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 ProblemsElementary Flow Problem: Elementary Potential Problem: min dT x + p max b T u − (c T y + q) s. t. c ≤ x, s. t. y ≤ d, AT x = b, b(V ) = 0 Au = yTheoremThe problems are dual to each other ifp + q = −c T d, (x − c)T (d − y ) = 0, c ≤ x, y ≤ dProof.Since b T u = (AT x)T u = x T Au = x T y , [min] − [max] = (d T x + p) −(b T u − [c T y + q]) = d T x + c T y − x T y + p + q = (x − c)T (d − y ) ≥ 0[min] - [max] when equality holds. W.-S. Luk (Fudan Univ.) Lecture 4: Min-Cost Linear Problems 2012 年 8 月 13 日 2 / 18

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. d T = [−1, −1, · · · , −1]T ) may be better to be solved by a dual method.W.-S. Luk (Fudan Univ.) Lecture 4: Min-Cost Linear Problems 2012 年 8 月 13 日 3 / 18

Linear Optimal ProblemsOptimal Flow Problem: Optimal Potential Problem: min dT x + p max b T u − (c T y + q) s. t. c − ≤ x ≤ c + , s. t. d − ≤ y ≤ d + , AT x = b, b(V ) = 0 Au = yBy 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. W.-S. Luk (Fudan Univ.) Lecture 4: Min-Cost Linear Problems 2012 年 8 月 13 日 4 / 18

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.W.-S. Luk (Fudan Univ.) Lecture 4: Min-Cost Linear Problems 2012 年 8 月 13 日 5 / 18

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.W.-S. Luk (Fudan Univ.) Lecture 4: Min-Cost Linear Problems 2012 年 8 月 13 日 6 / 18

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 algorithmW.-S. Luk (Fudan Univ.) Lecture 4: Min-Cost Linear Problems 2012 年 8 月 13 日 7 / 18

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 byletting x = x − c − . Algorithm idea: descent method: given a feasible x0 , ﬁnd a better solution x1 = x0 + αp, where α is positive. W.-S. Luk (Fudan Univ.) Lecture 4: Min-Cost Linear Problems 2012 年 8 月 13 日 8 / 18

Review of General Descent MethodInput: f (x), initial xOutput: x ∗while not converged do Choose descent direction p; Choose the step size α; x := x + αp;end W.-S. Luk (Fudan Univ.) Lecture 4: Min-Cost Linear Problems 2012 年 8 月 13 日 9 / 18

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 = − 2 f (x)−1 f (x)Note: Here, we have a natural way to choose p! W.-S. Luk (Fudan Univ.) Lecture 4: Min-Cost Linear Problems 2012 年 8 月 13 日 10 / 18

Min-Cost Flow Problem (MCFP1) Let x1 = x0 + αp, then we have: min d T x0 + αd T p ⇒ dT < 0 s. t. −x0 ≤ αp ≤ c − x0 ⇒ residual graph AT p = 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)W.-S. Luk (Fudan Univ.) Lecture 4: Min-Cost Linear Problems 2012 年 8 月 13 日 11 / 18

Primal Method for MCFP1Input: G (V , E ), c, dOutput: 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;endreturn OPTIMAL; W.-S. Luk (Fudan Univ.) Lecture 4: Min-Cost Linear Problems 2012 年 8 月 13 日 12 / 18

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

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.W.-S. Luk (Fudan Univ.) Lecture 4: Min-Cost Linear Problems 2012 年 8 月 13 日 14 / 18

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)W.-S. Luk (Fudan Univ.) Lecture 4: Min-Cost Linear Problems 2012 年 8 月 13 日 15 / 18

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 c T y0 + c T y ⇒ cT y < 0 s. t. y ≤ d − Au0 ⇒ residual graph βAq = y ⇒ q is a “cut”!W.-S. Luk (Fudan Univ.) Lecture 4: Min-Cost Linear Problems 2012 年 8 月 13 日 16 / 18

Method for MCPPInput: G (V , E ), c, dOutput: 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;endreturn OPTIMAL; W.-S. Luk (Fudan Univ.) Lecture 4: Min-Cost Linear Problems 2012 年 8 月 13 日 17 / 18

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

Lec04 min cost linear problems

by Danny Luk on Aug 13, 2012

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