the identity matrix. x ( t, p ) = M ( t ) p + N ( t ). 19

M ( t i ), N ( t i ) calculation 1. Apply x ( t i , p ) = M ( t i ) p + N ( t i ) for r + 1 values of p 2. Calculate x ( t i , p ) for the r + 1 values of p from the integration of the linear ODE 3. Solve n linear systems to find the r +1 unknowns for each one of the n dimensions of x 20

Example: Affine bounds Underestimating IVP x = − ( x + x ) x + xx + v ∀ t ∈ [0 , 1] ˙ x (0 , v ) = 9 Overestimating IVPs x 1 = − 2 xx 1 + x 2 + v ∀ t ∈ [0 , 1] ˙ x 1 (0 , v ) = 9 x 2 = − 2 xx 2 + x 2 + v ∀ t ∈ [0 , 1] ˙ x 2 (0 , v ) = 9 21

Example: Affine bounds for p = 0 10 p L =−5 8 p U =5 6 4 x(t,p=0) x 2 0 x −2 −4 0 0.2 0.4 0.6 0.8 1 t 22

α calculation for x k ( t i , p ) [ H x k ( t i ) ] ∋ H x k ( t i ) ( p ) = ∇ 2 x k ( t i , p ) , ∀ p ∈ [ p L , p U ] ⊂ ℜ r i = 0 , 1 , ..., NS, k = 1 , 2 , ..., n 1. 1st and 2nd order sensitivity equations 2. Create bounds using Differential Inequalities 3. Construct the interval Hessian matrix 4. Calculate using the scaled Gerschgorin α method 23

Example: All bounds 6 2 3 0 x(t=1,p) 0 x(t=1,p) −3 −6 −2 −9 −12 −15 −4 −5 −2.5 0 2.5 5 −5 −2.5 0 2.5 5 p p 24

Outline • Motivation • Dynamic Optimization: Problem and Methods • Convex Relaxation of the Dynamic Information • Deterministic Global Optimization Algorithm • Convergence of the Algorithm • Case Studies • Conclusions and Perspectives 25