Some modelling aspects for the Matlab implementation of MMA

Transcription

1 Some modellng aspects for the Matlab mplementaton of MMA Krster Svanberg Optmzaton and Systems Theory Department of Mathematcs KTH, SE Stockholm September Consdered optmzaton problem. The Matlab verson of the author s MMA code s based on the assumpton that the users optmzaton problem s wrtten on the followng form, where the optmzaton varables are x = (x 1,..., x n ) T, y = (y 1,..., y m ) T and z. m f 0 (x) a 0 z (c y 1 2 d y 2 ) subect to f (x) a z y 0, = 1,..., m y 0, z 0. x x max, = 1,..., n = 1,..., m (1.1) Here, x 1,..., x n are the true optmzaton varables, whle y 1,..., y m and z are artfcal optmzaton varables whch wll be motvated below. f 0, f 1,..., f m are gven, contnuously dfferentable, real-valued functons. and x max are gven real numbers whch satsfy < x max. a 0 and a are gven real numbers whch satsfy a 0 > 0 and a 0. c and d are gven real numbers whch satsfy c 0, d 0 and c d > Ordnary NLP problems. Assume that the user wants to solve a problem on the followng standard form for nonlnear programmng. f 0 (x) subect to f (x) 0, = 1,..., m x x max, = 1,..., n (2.1) To put (1.1) nto ths form (2.1), frst let a 0 = 1 and a = 0 for all. Then z = 0 n any optmal soluton of (1.1). Further, for each, let d = 0 and c = a large number, so that the varables y become expensve. Then typcally y =0 n any optmal soluton of (1.1), and the correspondng x s an optmal soluton of (2.1). 1

2 It should be noted that the problem (1.1) always has feasble solutons, and n fact also at least one optmal soluton. Ths holds even f the user s problem (2.1) does not have any feasble solutons, n whch case some y > 0 n the optmal soluton of (1.1). Ths s ust one advantage of the formulaton (1.1) compared to the formulaton (2.1). Now some practcal consderatons and recommendatons. In many applcatons, the constrants are on the form σ (x) σ max, where σ (x) stands for e.g. a certan stress, whle σ max s the largest permtted value on ths stress. Ths means that f (x) = σ (x) σ max (n (1.1) as well as n (2.1)). The user should then preferably scale the constrants n such a way that 1 σ max 100 for each (and not σ max = ). The obectve functon f 0 (x) should preferably be scaled such that 1 f 0 (x) 100 for reasonable values on the varables. The varables x should preferably be scaled such that 0.1 x max 100, for all. Concernng the large numbers on the coeffcents c (mentoned above), the user should for numercal reasons try to avod extremely large values on these coeffcents (lke ). It s better to start wth reasonably large values and then, f t turns out that not all y = 0 n the optmal soluton of (1.1), ncrease the correspondng values of c by e.g. a factor 100 and solve the problem agan, etc. If the functons and the varables have been scaled accordng to above, then resonably large values on the parameters c could be, say, c = 1000 or Fnally, concernng the smple bound constrants x x max, t may sometmes be the case that some varables x do not have any prescrbed upper and/or lower bounds. In that case, t s n practce always possble to choose artfcal bounds and x max such that every realstc soluton x satsfes the correspondng bound constrants. The user should then preferably avod choosng x max unnecessarly large. It s better to try some reasonable bounds and then, f t turns out that some varable x becomes equal to such an artfcal bound n the optmal soluton of (1.1), change ths bound and solve the problem agan (startng from the recently obtaned soluton), etc. 3. Least squares problems. (Mnmum 2 norm problems.) Assume that the user wants to solve a constraned least squares problem on the form p (h (x)) 2 subect to g (x) 0, = 1,..., q x x max, = 1,..., n (3.1) where h and g are gven dfferentable functons. 2

3 The functons f and the parameters a, c and d should then be chosen as follows n problem (1.1). m = 2p q, f p (x) = h (x), = 1,..., p f 2p (x) = g (x), = 1,..., q a = 0, = 1,..., m d = 2, = 1,..., 2p d 2p = 0, = 1,..., q c = 0, = 1,..., 2p c 2p = large number, = 1,..., q 4. Mnmum 1 norm problems. Assume that the user wants to solve a mnmum 1 norm problem on the form p h (x) subect to g (x) 0, = 1,..., q x x max, = 1,..., n (4.1) where h and g are gven dfferentable functons. The functons f and the parameters a, c and d should then be chosen as follows n problem (1.1). m = 2p q, f p (x) = h (x), = 1,..., p f 2p (x) = g (x), = 1,..., q a = 0, = 1,..., m d = 0, = 1,..., m c = 1, = 1,..., 2p c 2p = large number, = 1,..., q 5. Mnmax problem. (Mnmum norm problems.) Assume that the user wants to solve a mnmax problem on the form max (x) },..,p subect to g (x) 0, = 1,..., q where h and g are gven dfferentable functons. x x max, = 1,..., n (5.1) 3

4 The functons f and the parameters a, c and d should then be chosen as follows n problem (1.1). m = 2p q, f p (x) = h (x), = 1,..., p f 2p (x) = g (x), = 1,..., q a = 1, = 1,..., 2p a 2p = 0, = 1,..., q d = 0, = 1,..., m c = large number, = 1,..., m 6. The MMA subproblem MMA s a method for solvng problems on the form (1.1), usng the followng approach: In each teraton, the current teraton pont (x, y, z ) s gven. Then an approxmatng explct subproblem s generated. In ths subproblem, the functons f (x) are replaced by approxmatng convex functons f (x). These approxmatons are based manly on gradent nformaton at the current teraton pont, but also (mplctly) on nformaton from prevous teraton ponts. The subproblem s solved, and the unque optmal soluton becomes the next teraton pont (x (k1), y (k1), z (k1) ). Then a new subproblem s generated, etc. The subproblem mentoned above looks as follows. subect to m f 0 (x) a 0 z (c y 1 2 d y 2 ) f (x) a z y 0, = 1,..., m α y 0, z 0. x β, = 1,..., n = 1,..., m (6.1) The approxmatng functons f (x) = n =1 f p u x (x) are chosen as q x l r, = 0, 1,..., m, where p q = (u = (x ) x ) ( 2 f (x ) κ, x ) l ) ( 2 f (x ) κ, x 4

5 Here, r = f (x ) α β n =1 u p x x q = max{, 0.9l 0.1x }, = mn{x max, 0.9u 0.1x }. l, ( ) f (x ) = max{0, f (x )} and x x ( ) f (x ) = max{0, f (x )}. x x The default rules for updatng the lower asymptotes l and the upper asymptotes u are as follows. The frst two teratons, when k = 1 and k = 2, l u In later teratons, when k 3, l u = x = x = x = x 0.5(x max 0.5(x max ), ). γ (x (k 1) l (k 1) ), γ (u (k 1) ), where γ = 0.7 f (x 1.2 f (x 1 f (x )(x (k 1) x (k 2) ) < 0, )(x (k 1) x (k 2) ) > 0, )(x (k 1) x (k 2) ) = 0. The default values of the parameters κ are κ = 10 3 f (x ) x 10 6 u l, for = 0, 1,.., m and = 1,.., n. (6.2) Ths mples that all the approxmatng functons are strctly convex, whch n turn mples that there s always a unque optmal soluton of the MMA subproblem. Regardless of the values of the parameters κ, the functons f are always frst order approxmatons of the orgnal functons f at the current teraton pont,.e. f f (x ) = f (x ) and f x (x ) = f x (x ). (6.3) 5