11 3 1 stsp by column generation
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11.3.1 STSP by Column Generation PowerPoint PPT Presentation


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11.3.1 STSP by Column Generation. Recall the formulation of STSP (for 1-tree relaxation) minimize eE c e x e subject to  e({ i }) x e = 2 , i  N\{ 1}  e({1}) x e = 2 ,  eE (S) x e  |S| - 1 ,S  N\{ 1}, S  , N\{ 1},  e E (N\{ 1}) x e = |N| - 2.

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11.3.1 STSP by Column Generation

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11 3 1 stsp by column generation

11.3.1 STSP by Column Generation

  • Recall the formulation of STSP (for 1-tree relaxation)

    minimizeeEcexe

    subject toe({i})xe = 2 ,i  N\{1}

    e({1})xe = 2 ,

    eE(S)xe  |S| - 1 ,S  N\{1}, S  , N\{1},

    eE(N\{1})xe = |N| - 2.

    xe  { 0, 1 }.

    (note that we use N to denote the set of nodes instead of V)

  • min { eEcexe : e(i)xe = 2 for iN\{1}, xX1 }

    where X1 = { xZ+m : e(1)xe = 2, eE(S)xe  |S|-1 for   S  N\{1},

    eE(N\{1})xe = |N| - 2 }

    is the set of incidence vectors of one-trees.


11 3 1 stsp by column generation

  • STSP:

    min { eEcexe : e(i)xe = 2 for iN\{1}, xX1 }

    where X1 = { xZ+m : e(1)xe = 2, eE(S)xe  |S|-1 for   S  N\{1},

    eE(N\{1})xe = |N| - 2 }

    is the set of incidence vectors of one-trees.

  • Write xe = t : eE(t) t , t {0, 1}, where Gt = (N, Et) is the tth one-tree.

     e(i)xe = e(i) t : eE(t) t = t ditt = 2

    (dit is degree of node i in the one-tree Gt.)

    (In matrix notation, we have Ax = 2, where A is the node-edge incidence matrix (for edges E(N\{1})).

    x = T, where each column of T is incidence vector of a one-tree.

     AT = 2 (each column of AT is ATt ) )


11 3 1 stsp by column generation

  • min t=1T1 (cxt)t

    (DW) t=1T1ditt = 2, for i  N\{1}

    t=1T1 t = 1

    B|T1|

  • LP relaxation is:

    min t=1T1 (cxt)t

    (LPM) t=1T1ditt = 2, for i  N\{1}

    t=1T1 t = 1

    R+T1

  • Note that the convexity constraintt=1T1 t = 1 (t=1T12t = 2) may be regarded as d1t= 2 for node 1.

  • Subproblem: 1= min{eE(ce- ui- uj)xe - : xX1} (e=(i, j)E, i, jN\{1})

    ( cxt - iN\{1}ditui -  = cxt - iN\{1}ui  e(i)xet -  = eE (ce - ui - uj)xet -  )


11 3 2 strength of the lp master

11.3.2 Strength of the LP Master

  • Prop 11.1:

    zLPM = max { k=1Kckxk : k=1KAkxk = b, xk conv(Xk) for k = 1, … , K }

    Pf) LPM is obtained from IP by substituting

    xk = t=1Tkxk, tk, t, t=1Tkk, t = 1, k, t  0 for t = 1, … , Tk .

    This is equivalent to substituting xk  conv(Xk).

  • Prop 11.1 implies that the previous column generation approach for STSP provides the same bound as Lagrangean dual.

  • Let wLD be the value of the Lagrangian dual when the joint constraint k=1KAkxk = b are dualized.

    Let zCUT be the optimal value obtained when cutting planes are added to the LP relaxation of IP using exact separation algorithm for each of conv(Xk) for k = 1, … , K.

    Then Thm 11.2:zLPM= wLD = zCUT


11 3 1 stsp by column generation

  • Hence column generation approach (if it can be used for the problem) usually provides strong bounds.

  • It uses the dual variables obtained from LP relaxation as guides compared to simple rule of subgradient method for Lagrangian dual. (Some more discussions on this later.)

  • Generated columns can be kept for overall optimization. (compare to Lagrangian dual)

  • Consider column generation algorithm for generalized assignment problem and its merits compared to Lagrangian dual and cutting plane algorithms.

    (See Savelsbergh, A Branch and Price Algorithm for the Generalized Assignment Problem, Operations Research, 1997, Vol 45, No 6, p831-841)


11 4 ip column generation for 0 1 ip

11.4 IP Column Generation for 0-1 IP

  • Branch-and-price algorithm

    How to branch after column generation?

  • (IP)z = max { k=1Kckxk : k=1KAkxk = b,

    Dkxk  dk for k = 1, … , K, xk for k = 1, … , K},

    reformulation

    z = max k=1K t=1Tk ( ckxk, t ) k, t

    (IPM) k=1K t=1Tk ( Akxk, t ) k, t = b

    t=1Tkk, t = 1 for k = 1, … ,K

    k, t  {0, 1} for t = 1, … , Tk and k = 1, … , K

    if and only if is integer


11 3 1 stsp by column generation

  • If LP relaxation of IPM has fractional solution, two choices to branch : (1) on x variable, (2) on  variable.

    (Following are general schemes. Particular implementation may vary and/or even very difficult.)

  • Consider branch on x variables

    If fractional in LP optimal, there is some  and j such that the corresponding 0-1 variable xj has LP value that is fractional.

    Split the set S of all feasible solutions into S0 = S  { x: xj = 0} and S1 = S  { x: xj = 1}

    Now as xjk = t=1Tkk, txjk, t  {0, 1}, xjk =   {0, 1} implies that xjk, t =  for all k, t with k, t > 0


11 3 1 stsp by column generation

Hence the Master Problem at node Si = S  { x: xj = i} for i = 0, 1 is

z(Si) = max k   t ( ckxk, t ) k, t +

(IPM(Si)) k   t ( Akxk, t ) k, t + = b

t k, t = 1 for k  

= 1

k, t  {0, 1} for t = 1, … , Tk , k = 1, … , K

The columns for k =  are affected.

Column generation for subproblem  and i = 0, 1

(Si) = max { (c - A) x -  : x  X , xj = i }


11 3 1 stsp by column generation

  • Branch on  variables

    If k, t fractional, fix it to 0 and 1 respectively.

    If fixed to 1, no problem in column generation.

    ( usually implemented by setting the lower and upper bound on k, t as 1)

    If fixed to 0 ( set upper bound of k, t to 0), we need a scheme to prevent the generation of the same column again in the column generation procedure.

  • Also the branching may partition the set of feasible solutions unbalanced.


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