Lecture 5: Convex Piecewise Linear Problems

Wai-Shing Luk (陆伟成)

Fudan University

2012年 8月 13日

W.-S. Luk (Fudan Univ.) Lecture 5: Convex Piecewise Linear Problems 2012年 8月 13日 1 / 3

Remark 1

Can be reduced to problems with linear costs [?, p.239,p.260] by

modifying the underlying network.

Potential problem: partition the interval [d−(j), d+(j)] into

consecutive closed subintervals D1,D2, · · · ,Dr .

Flow problem: partition the interval [c−(j), c+(j)] into consecutive

closed subintervals C1,C2, · · · ,Cr .

W.-S. Luk (Fudan Univ.) Lecture 5: Convex Piecewise Linear Problems 2012年 8月 13日 2 / 3

Applications

Power minimization with piecewise linear power-delay curve [?]

l1-norm minimization (c.f. [?, Ex. 7.18])

min ||x ||1s. t. c− ≤ x ≤ c+,

AT x = b, b(V ) = 0

And hence, heuristics method for cardinality problems (c.f. EE364b

Boyd)

min card(x)

s. t. c− ≤ x ≤ c+,

AT x = b, b(V ) = 0

W.-S. Luk (Fudan Univ.) Lecture 5: Convex Piecewise Linear Problems 2012年 8月 13日 3 / 3