Lec05 convex pwl problems
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This document summarizes Lecture 5 given by W.-S. Luk from Fudan University on August 13, 2012 about convex piecewise linear problems. The lecture discusses how such problems can be reduced to problems with linear costs by modifying the underlying network. Applications mentioned include power minimization with piecewise linear power-delay curves and l1-norm minimization, which can provide heuristics for cardinality problems.
![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](https://image.slidesharecdn.com/lec05convexpwlproblems-120813015505-phpapp02/85/Lec05-convex-pwl-problems-2-320.jpg)
![Applications
Power minimization with piecewise linear power-delay curve [?]
l1 -norm minimization (c.f. [?, Ex. 7.18])
min ||x||1
s. 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](https://image.slidesharecdn.com/lec05convexpwlproblems-120813015505-phpapp02/85/Lec05-convex-pwl-problems-3-320.jpg)
Lec05 convex pwl problems
- 1. 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
- 2. 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
- 3. Applications Power minimization with piecewise linear power-delay curve [?] l1 -norm minimization (c.f. [?, Ex. 7.18]) min ||x||1 s. 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