NSIPS: Nonlinear Semi-Infinite Programming Solver A. Ismael F. Vaz Edite M.G.P. Fernandes M. Paula S.F. Gomes Production and Systems Department Mechanical Engineering Department Engineering School Imperial College of Science, Minho University Technology and Medicine Braga - Portugal London SW7 2BX - UK { aivaz,emgpf } @dps.uminho.pt p.gomes@ic.ac.uk 18-20 June, 2004 Talk partially financed by the Portuguese Calouste Gulbenkian Foundation
HPSNO’04 - Ischia - A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes 1 Contents • Semi-Infinite Programming (SIP) • Nonlinear SIP Solver - (NSIPS) ⋆ Discretization method ⋆ Sequential quadratic programming method ⋆ Constraint transcription methods ∗ Interior point method ∗ Penalty (multiplier) method • Numerical results / Conclusions
HPSNO’04 - Ischia - A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes 2 Semi-Infinite Programming x ∈ R n f ( x ) min s.t. g i ( x, t ) ≤ 0 , i = 1 , ..., m (1) h i ( x ) ≤ 0 , i = 1 , ..., o h i ( x ) = 0 , i = o + 1 , ..., q ∀ t ∈ T where f ( x ) is the objective function, h i ( x ) are the finite constraint functions, g i ( x, t ) are the infinite constraint functions and T ⊂ R r is, usually, a cartesian product of intervals ( [ α 1 , β 1 ] × [ α 2 , β 2 ] × · · · × [ α r , β r ] ).
HPSNO’04 - Ischia - A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes 3 Solver NSIPS (Nonlinear Semi-Infinite Programming Solver)
HPSNO’04 - Ischia - A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes 3 Solver NSIPS (Nonlinear Semi-Infinite Programming Solver) Version 2.1 publicly available in the internet http://www.norg.uminho.pt/aivaz/ , implementing the four methods in a total of seven algorithms.
HPSNO’04 - Ischia - A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes 3 Solver NSIPS (Nonlinear Semi-Infinite Programming Solver) Version 2.1 publicly available in the internet http://www.norg.uminho.pt/aivaz/ , implementing the four methods in a total of seven algorithms. NSIPS receives the problem in the (SIP)AMPL format.
HPSNO’04 - Ischia - A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes 3 Solver NSIPS (Nonlinear Semi-Infinite Programming Solver) Version 2.1 publicly available in the internet http://www.norg.uminho.pt/aivaz/ , implementing the four methods in a total of seven algorithms. NSIPS receives the problem in the (SIP)AMPL format. The method is select by the nsips options , and there are many other options for each method.
HPSNO’04 - Ischia - A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes 3 Solver NSIPS (Nonlinear Semi-Infinite Programming Solver) Version 2.1 publicly available in the internet http://www.norg.uminho.pt/aivaz/ , implementing the four methods in a total of seven algorithms. NSIPS receives the problem in the (SIP)AMPL format. The method is select by the nsips options , and there are many other options for each method. The NPSOL software is used to solve the finite subproblems (discretization and SQP methods).
HPSNO’04 - Ischia - A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes 4 Discretization method - Three versions A sequence of finite problems are solved. The finite problems are obtained from the SIP problem where the infinite constraints are evaluated at a finite set of points ˜ T [ h k ] ⊆ T [ h k ] , where T [ h k ] ⊆ T is a uniform grid of points with space h k . Versions adapted for nonlinear SIP and implemented: • Hettich (1986, 1990) • Reemtsen (1991) • Hettich with pseudo-number generation.
HPSNO’04 - Ischia - A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes 5 Discretization method • Step 0: Define T [ h 0 ] . Let � T [ h 0 ] = T [ h 0 ] . Solve the NLP( � T [ h 0 ] ) and let x 0 be the solution found. • Step k : If x k − 1 is not feasible for all the points in the set T [ h k − 1 ] ⋆ then: Insert all the infeasible points in the set � T [ h k − 1 ] . Solve the NLP( � T [ h k − 1 ] ) and let x k − 1 be the solution found. Continue with step k . ⋆ else: If the maximum number of refinements is reached then stop. Else build the set � T [ h k ] from T [ h k ] and � T [ h k − 1 ] . Solve the NLP( � T [ h k ] ) and let x k be the solution found. Go to step k + 1 .
HPSNO’04 - Ischia - A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes 6 Reduced problem Problem with no finite constraints and only one infinite variable. x ∈ R n f ( x ) min (2) s.t. g i ( x, t ) ≤ 0 , i = 1 , ..., m ∀ t ∈ T ≡ [ a, b ]
HPSNO’04 - Ischia - A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes 7 Sequential Quadratic Programming Considering the reduced problem (2), the sequential quadratic programming is based on the quadratic semi-infinite programming (QSI) d ∈ R n f Q ( d ) = 1 2 d T H k d + d T ∇ f ( x k ) min s.t. d T ∇ x g i ( x k , t ) + g i ( x k , t ) ≤ 0 , i = 1 , . . . , m, ∀ t ∈ [ a, b ] , where H k is an approximation to ∇ 2 xx L ( x k , v ) .
HPSNO’04 - Ischia - A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes 8 SQP The solution of the QSI problem is d k and x k +1 = x k + α k d k , k = 1 , 2 , . . . { x k } → x ∗ , solution to the initial SIP problem. The Lagrangian of the QSI problem is L Q ( d, v ) = 1 2 d T H k d + d T ∇ f ( x k ) � b m � � � d T ∇ x g j ( x k , t ) + g j ( x k , t ) + dv j ( t ) a j =1
HPSNO’04 - Ischia - A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes 9 Solving the QSI The dual problem min v ∈V ∗ L ∗ Q ( v ) ≡ −L Q ( d ( v ) , v ) is solved by a linear parametrization of the dual variables. w 1 j ( t − a ) , for t ∈ [ a, t 1 ); a ij + w i +1 j ( t − t i ) , for t ∈ [ t i , t i +1 ) , v j ( t ) = i = 1 , 2 , . . . , l − 1; a lj + w l +1 j ( t − t l ) , for t ∈ [ t l , b ] ; j = 1 , . . . , m , where i i � � a ij = h pj + w pj ( t p − t p − 1 ) , i = 1 , . . . , l p =1 p =1
HPSNO’04 - Ischia - A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes 10 Example with m = 2 , l = 2 w3 1 w 3 2 } h w2 2 1 1 w22 = 0 } h 1 1 w11 h 22 = 0 } h 12 w 1 2 t1 t t1 t2 a = t0 2 b = t3 a = t0 b = t3 w are the linear segments slope, h are the jumps and t are the discontinuity points.
HPSNO’04 - Ischia - A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes 11 Merit function � b m � φ ( x, µ ) = f ( x ) + 1 [ g i ( x, t )] 2 2 µ + dt a i =1 A strategy for computing the penalty parameter. Numerical integrals computation - Numerical adaptative formulae (Gaussian or trapezoid).
HPSNO’04 - Ischia - A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes 12 SQP - Dual method 1. Given x 0 . Let k = 0 and H 0 = I . 2. Compute H k using a BFGS quasi-Newton updating formula. 3. Solve the QSI problem to obtain the search direction d k . 4. If d k = 0 then stop. 5. Find α k such that x k +1 = x k + α k d k sufficiently decreases the merit function. 6. If there is not a major difference between x k +1 and x k then stop with x k +1 as an approximated solution. Otherwise do k = k + 1 and go to step 2.
HPSNO’04 - Ischia - A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes 13 Constraint transcription Considering the reduced problem (2), the infinite constraints � g i ( x, t ) ≤ 0 , ∀ t ∈ T , are transformed into T [ g i ( x, t )] + dt = 0 where [ z ] + = max { 0 , z } . The SIP is then transformed into x ∈ R n f ( x ) min � s.t. G i ( x ) ≡ [ g i ( x, t )] + dt = 0 T i = 1 , . . . , m Constraint functions not differentiable.
HPSNO’04 - Ischia - A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes 14 Approximate problem x ∈ R n f ( x ) min � s.t. G i,ǫ ( x ) ≡ g i,ǫ ( x, t ) dt = 0 T i = 1 , . . . , m with ǫ → 0 and 0 , if g i ( x, t ) < − ǫ ; ( g i ( x,t )+ ǫ ) 2 g i,ǫ ( x, t ) = if − ǫ ≤ g i ( x, t ) ≤ ǫ ; , 4 ǫ g i ( x, t ) , if g i ( x, t ) > ǫ, Once differentiable constraint functions.
HPSNO’04 - Ischia - A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes 15 Penalty method A sequence of subproblems is solved, parameterized by µ x ∈ R n φ S ( x, µ ) min for a sequence of increasing µ > 0 values.
HPSNO’04 - Ischia - A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes 16 Simple penalty functions � m � φ 1 S ( x, µ ) = f ( x ) + µ g i,ǫ ( x, t ) dt T i =1
HPSNO’04 - Ischia - A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes 16 Simple penalty functions � m � φ 1 S ( x, µ ) = f ( x ) + µ g i,ǫ ( x, t ) dt T i =1 � m � S ( x, µ ) = f ( x ) + µ φ 2 g i,ǫ ( x, t ) 2 dt 2 T i =1
HPSNO’04 - Ischia - A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes 16 Simple penalty functions � m � φ 1 S ( x, µ ) = f ( x ) + µ g i,ǫ ( x, t ) dt T i =1 � m � S ( x, µ ) = f ( x ) + µ φ 2 g i,ǫ ( x, t ) 2 dt 2 T i =1 and � � � m � e g i,ǫ ( x,t ) − 1 φ 3 S ( x, µ ) = f ( x ) + µ dt T i =1
HPSNO’04 - Ischia - A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes 17 Relaxed problem to satisfy LICQ x ∈ R n f ( x ) min s.t. G i,ǫ ( x ) ≤ τ i = 1 , . . . , m τ > 0 ( τ ( ǫ ) → 0)
HPSNO’04 - Ischia - A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes 18 Multiplier method A sequence of subproblems is solved x ∈ R n φ AL ( x, λ, µ ) min where φ AL is the augmented Lagrangian penalty function
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