And the full matrix is the concatenation of the two:
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1 Appendx A--A Step n the Dual Newton Method Alesandar Donev, 1/17/00 Ths Maple worsheet llustrates the basc step n the Dual Newton Method for convex networ optmzaton when the arcs have a superconductor-le cost functon. > restart > wth( lnalg > wth( plots Part I: Mang the networ node-arc ncdence matrx The actual node-arc ncdence matrx of ths graph s composed of two parts: The bass part correspondng to a spannng tree: > A B := A non-bass part correspondng to the non-tree arcs: > A N := And the full matrx s the concatenaton of the two: > A G := concat ( A B, A N Ths matrx has ran 6 even though t has 7 rows. So we should throw away one row correspondng to the root node: > ran( A G 6 Tae Root to be 1: > Root := 1 So the actual node-arc ncdence matrx we use s: > A := delrows ( A G, Root.. Root A :=
2 Ths s now ndeed a full-ran matrx: > ran( A 6 The number of arcs and nodes s thus: > m := 9 ; n := 6 Part II Cost and conjugate functons and dervatves: In the case of a superconductor, the cost functon s: ( x j j ξ ξ ( x j 1 R ξ dlog e + 1 dlog e + 1 ξ > f := ( x, + R ξ ln e + 1 x It's dervatve f x ( x (the voltage V ( I s: x j > Df := ( x, evalf sgnum( x R x 1 + tanh ξ The second dervatve x f( x s: x j R x j x 1 tanh > Df := ( x, R 1 + tanh ξ + ξ ξ And the most mportant one s the nverse of the voltage-current functon, ( f' ( 1 ( t, whch nvolves the LambertW functon: t j R ξ 1 ξ LambertW t e 1 t R ξ > nvdf := ( t, evalf sgnum( t + R The networ cost functon s the sum of the costs over all arcs: > F := x VecSum ( Invoe ( x, f Part III Random values for parameters: The source vector: > ST := [ seq (.01 randvector( 9, = 1.. n ] ST := [.85,.57,.54,.01,.94,.49] Resstances, crtcal currents and transton wdths: > R := [ seq (.01 randvector( 9, = 1.. m ] R := [.99,.19,.3,.61,.17,.04,.6,.10,.14]
3 > j := [ seq (.01 randvector( 9, = 1.. m ] j := [.35,.43,.14,.59,.93,.91,.03,.68,.66 ] > ξ := [ seq (.001 randvector( , = 1.. m ] ξ := [.037,.033,.059,.044,.090,.039,.036,.106,.09] Here s the random vector of voltages for the ntal guess: > V := [ seq (.01 randvector( 9, = 1.. n ] V := [.71,.40,.49, 0,.90,.0] Part IV Vectors and matrces used n the optmzaton The supply-demand vector: > b := evalm( ST For now, we do not assgn a specfc value to the vector of Lagrange multplers (voltages: > λ := vector( n The tenson vector (potental drops across arcs: > t := multply ( transpose( A, λ The flow vector (currents for the gven potentals can now be found as = ( f' ( 1 ( t : > := Invoe ( t, nvdf And the gradent s: > G := evalm ( multply ( A, b The dagonal of the conjugate Hessan s the nverse of the dagonal of the cost-functon Hessan: 1 > DH * := DagonalInvoe x, * ( x, Df ( x, And the full Hessan s: > H := multply ( A, DH *, transpose( A Part IV Tang a step toward the mnmum Ths pece of code would be repeated untl convergence to the global mnmum. We start by assgnng a random vector of voltages to the vector of Lagrange multplers: > λ := evalm( V The drecton vector s the soluton to the Newton system H d = G: > d := lnsolve ( map ( eval, H, evalm( G Now we do a lne search along ths drecton λ = λ + α d: > λ := evalm ( V + α d The objectve functon s the Lagrangan h( α = L λ + α d, where L λ = T t F( x* λ T d: > L λ := unapply ( dotprod ( map ( eval,, map ( eval, t, ' orthogonal ' F ( map ( eval, dotprod ( λ, b, ' orthogonal', α Ths functon should be concave and contnous: > plot ( L λ ( α, α = 0.. 1, numponts = 10, adaptve = false, labels = [ "alpha", "L[lambda](alpha" ] To fnd the mnmum (.e. perform the lne search, we fnd the dervatve h( α = α L λ ( α, where
4 α = α λ α h( α = d T G, and set t to zero: > h := unapply ( dotprod ( d, map ( eval, G, ' orthogonal', α The plot below shows that h(alpha s not contnous. But ths s an optcal lluson, t comes from the fact that there s a very sharp non-lnear transton n the V( I curve. The dervatve of ths functon = α h( α dt H d s ths practcally undefned (nfnte at certan ponts. The theory guarantees that h( α s pecewse convex, contnous, and non-decreasng. So Newton's method for the lne search mnmzaton, f used, must be used wth great care and n combnaton wth secant-le or bsecton methods! > dh := unapply ( dotprod ( d, multply ( map ( eval, H, d, ' orthogonal', α > plot ( h( α, α = , labels = ["alpha", "h(alpha" ] Now we fnd the zero of h( α --Beware, fsolve often fals here: > α := fsolve ( h( α = 0, α = 0..
5 α := And the new estmate for the potentals s: > V := map ( eval, λ [ , , , , , ] Ths of course has not converged yet.
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