Chapter 4 Duality in integer optimization (B&W)

Chapter 4Duality in integer optimization (B&W)

4.3 Lagrangean duality • Consider minimize c’x subject to Ax  b Dx  d (4.23) x  Zn Let F = { x  Zn : Dx  d} and consider the problem: minimize c’x + ’(b-Ax) subject to x  F (4.24) and denote its optimal cost by Z(). • Note that (4.24) is a relaxation of (4.23) if   0 (Lagrangean relaxation). ( recall relaxation: (i) S  SR, (ii) c’x zR(x) for all x  S (for max prob.) )

Prop 4.4: • (a) If (4.23) has an optimal solution, then Z()  ZIP for   0. • (b) The function Z() is concave. Pf) (a) Let x* be an optimal solution to (4.23). Then b-Ax*  0 and x*  F. Hence Z()  c’x* + ’(b-Ax*)  c’x* = ZIP (b) Let x*  F be an optimal solution to (4.24) for  ( = 1 + (1-) 2 ) Z(1 + (1-) 2) = c’x* + (1 + (1-) 2)’(b-Ax* ). Z(1)  c’x* + 1’(b-Ax*) Z(2)  c’x* + 2’(b-Ax*)  Z(1) + (1-)Z(2)  c’x* + (1 + (1-) 2)’(b-Ax* ) = Z(1 + (1-) 2)  • Lagrangean dual: maximize Z() subject to   0. (4.25) • Thm 4.8: (Weak duality) We have ZD  ZIP .

If equality constraints are dualized, corresponding dual variables are unrestricted in sign. (Check that weak duality holds) • minimize c’x + ’(b-Ax) subject to x  F (4.24) • Two view points for Z() : • Optimize affine function of x for fixed , i.e. solve min (c’-’A)x + ’b, xconv(F) • Z() = minxk  F (c’xk + ’(b - Axk )) : min of affine functions of .  Z() is concave.

Thm 4.9: The optimal value ZD of the Lagrangean dual (4.25) is equal to the optimal cost of the following linear optimization problem: minimize c’x subject to Ax  b x  conv(F) (4.27) Pf) By definition, Z() = min x  F (c’x + ’(b-Ax)) = min x  conv(F) (c’x + ’(b-Ax)) (since obj fn. is linear in x)  ZD = max   0min x  conv(F) (c’x + ’(b-Ax)) Let xk , kK, wjjJ, be the extreme pts and extreme rays of conv(F). Then Z() = - if (c’-’A)wj < 0, for some jJ = min kK (c’xk + ’(b-Axk)) otherwise Hence Lagrangean dual is equivalent to maximize min kK (c’xk + ’(b-Axk)) subject to (c’-’A)wj  0, for all jJ   0

or equivalently maximize y subject to y + ’(Axk- b)  c’xk , kK ’Awj  c’wj , jJ   0 Taking the dual, and using strong duality for LP, ZD is equal to minimize c’( kKkxk + jJ jwj ) subject to kKk = 1 A( kKkxk + jJ jwj )  b, k, j  0, kK , jJ Since conv(F) = { kKkxk + jJ jwj : kKk =1, k, j  0, kK , jJ}, the result follows. 

Ex 4.3: Consider the problem: minimize subject to ,

conv(F) • Ex: 3 2 c 1 0 2 1

Cor 4.1: (a) We have ZIP = ZD , for all cost vectors c, if and only if conv( F {x: Ax  b}) = conv(F)  {x: Ax  b} (b) We have ZLP = ZD ,for all cost vectors c, if and only if conv(F) = { x: Dx d}. • Ex 4.5: One tree relaxation of the traveling salesman problem e({i})xe = 2 , i  V eE(S)xe  |S| - 1 , S  V, S  , V, xe  { 0, 1 }. We choose node 1 as a special node, called the root node, and add the redundant equality eE(V\{1})xe = |V| - 2.

minimize eE ce xe subject to e({i}) xe = 2 , i  V\{1} e({1}) xe = 2 , eE(S) xe  |S| - 1 , S  V\{1}, S  , V\{1}, eE(V\{1}) xe = |V| - 2. xe  { 0, 1 }. • Dualize the constraints e({i}) xe = 2 , i  V\{1} (4.28) • One tree: A tree involving all nodes in V\{1}, and two additional edges incident to node1. (relaxation of tours) • After dualizing, F is the set of one-trees and, moreover, conv(F) is the same as the LP relaxation of the constraints except (4.28) (note that the constraints involving node 1 is independent of the other inequalities) • By Cor 4.1(b), ZD = ZLP , i.e. solving the Lagrangean dual using one tree relaxation provides the same bound obtained from the LP relaxation of subtour elimination formulation.

Note that xe 1 is implied by the subtour elimination constraints for S = {i, j} and, because of symmetry, the subtour elimination constraints for V\{1} are enough for the formulation. ( |S| - eE(S) xe = ½ (iS e(i) xe) - eE(S) xe = ½ e(S:S) xe (S:S) = (S:S) |S| - eE(S) xe = |S| - eE(S) xe eE(S) xe  |S|-1  eE(S) xe  |S|-1 )

Finding minimum one tree when  is fixed can be solved easily. We update edge costs for given , and find min spanning tree on V\{1}. Then we just choose two cheapest edges among edges incident to node 1 and add them to the constructed tree. • Consider how we can maximize Z(). (later) • First application of Lagrangean dual and subgradient optimization for integer optimization problems. (Held and Karp) • Held and Karp, The traveling salesman problem and minimum spanning trees, Operations Research, 18: 1138-1162, 1970 • Held and Karp, The traveling salesman problem and minimum spanning trees: Part II, Mathematical Programming, 1: 6-25, 1971

Chapter 4 Duality in integer optimization (B&W)