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11.3.1 STSP by Column Generation

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

subject toe({i})xe = 2 ,i N\{1}

e({1})xe = 2 ,

eE(S)xe |S| - 1 ,S N\{1}, S , N\{1},

eE(N\{1})xe = |N| - 2.

xe { 0, 1 }.

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

- min { eEcexe : e(i)xe = 2 for iN\{1}, xX1 }
where X1 = { xZ+m : e(1)xe = 2, eE(S)xe |S|-1 for S N\{1},

eE(N\{1})xe = |N| - 2 }

is the set of incidence vectors of one-trees.

- STSP:
min { eEcexe : e(i)xe = 2 for iN\{1}, xX1 }

where X1 = { xZ+m : e(1)xe = 2, eE(S)xe |S|-1 for S N\{1},

eE(N\{1})xe = |N| - 2 }

is the set of incidence vectors of one-trees.

- Write xe = t : eE(t) t , t {0, 1}, where Gt = (N, Et) is the tth one-tree.
e(i)xe = e(i) t : eE(t) t = t ditt = 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 ) )

- min t=1T1 (cxt)t
(DW) t=1T1ditt = 2, for i N\{1}

t=1T1 t = 1

B|T1|

- LP relaxation is:
min t=1T1 (cxt)t

(LPM) t=1T1ditt = 2, for i N\{1}

t=1T1 t = 1

R+T1

- Note that the convexity constraintt=1T1 t = 1 (t=1T12t = 2) may be regarded as d1t= 2 for node 1.
- Subproblem: 1= min{eE(ce- ui- uj)xe - : xX1} (e=(i, j)E, i, jN\{1})
( cxt - iN\{1}ditui - = cxt - iN\{1}ui e(i)xet - = eE (ce - ui - uj)xet - )

- 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, tk, t, t=1Tkk, 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

- 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)

- 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=1Tkk, t = 1 for k = 1, … ,K

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

if and only if is integer

- 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=1Tkk, txjk, t {0, 1}, xjk = {0, 1} implies that xjk, t = for all k, t with k, t > 0

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 }

- 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.