Deterministic Global Optimization Algorithm and Nonlinear Dynamics

Transcription

1 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

2 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

3 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

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

5 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

6 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): αbb algorithm 6

7 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 αbb 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

8 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

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

10 Convex Relaxation of J and g i (1) f(z) = f CT (z) + bt i=1 b iz Bi,1z Bi,2 + ut i=1 f UT,i(z i ) + nt i=1 f NT,i(z) Underestimating Bilinear Terms (McCormick, 1976) w = z 1 z 2 over [z L 1, zu 1 ] [zl 2, zu 2 ] w z L 1 z 2 + z L 2 z 1 z L 1 zl 2 w z U 1 z 2 + z U 2 z 1 z U 1 zu 2 w z L 1 z 2 + z U 2 z 1 z L 1 zu 2 w z U 1 z 2 + z L 2 z 1 z U 1 zl 2 Underestimating Univariate Concave Terms f UT (z) = f UT (z L ) + f UT (z U ) f UT (z L ) z U z L (z z L ) over [z L, z U ] R 10

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

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

13 Illustrative example ẋ(t) = x(t) 2 + p, t [0, 1] x(0) = 9 p [ 5, 5] 10 8 p L = 5 p U =5 x p=5 p=2 p=0 p= 2 How do we find bounding trajectories? t p= 5 13

14 Differential Inequalities (1) Consider the following parameter dependent ODE: ẋ = f(t, x, p), t [t 0, t NP ] x(t 0, p) = x 0 (p) p [p L, p U ] R r where x and ẋ R n, f : (t 0, t NP ] R n [p L, p U ] R n 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

15 Differential Inequalities (2) Bounds on the solutions of the parameter dependent ODE: ẋ k = inf f k (t, x k, [x k, x k ], [p L, p U ]) ẋ k = sup f k (t, x k, [x k, x k ], [p L, p U ]) x(t 0 ) = inf x 0 ([p L, p U ]) x(t 0 ) = sup x 0 ([p L, p U ]) t [t 0, t NP ] and k = 1,..., n 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

16 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

17 Example: Constant bounds ẋ(t) = x(t) 2 5 ẋ(t) = x(t) x(0) = 9 x(0) = p L = 5 p U =5 x 4 2 x 0 2 x t 17

18 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: ẋ = f(t, x, p), t [t 0, t NP ] x(t 0, p) = x 0 (p) p [p L, p U ] then x(t, p) x(t, p), p [p L, p U ], t [t 0, t NP ]. 18

19 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): x(t, p) = { Φ(t, t 0 )D + +Φ(t, t 0 )E + where Φ(t, t 0 ) is the transition matrix: t t 0 Φ(t, τ)b(τ)dτ t } t 0 Φ(t, τ)c(τ)dτ, 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). p 19

20 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

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

22 Example: Affine bounds for p = p L = 5 p U =5 x(t,p=0) 4 2 x 0 x t 22

23 α calculation for x k (t i, p) [H xk (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 α using the scaled Gerschgorin method 23

24 Example: All bounds x(t=1,p) 3 6 x(t=1,p) p p 24

25 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

26 Spatial BB Algorithm (Horst and Tuy, 1996) Initialisation Upper and Lower Bounds Upper Bound Branch NO Lower Bound Subregion Selection u J - J l J R l R < ε r YES Terminate 26

27 Global optimization algorithm Step 1 Initialization: empty list of subregions, bounds on solution Step 2 First upper bound calculation (local optimization) Step 3 First lower bound calculation, including relaxation. Add subregions to list. Step 4 Subregion selection Terminate if list is empty Choose region with lowest lower bound otherwise Step 5 Check for convergence (relative tolerance, max iter) Step 6 Branch with standard rule (least reduced axis) Step 7 Upper bound for new regions (not needed at every iteration) Step 8 Lower bound calculation for each region. Go to Step 4. 27

28 Lower bound calculation for region R (Step 8) Let J L be the lower bound on the parent region of R. Let J U be the best known upper bound. Obtain bounds on the differential variables If affine bounds are used, obtain necessary matrices Form convex relaxation of problem for region R If a feasible solution with objective function JR L is obtained, then If affine bounds are used and JR L < JL, set JR L = JL If JR L JU, add R to the list of subregions 28

29 Outline of proof of convergence Three main properties are needed: Bound improvement after branching Constant bounds improve after branching α-based bounds improve after branching Affine bounds do not improve after branching. Ensure improvement through test in Step 8: if J L R < JL, set J L R = JL Bound improving selection operation Consequence of region selection criterion (Step 6) Consistent bounding operation Maximum distance between objective function and its relaxation converges to zero. Maximum distance between any constraint and its relaxation converges to zero. 29

30 Key elements of proof Bounds on the solutions of the ODE are such that: ẋ k = inf f k (t, x k, [x k, x k ], [p L, p U ]) inf f k (t, [x, x], [p L, p U ]) ẋ k = sup f k (t, x k, [x k, x k ], [p L, p U ]) sup f k (t, [x, x], [p L, p U ]) t [t 0, t NP ] and k = 1,..., n Inclusion monotonicity of interval operations ensures consistency of bounding operation with constant bounds. Similar approach can be taken to show α-based underestimators yield a consistent bounding operation: Interval Hessian matrix obtained through differential inequalities has desired properties. Hence, α and α-based bounds have desired properties. 30

31 Implementation of algorithm MATLAB 5.3 implementation NLPs: Use fmincon function (Optimization Toolbox) IVP solution: ode45 (Runge-Kutta based on Dormand-Prince pair) Interval calculations: INTLAB with directed outward rounding. Runs performed on an Ultra 60 workstation. 31

32 Case Study I: Parameter estimation for 1 st order reaction (Tjoa and Biegler, 1991) A k 1 B k 2 C subject to: min k 1,k j=1 i=1 (x i (t j ) x exp i (t j )) 2 ẋ 1 = k 1 x 1 t [0, 1] ẋ 2 = k 1 x 1 k 2 x 2 x 1 (0) = 1 x 2 (0) = 0 0 k k 2 10 Up to 8 affine underestimators and 8 affine overestimators can be constructed. α-values 0.5. Convexity is identified. 32

33 Results for case study I Obj. fun. = e-06; k 1 = ; k 2 = Underestimation Branching CPU time ɛ r Iter. scheme strategy (sec) C e-02 3,501 2,828 C e-03 34,508 22,959 C & A e C & A e C & α e C & α e C & α e C & α e C & A & α e C & A & α e

34 Case Study II: Parameter estimation for catalytic cracking of gas oil (Tjoa and Biegler, 1991) A k 1 Q k 3 k 2 S subject to: min k 1,k 2,k j=1 i=1 (x i (t j ) x exp i (t j )) 2 ẋ 1 = (k 1 + k 3 )x 21 t [0, 0.95] ẋ 2 = k 1 x 2 1 k 2x 2 x 1 (0) = 1 x 2 (0) = 0 0 k k k

35 Results for case study II Obj. fun. = e 03; k = ( , , ) T Underestimation Branching CPU time ɛ r Iter. scheme strategy (sec) C e-02 10,000 16,729 C e , ,816 C & A e ,597 C & A e ,478 C & α e ,415 C & α e ,524 C & α e ,116 C & α e , affine underestimators + 64 affine overestimators 35

36 Case Study III: Parameter estimation for reversible gas phase reaction (Bellman, 1967): 2NO + O 2 2NO 2 min k 1,k 2 14 j=1 (x(t = t j, k 1, k 2 ) x exp (t j )) 2 s.t. ẋ = k 1 (126.2 x)(91.9 x) 2 k 2 x 2 t [0, 39] x(t = 0, k 1, k 2 ) = 0 0 k k x is the difference of the pressure of the system from the initial pressure. k 1 and k 2 are the rate constants of the forward and reverse reactions. 36

37 Results for case study III Obj. fun.= ; k 1 = e-06; k 2 = e-04 Underestimation Branching CPU time ɛ r Iter. scheme strategy (sec) constant e-02 75,441 58,513 only constant bounds used 32 affine underestimators affine overestimators stiff systems, expensive to integrate quality of bounds not better than constant bounds quality of α-based bounds poor due to wrapping effect 37

38 Case Study IV: Optimal control with end-point constraint (Goh and Teo, 1988) Problem formulation using control vector parameterization: min u 1,u 2 x 2 (t = 1, u 1, u 2 ) s.t. ẋ 1 = u 1 (1 t) + u 2 t t ẋ 2 = x (u 1(1 t) + u 2 t) 2 x 1 (t = 0, u 1, u 2 ) = 1 x 2 (t = 0, u 1, u 2 ) = 0 x 1 (t = 1, u 1, u 2 ) 1 x 1 (t = 1, u 1, u 2 ) 1 1 u u 2 1 [0, 1] 16 affine underestimators + 2 affine overestimators 38

39 Results for case study IV Obj. fun.= e-01; u 1 = ; u 2 = Underestimation Branching CPU time ɛ r Iter. scheme strategy (sec) C e C e-03 1,062 1,106 C & A e C & A e C & α 1 or e α-based bounds recognize convexity of problem at root node. 39

40 Conclusions Three types of rigorous convex relaxations have been developed Convergence of the algorithm has been proved A BB global optimization algorithm has been applied successfully to case studies in parameter estimation and optimal control References: Papamichail and Adjiman, J. Glob Opt, Papamichail and Adjiman, Comp Chem Eng,

41 Perspectives Basic theoretical developments of recent years make global optimization of problems with nonlinear IVPs in the constraints possible. Practical applicability limited by cost of constructing underestimators and overestimators, quality of estimators for highly nonlinear systems (e.g. oscillatory) and long time horizons. Further research needed to identify classes of IVPs for which current estimators are effective, develop new estimators for other problem classes, establish basic theory for DAE systems. 41