Lec04 min cost linear problems

Lecture 4: Min-Cost Linear Problems

Wai-Shing Luk (陆伟成)

Fudan University

2012年 8月 13日

W.-S. Luk (Fudan Univ.) Lecture 4: Min-Cost Linear Problems 2012年 8月 13日 1 / 18

Elementary Optimal Problems

Elementary Flow Problem:

min dT x + p

s. t. c ≤ x ,

AT x = b, b(V ) = 0

Elementary Potential Problem:

max bTu − (cT y + q)

s. t. y ≤ d ,

Au = y

Theorem

The problems are dual to each other if

p + q = −cTd , (x − c)T (d − y) = 0, c ≤ x , y ≤ d

Proof.

Since bTu = (AT x)Tu = xTAu = xT y , [min]− [max] = (dT x + p)−(bTu − [cT y + q]) = dT x + cT y − xT y + p + q = (x − c)T (d − y) ≥ 0

[min] - [max] when equality holds.


Remark 1

We can formulate a linear problem either in its primal form or in its

dual form, depending on which one is more appropriate, for example,

whether design variables are in integral domain:

max-flow problem (i.e. dT = [−1,−1, · · · ,−1]T ) may be better to be

solved by a dual method.


Linear Optimal Problems

Optimal Flow Problem:

min dT x + p

s. t. c− ≤ x ≤ c+,

AT x = b, b(V ) = 0

Optimal Potential Problem:

max bTu − (cT y + q)

s. t. d− ≤ y ≤ d+,

Au = y

By modifying the network:

This problem can be reduced to the elementary case [?, pp.275–276]

Piecewise linear convex cost can be reduced to this linear problem [?,

p.239,p.260]

The problem has been studied extensively with a lot of applications.


Remark

We can transform the cost function to be non-negative by reversing

the orientation of the negative cost edges.

Then reduce the problem to the elementary case.

Most software packages only solve the problems in primal forms,

usually with c− = 0.


Algorithms for Optimal Flow Problems

Successive shortest path algorithm

Cycle cancellation method

iteratively insert additional minimal flows according to a negative cycle

of the residual network, until no negative cycle is found.


For Special Cases

Max flow problem (d = −[1, · · · , 1])

Ford-Fulkerson algorithm: iteratively insert additional minimal flows

according to an argument path of the residual network, until no

argument path of the residual network is found.

Preflow Push-Relabel algorithm (dual method???)

Matching problems ([c−, c+] = [0, 1])

Edmond’s blossom algorithm


Min-Cost Flow Problem (MCFP1)

Consider:min dT x

s. t. 0 ≤ x ≤ c ,

AT x = b, b(V ) = 0

where d is assumed to be non-negative.

Note: Optimal flow problem can be transformed into this problem by

letting x = x ′ − c−.

Algorithm idea: descent method: given a feasible x0, find a better

solution x1 = x0 + αp, where α is positive.


Review of General Descent Method

Input: f (x), initial x

Output: x∗

while not converged do

Choose descent direction p;

Choose the step size α;

x := x + αp;

end


Some Common Descent Directions

For convex problems, the search direction must satisfy ∇′f (x) · p < 0

Gradient descent: p = −∇f (x)T

Steepest descent: (complicated)

Newton’s method: p = −∇2f (x)−1∇f (x)

Note: Here, we have a natural way to choose p!


Min-Cost Flow Problem (MCFP1)

Let x1 = x0 + αp, then we have:

min dT x0 + αdTp ⇒ dT < 0

s. t. −x0 ≤ αp ≤ c − x0 ⇒ residual graph

ATp = 0 ⇒ p is a cycle!

In other words, choose p to be a negative cycle!

Simple negative cycle, or

Minimum mean cycle (c.f. Lecture 3)


Primal Method for MCFP1

Input: G (V ,E ), c , d

Output: optimal opt x∗

Initialize a feasible x and certain data structure;

while ( a negative cycle p found in G (x)) do

Choose a step size α;

If α is unbound, return UNBOUNDED;

If α = 0, break;

x := x + αp;

Update corresponding data structures;

end

return OPTIMAL;


Remarks

G (x) denotes the residual graph.

Negative cycle can be found using Bellman-Ford-like algorithms (c.f.

Lecture 2).

A precede graph or other data structures are used for finding negative

cycles efficiently.

Usually α is chosen such that one constraint is tight.


Dual Problem of MCFP1

It is well known that the dual problem of MCFP1 is given by

min cT I

s. t. y ≤ d + I ,

Au = y

I ≥ 0

where I is a slack variable.

Note: Delay padding problem can be formulated as this problem

where I represents the buffer delay.


Dual Problem of MCFP1 (cont’d)

Note: The problem can be transformed into the elementary potential

problem by splitting an edge between u(i) and u(j):

u(i)− u(h) = I (i , j) (1)

u(h)− u(j) ≤ d(i , j) (2)

u(i)− u(h) ≥ 0 (3)


Min-Cost Potential Problem (MCPP)

Consider:min cT y

s. t. y ≤ d ,

Au = y

where c is assumed to be non-negative.

Algorithm idea: given an initial feasible u0, find a better solution

u1 = u0 + βq, where β is positive:

min cT y0 + cT y ⇒ cT y < 0

s. t. y ≤ d − Au0 ⇒ residual graph

βAq = y ⇒ q is a “cut”!


Method for MCPP

Input: G (V ,E ), c , d

Output: optimal value OPT, u∗

Initialize a feasible u and certain data structure;

while ( a negative cut q found in G (u)) do

Choose a step size β;

If β is unbounded,return UNBOUNDED;

If β = 0, break;

u := u + βq;

Update corresponding data structures;

end

return OPTIMAL;


Remarks

Usually β is chosen such that one constraint is tight.

Min-cost potential problem is a dual of min-cost flow problem, hence

algorithms can solve both problems.


Lec04 min cost linear problems

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