1. Chapter 6. Large Scale Optimization
6.1 Delayed column generation
min c’x, Ax = b, x  A is full row rank with a large number of columns. Impractical to have all columns initially.
Want to generate (find) entering nonbasic variable (column) as needed. ( column generation technique )
If ci < 0, then xi can enter basis.
Hence solve min ci over all i.
If min ci < 0, have found an entering variable (column).
If min ci  0, no entering column exists, hence current
basis optimal.
If entering col. found, solve restricted problem to optimality.
min i  I , i  I Aixi = b, x  0
( I : index set of variables we have at hand)
Then continue to find entering columns.
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2. 6.2. Cutting stock problem W = 70 17 17 17 15 scrap
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3. bi rolls of width wi , i = 1, 2, … , m need to be produced.
Rolls of paper with width W ( called raw) to be cut into small pieces (called final).
bi rolls of width wi , i = 1, 2, … , m need to be produced.
How to cut the raws to minimize the number of raws used while satisfying order?
ex) W = 70, then 3 of w1 = 17 and 1 of w2 = 15 can be produced from a raw. This way of production can be represented as pattern ( 3, 1, 0, 0, … , 0 )
( a1j, a2j, … , amj )’ for j-th pattern is feasible if i aijwi  W.
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4. , where aij is the number of i-th finals produced in j-th pattern.
Formulation
, where aij is the number of i-th finals produced in j-th pattern.
Note that the number of possible cutting patterns can be very large.
We need integer solution, but LP relaxation can be used to find good approximate solution if solution value large.
For initial b.f.s., for j = 1, … , m, let j-th pattern consists of one final of width wj and none of the other widths.
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5.  max p’Aj over all patterns ( integer knapsack problem )
After computing p vector, we try to find entering nonbasic variable (column).
Candidate for entering column (pattern) is any nonbasic variable with reduced cost ( 1 – p’Aj ) < 0, hence solve min ( 1 – p’Aj ) over all possible patterns.
 max p’Aj over all patterns ( integer knapsack problem )
max
Dynamic programming algorithm for integer knapsack prob.
( can assume pi, wi > 0, integer.
knapsack is NP-hard, so no polynomial time alg. is known. )
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6. Let F(v) be optimal value of the problem when knapsack capacity is v.
wmin = mini { wi }
For v < wmin , F(v) = 0
For v  wmin , F(v) = maxi = 1, … , m { F( v – wi ) + pi : v  wi }
F(v) is true optimal value when knapsack capacity is v.
Suppose a0 is opt. solution when r.h.s. is v – wi , then a0 + ei is a feasible solution when r.h.s. is v.
Hence F(v)  F(v – wi) + pi , i = 1, … , m, v  wi
Suppose a* is optimal solution when r.h.s. is v  wmin , then there exists some k with a*k > 0 and v  wk .
Hence a* - ek is a feasible solution when r.h.s. is v – wk .
So F( v-wk )  F(v) – pk ( F(v)  F( v-wk ) + pk for some k )
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7. Actual solution recovered by backtracking the recursion.
Running time of the algorithm is O(mW) which is not polynomial of the length of encoding. Called pseudopolynomial running time ( polynomial of data W itself).
Note : the running time becomes polynomial if it is polynomial with respect to m and log2 W, but W = 2logW which is not polynomial of log2 W.
Many practical problems can be naturally formulated similar to the cutting stock problem. Especially in 0-1 IP with many columns. For cutting stock problem, we only obtained approximate fractional solution. But for 0-1 IP, fractional solution can be of little help and we need a mechanism to find optimal integer solution ( branch-and-price approach, column generation combined with branch-and-bound ).
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8. 6.3. Cutting plane methods
Dual of column generation (constraint generation)
Consider max p’b, p’Ai  ci , i = 1, … , n (1)
( n can be very large )
Solve max p’b, p’Ai  ci , i  I, I  { 1, … , n } (2)
and get optimal solution p* to (2)
If p* is feasible to (1), then it is also optimal to (1)
If p* is infeasible to (1), find a violated constraint in (1) and add it to (2), then reoptimize (2) again. Repeat it.
Recall the TSP formulation with subtour elimination constraints.
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9. Solve min ci – (p*)’Ai over all i. If optimal value  0  p*  P
Separation problem :
Given a polyhedron P (described with possibly many inequalities) and a vector p* , determine if p*  P. If p*  P, find a (valid) inequality violated by p*.
Solve min ci – (p*)’Ai over all i.
If optimal value  0  p*  P
If optimal value < 0  ci < (p*)’Ai (violated)
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10. 6.4. Dantzig-Wolfe decomposition
Use of decomposition theorem to represent a specially structured LP problem in different form. Column generation is used to solve the problem. Consider a LP in the following form
min
x1, x2 : dimension n1, n2, b0, b1, b2 : m0, m1, m2
Let Pi = { xi  0 : Fix = bi }, i = 1, Assume Pi  .
Note that the nonnegativity constraints guarantee that Pi is pointed, hence S = {0} ( P = S + K + Q )
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11. xi  Pi can be represented as
 min
xi  Pi can be represented as
Plug into (2)  get master problem
min
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12. Alternatively, its columns can be viewed as
The new formulation has many variables (columns), but it can be solved by column generation technique.
Actual solution x1, x2 can be recovered from  and .
xi is expressed as convex combination of extreme points of Pi + conical combination of extreme rays of Pi.
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13. Decomposition algorithm
Suppose having a b.f.s. to the master problem, dual value p = ( q, r1, r2), qRm0, r1, r2  R. Then reduced costs are (for 1, 1 )
Entering variable if reduced cost < 0.
Hence solve min ( c1’ – q’D1 )x1, x1  P1 (subproblem)
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14. simplex returns extreme ray w1k with ( c1’ – q’D1 )w1k < 0.
(a) optimal cost is -  
simplex returns extreme ray w1k with ( c1’ – q’D1 )w1k < 0.
Generate column for 1k, i.e. [ D1w1k ’, 0 , 0 ]’
(b) optimal finite and < r1 
returns extreme point x1j with ( c1’ – q’D1 )x1j < r1.
Generate column for 1j, i.e. [ D1x1j ’, 1 , 0 ]’
(c) optimal cost  r1 
( c1’ – q’D1 )x1j  r1  x1j , ( c1’ – q’D1 )w1k  0  w1k
no entering variable among 1j , 1k
Perform the same for 2j , 2k
The method can also be used when there are more than 2 blocks or just one block in the constraints.
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15. Starting the algorithm
Find extreme points x11, x21 of P1 and P2
May assume that D1x11 + D2x21  b, then solve.
min
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16. Termination and computational experience
Fast improvement in early iterations, but convergence becomes slow in the tail of the sequence.
Revised simplex is more competitive in terms of running time.
Suitable for large, structured problems.
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17. Bounds on the optimal cost
Thm 6.1 : Suppose optimal z* is finite. Let z be the current best solution (upper bound on z* ), ri dual variable value for i-th convexity constraint and zi finite optimal cost for i-th subproblem. Then z + i ( zi – ri )  z*  z.
pf) Modify the current dual solution to a dual feasible solution by decreasing the value of ri to zi .
Dual of master problem is
max
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18. Suppose have a b.f.s. to master problem with z and ( q, r1, r2 ).
Have q’b0 + r1 + r2 = z
Optimal cost z1 to the first subproblem finite
 minj  J1 (c1’x1j – q’D1x1j ) = z1
mink  k1 (c1’w1k – q’D1w1k )  0
Note that currently we have minj ( c1’ – q’D1 )x1j < r1 ( reduced cost (c1’ – q’D1 )x1j - r1 < 0 for entering variable).
If we use z1 in place of r1, get dual feasibility for the first two dual constraints.
Similarly, use z2 in place of r2.
Cost is q’b0 + z1 + z2  z*  q’b0 + z1 + z2
= q’b0 +r1 + r2 + ( z1 – r1 ) + ( z2 – r2 )
= z + ( z1 – r1 ) + ( z2 – r2 ) 
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