Deterministic Global Optimization for Dynamic Systems Using Interval Analysis Youdong Lin and Mark A. Stadtherr Department of Chemical and Biomolecular Engineering University of Notre Dame, Notre Dame, IN, USA 12th GAMM–IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics Duisburg, Germany, 26–29 September 2006
Outline • Background • Tools – Interval Analysis – Taylor Models – Validated Solutions of Parametric ODE Systems • Algorithm Summary • Computational Studies • Concluding Remarks 2
Background • Many practically important physical systems are modeled by ODE systems. • Optimization problems involving these models may be stated as min φ [ x µ ( θ ) , θ ; µ = 0 , 1 , . . . , r ] θ , x µ s .t. x = f ( x , θ ) ˙ x 0 = x 0 ( θ ) x µ ( θ ) = x ( t µ , θ ) t ∈ [ t 0 , t r ] θ ∈ Θ • Sequential approach: Eliminate x µ using parametric ODE solver, obtaining an unconstrained problem in θ • May be multiple local solutions – need for global optimization 3
Deterministic Global Optimization of Dynamic Systems Much recent interest, mostly combining branch-and-bound and relaxation techniques, e.g., • Esposito and Floudas (2000): α -BB approach – Rigorous values of α not used: no theoretical guarantees • Chachuat and Latifi (2003): Theoretical guarantee of ǫ -global optimality • Papamichail and Adjiman (2002, 2004): Theoretical guarantee of ǫ -global optimality • Singer and Barton (2006): Theoretical guarantee of ǫ -global optimality – Use convex underestimators and concave overestimators to construct two bounding IVPs, which are then solved to obtain lower and upper bounds on the state trajectories. – Bounding IVPs are not solved rigorously, so state bounds are not computationally guaranteed. 4
Deterministic Global Optimization of Dynamic Systems Our approach: Branch-and-reduce algorithm based on interval analysis and using Taylor models • Basic ideas – Use local optimizations to obtain an upper bound � φ on the global minimum – Compute Taylor models of the state variables using a new validated solver for parametric ODEs (VSPODE) (Lin and Stadtherr, 2006) – Compute the Taylor model T φ of the objective function – Perform constraint propagation procedure using T φ ≤ � φ , to reduce the parameter (decision variable) space Θ – Use branch-and-bound • Can implement to obtain either an ǫ -global minimum, or (using interval Newton approach) the exact ( ǫ = 0 ) global minimum 5
Interval Analysis • A real interval X = [ a, b ] = { x ∈ ℜ | a ≤ x ≤ b } is a segment on the real number line. • An interval vector X = ( X 1 , X 2 , ..., X n ) T is an n -dimensional rectangle. • Basic interval arithmetic for X = [ a, b ] and Y = [ c, d ] is X op Y = { x op y | x ∈ X, y ∈ Y } • Interval elementary functions (e.g., exp( X ) , sin( X ) ) are also available. • The interval extension F ( X ) encloses all values of f ( x ) for x ∈ X ; that is, { f ( x ) | x ∈ X } ⊆ F ( X ) • Interval extensions computed using interval arithmetic may lead to overestimation of function range (the interval “dependency” problem). 6
Taylor Models • Taylor Model T f = ( p f , R f ) : Bounds a function f ( x ) over X using a q -th order Taylor polynomial p f and an interval remainder bound R f . • Could obtain T f using a truncated Taylor series: q � i ! [( x − x 0 ) · ▽ ] i f ( x 0 ) 1 p f = i =0 ( q +1)! [( x − x 0 ) · ▽ ] q +1 F [ x 0 + ( x − x 0 ) ζ ] 1 R f = where, x 0 ∈ X ; ζ ∈ [0 , 1] � [ g · ▽ ] k = ∂ k j 1 ! ··· j m ! g j 1 k ! 1 · · · g j m m ∂x j 1 1 ··· ∂x jm m j 1+ ··· + jm = k 0 ≤ j 1 , ··· ,jm ≤ k • Can also compute Taylor models by using Taylor model operations (Makino and Berz, 1996) 7
Taylor Model Operations • Let T f and T g be the Taylor models of the functions f ( x ) and g ( x ) , respectively, over the interval x ∈ X . • Addition: T f ± g = ( p f ± g , R f ± g ) = ( p f ± p g , R f ± R g ) • Multiplication: T f × g = ( p f × g , R f × g ) with p f × g = p f × p g − p e and R f × g = B ( p e ) + B ( p f ) × R g + B ( p g ) × R f + R f × R g • B ( p ) indicates an interval bound on the function p . • Reciprocal operation and intrinsic functions can also be defined. • Store and operate on coefficients of p f only. Floating point errors are accumulated in R f . • Beginning with Taylor models of simple functions, Taylor models of very complicated functions can be computed. • Compared to other rigorous bounding methods (e.g., interval arithmetic), Taylor models often yield sharper bounds for modest to complicated functional dependencies (Makino and Berz, 1999). 8
Taylor Models – Range Bounding • Exact range bounding of the interval polynomials – NP hard • Direct evaluation of the interval polynomials – overestimation • Focus on bounding the dominant part (1st and 2nd order terms) • Exact range bounding of a general interval quadratic – also worst-case exponential complexity • A compromise approach – Exact bounding of 1st order and diagonal 2nd order terms � � m � a i ( X i − x i 0 ) 2 + b i ( X i − x i 0 ) B ( p ) = + S i =1 � � � � 2 m � − b 2 X i − x i 0 + b i i = a i + S, 2 a i 4 a i i =1 where S is the interval bound of other terms by direct evaluation 9
Taylor Models – Constraint Propagation • Consider constraint c ( x ) ≤ 0 , x ∈ X . Goal – Eliminate parts of X in which constraint cannot be satisfied • For each i = 1 , 2 · · · , m , shrink X i using: � � 2 − b 2 X i − x i 0 + b i i B ( T c ) = B ( p c ) + R c = a i + S i ≤ 0 2 a i 4 a i and V i = b 2 with U i = X i − x i 0 + b i a i U 2 i ⇒ i ≤ V i , − S i = 2 a i 4 a i ∅ if a i > 0 and V i < 0 � � � � V i V i − if a i > 0 and V i ≥ 0 a i , a i = ⇒ U i = [ −∞ , ∞ ] if a i < 0 and V i ≥ 0 � � �� � � V i V i −∞ , − ∪ a i , ∞ if a i < 0 and V i < 0 a i � � U i + x i 0 − b i ⇒ X i = X i ∩ = 2 a i 10
Validated Solution of Parametric ODE Systems • Consider the parametric ODE system ˙ x = f ( x , θ ) x ( t 0 ) = x 0 ∈ X 0 θ ∈ Θ • Validated methods: – Guarantee there exists a unique solution x ( t ) in [ t 0 , t f ] , for each θ ∈ Θ and x 0 ∈ X 0 – Compute an interval X j that encloses all solutions of the ODE system at t j for θ ∈ Θ and x 0 ∈ X 0 • Tools are available – VNODE, COSY VI, AWA, etc. – May need to treat parameters as additional state variables with zero derivative • New tool – VSPODE (Lin and Stadtherr, 2006): Deals directly with interval-valued parameters (and also interval-valued initial states) 11
New Method for Parametric ODEs • Use interval Taylor series to represent dependence on time. Use Taylor models to represent dependence on uncertain quantities (parameters and initial states). • Assuming X j is known, then – Phase 1: Same as “standard” approach (e.g., VNODE). Compute a coarse enclosure ˜ X j and prove existence and uniqueness. Use fixed point iteration with Picard operator using high-order interval Taylor series. – Phase 2: Refine the coarse enclosure to obtain X j +1 . Use Taylor models in terms of the uncertain parameters and initial states. • Implemented in VSPODE (Validating Solver for Parametric ODEs) (Lin and Stadtherr, 2006). 12
Method for Phase 2 • Represent uncertain initial states and parameters using Taylor models T x 0 and T θ , with components T x i 0 = ( m ( X i 0 ) + ( x i 0 − m ( X i 0 )) , [0 , 0]) , i = 1 , · · · , m T θ i = ( m (Θ i ) + ( θ i − m (Θ i )) , [0 , 0]) , i = 1 , · · · , p. • Bound the interval Taylor series coefficients f [ i ] by computing Taylor models T f [ i ] . – Use mean value theorem. – Evaluate using Taylor model operations. • Reduce “wrapping effect” by using a new type of Taylor model involving a parallelpiped remainder bound. • This results in a Taylor model T x j +1 in terms of the initial states x 0 and parameters θ . • Compute the enclosure X j +1 = B ( T x j +1 ) by bounding over X 0 and Θ . 13
VSPODE Example • Lotka-Volterra Problem x 1 = θ 1 x 1 (1 − x 2 ) ˙ x 2 = θ 2 x 2 ( x 1 − 1) ˙ x 0 = (1 . 2 , 1 . 1) T θ 1 ∈ [2 . 99 , 3 . 01] θ 2 ∈ [0 . 99 , 1 . 01] • Integrate from t 0 = 0 to t N = 10 . • VSPODE run using q = 5 (order of Taylor model), k = 17 (order of interval Taylor series) and QR for wrapping. • For comparison, VNODE was used, with interval parameters treated as additional state variables, and run using k = 17 order interval Hermite-Obreschkoff and QR for wrapping. • Constant step size of h = 0 . 1 used in both VSPODE and VNODE (step size may be reduced if necessary in Phase 1). 14
Lotka-Volterra Problem 1.5 1.4 1.3 ← X 1 1.2 ← X 2 1.1 x 1 0.9 0.8 0.7 0.6 VSPODE VNODE 0.5 0 1 2 3 4 5 6 7 8 9 10 t (Eventual breakdown of VSPODE at t = 31 . 8 ) 15
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