ESI 6448 Discrete Optimization Theory

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ESI 6448Discrete Optimization Theory

• Lecture 27

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Last class

• Optimal subtour problem
• separation to deal with (exponential number of)
  generalized subtour constraints
• Lagrangian relaxation
• Lagrangian dual problem
• optimality

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Optimal subtour problem

• A variant of TSP
• The salesman makes a profit fj if he visits
  city j ? N, he pays travel costs ce if he
  traverses edge e ? E, but he is not obligated to
  visit all the cities.His subtour must start and
  end at city 1, and include at least two other
  cities.
• subtour elimination constraints
• Si?SSj?S xij S - 1 for S ? N, 2 S n 1

exponential number of inequalities
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Separation for generalized subtour constraints

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Separation (contd)

• If xe ? 0 for e ? E, the linear program
  consisting of (1), (2) and the bound constraints
  obtained by relaxing integrality in (4) always
  has an integer optimal solution solving IP(14).
• The constraint matrix in the dual of the linear
  program consisting of (1), (2), (4) is a node-arc
  incidence matrix, so the problem can be solved as
  a max flow problem.
• The separation problem can be solved by solving n
  1 max flow problems.

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Branch and cut

• Branch and bound algorithm in which cutting
  planes are generated
• reoptimizing fast at each node ? do as much work
  as is necessary to get a tight dual bound at each
  node
• includes preprocessing at each node
• primal heuristics at each node
• cut pool is used
• pointers to the appropriate constraints in the
  cut pool are kept.

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Flowchart
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Lagrangian relaxation

• z maxcx Ax ? b, Dx ? d, x ? Zn
• if Dx ? d are complicating constraints
• drop Dx ? d and get relaxation
• provides (weak) bounds
• Not just dropping but incorporating those into
  objective function (Lagrangian relaxation)
• Transform IP z maxcx Dx ? d, x ? X
  intoIP(u) z(u) maxcx u(d Dx) x ? X for
  any value ofu (u1, , um) ? 0

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Lagrangian dual problem

• Problem IP(u) is a relaxation problem for IP for
  all u ? 0.
• z(u) ? z (upper bound on the optimal value of IP)
• To find the tightest upper bound over u, solve
  the Lagrangian dual problem wLD minz(u) u
  ? 0
• If the constraints are Dx d, wLD min z(u)

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Optimality

• If u ? 0, and(i) x(u) is an optimal solution,
  and(ii) Dx(u) ? d, and(iii) (Dx(u))i di
  whenever ui gt 0,then x(u) is optimal in IP.

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Application to UFL

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Application to STSP

1-tree constraints
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Example

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1
2
39 (24)
40
3
5
39 (24)
30
4
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2 viewpoints for Lagrangian relaxation

• X x ? Zn Ax ? b
• conv(X) x ? Rn Ax ? b
• z maxcx Dx ? d, x ? X
• z(u) maxcx u(d Dx) x ? conv(X)
• 2 viewpoints for z(u, x) cx u(d Dx)
• affine function of x for u fixed
• affine function of u for x fixed
• wLD minz(u) u ? 0

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Example

X
x2
x1
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Example(contd)

x2
x1
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Example(contd)

z(u, xi)
u
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Strength of Lagrangian dual

• wLD maxcx Dx ? d, x ? conv(X)
• LD can be viewed as the problem of minimizing the
  piecewise linear convex, but nondifferentiable
  function z(u)
• If X x ? Zn Ax ? b andconv(X) x ? Rn
  Ax ? b, thenwLD maxcx Ax ? b, Dx ? d, x
  ? Rn

z(u)
u
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Today

• Lagrangian relaxation
• application to STSP
• Strength of Lagrangian dual
• 2 viewpoints for Lagrangian relaxation
• wLD maxcx Dx ? d, x ? conv(X)