Gradient Method

1
Gradient Method

• Gradient of f(x)
• For nonlinear objective function f(x) of
  n-dimensional vector,
• Gradient ?f(x) defined as
• Gradient Method
• If we choose d so that ?f(x)d gt0 and change x to
  x ?d, for enough small ?, we move uphill.
• Only useful when the objective function is
  differentiable.
• However, L(?) is not differentiable when the
  optimal solutions are not unique

• d direction( n-dimensional vector)
• ? step length(scalar)

2
Subgradient Optimization Technique

• Subgradient optimization technique
• Suppose L(?) min cx ?(Ax b) x?X has a
  unique solution x and is differentiable.
• The solution x remains optimal for small change
  of ?
• ? ? ? ? (Ax b)
• Intuitive interpretation
• When (Ax b)i 0, the solution x uses up
  exactly the required units of the ith resource,
  we hold ?i.
• When (Ax b)i lt 0, the solution x uses up less
  than the available units of the ith resource, we
  decrease ?i.
• When (Ax b)i gt 0, the solution x uses up more
  than the available units of the ith resource, we
  increase ?i.

Ax b direction ? step size
3
Subgradient (Ax b)

• Def Subgradient
• Let f be concave, then is ? called a subgradient
  of f at x?S
• If f(x) ?f(x) ?t(x-x) for
  all x ?S
• Need not to be differential
• Meaning of Ax b
• Subgradient Steepest decent direction

4
Lagrangian Multiplier Updating

• Lagrangian Multiplier Updating
• ?k1 ? ? k ?k (Axk b)
• ?0 any initial choice of the Lagrangian
  multiplier
• xk any solution to the Lagrangian subproblem
  when ? ?k
• ?k step length at the kth iteration
• Choice of Step Sizes(?k )
• Condition for convergence to an optimal solution
  of the multiplier problem
• For example, ?k 1/k.

5
Choice of Step Sizes(?k )(1)

• An adaptation of Newtons method
• L(?k) cxk ?k (Axk b) where xk solves
  Lagrangian subproblem when ? ?k
• L(?) ? r(?) cxk ?(Axk b) linear
  approximation
• Suppose we know the optimum value L of the
  Lagrangian multiplier problem
• r(?k1) cxk ?k1 (Axk b) L we hope
• ?k1 ? ? k ?k (Axk b)
• ?r(?k1) cxk ? k ?k (Axk b) (Axk b)
  L

6
Choice of Step Sizes(?k )(2)

• Practically
• However, we dont know the objective function
  value of L Lagrangian multiplier problem.
• Popular heuristic for selecting the step length
• UB upper bound on the optimal objective
  function z of the problem (P)
• ?k scalar chosen (strictly) between 0 and 2
• Choosing ?k
• Starting ?k with 2 and then reducing ?k by a
  factor of 2 until failing to find the better
  solution.

7
Cases of Inequality Constraints

• Subgradient Optimization and Inequality
  Constraints
• The update formula ?k1 ? ? k ?k (Axk b)
  might cause ? to become negative. To avoid this
  possibility,
• ?k1 ? ? k ?k (Axk b) where (y)i
  max(yi,0)
• The other steps (i.e. the choice of ?k) with
  inequality constraints are the same as that with
  equality constraints.

8
Overview of Lagrangian Technique
LS(?k) Lagrangian Subproblem at ? ?k
Solve LS(?k) ? xk
If xk feasible, UB updating
?k1 ? ?k ?k (Axk b)
Yes
No
Done
9
Example of CSPP

• CSPP with T14

• Initial parameter setting
• ?00, ?00.8

(1, 1)
2
4
(1, 7)
(1, 10)
(2, 3)
1
6
(1, 2)
(10, 1)
(5, 7)
(2, 2)
(10, 3)
3
5
(12, 3)
10
Example of CSPP

• CSPP with T14

• Initial parameter setting
• ?00, ?00.8
• Iteration 1
• Path 1-2-4-6, L(0)3
• UB 24 (path 1-3-5-6)
• ?0 0.8(24-3)/16 1.05
• ?1 0 1.05(4) 4.2

?00
1
2
4
1
1
2
1
6
1
10
5
2
10
3
5
12
11
Example of CSPP

• CSPP with T14

• Initial parameter setting
• ?00, ?00.8
• Iteration 1
• Path 1-2-4-6, L(0)3
• UB 24 (path 1-3-5-6)
• ?0 0.8(24-3)/16 1.05
• ?1 0 1.05(4) 4.2
• Iteration 2
• Path 1-3-2-5-6, L(4.2)57
• UB 15 (path 1-3-2-5-6)
• ?1 0.8(151.8)/16 0.84
• ?2 4.2 0.84(-4)0.84

?14.2
5.2
2
4
23.2
43
14.6
1
6
9.4
14.1
34.4
10.4
22.6
3
5
24.6
12
Duality Gap of CSPP

• Convergence in CSPP
• Lagrangian obj. ftn. value ? L 7
• Lagrangian multiplier ? ? 2
• Decreasing ?k ? 0
• Duality(relaxation) Gap
• L 7 lt the length of the shortest constrained
  path 13
• In these instance, we say that the Lagrangian
  relaxation has a duality(relaxation) gap.
• To solve problems with a duality gap to
  completion
• (i.e. to find an opt. Sol. and guarantee that it
  is optimal)
• ? enumeration procedure (ex. branch and bound)