Lecture 2 Piecewise-linear optimization

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equivalent LP (variables a ∈ R n , b ∈ R, u ∈ R N ) minimize N∑ u i i=1 subject to 1−s i (v T i a+b) ≤ u i, i = 1,...,N u i ≥ 0, i = 1,...,N in matrix notation: ⎡ ⎤ T ⎡ ⎤ minimize subject to ⎣ 0 0 ⎦ ⎣ a b ⎦ 1 u ⎡ −s 1 v1 T −s 1 −1 0 ··· 0 −s 2 v2 T −s 2 0 −1 ··· 0 . . . . ... . −s N vN T −s N 0 0 ··· −1 0 0 −1 0 ··· 0 ⎢ 0 0 0 −1 ··· 0 ⎣ . . . . ... . 0 0 0 0 ··· −1 ⎤ ⎡ ⎢ ⎣ ⎥ ⎦ ⎤ a b u 1 u 2 ⎥ . ⎦ u N ≤ ⎡ ⎢ ⎣ −1 −1 . −1 0 0. 0 ⎤ ⎥ ⎦ Piecewise-linear optimization 2–22
Modeling software modeling tools simplify the formulation of LPs (and other problems) • accept optimization problem in standard notation (max, ‖·‖ 1 , . . . ) • recognize problems that can be converted to LPs • express the problem in the input format required by a specific LP solver examples of modeling packages • AMPL, GAMS • CVX, YALMIP (MATLAB) • CVXPY, Pyomo, CVXOPT (Python) Piecewise-linear optimization 2–23
Page 1 and 2: EE236A (Fall 2013-14) Lecture 2 Pie
Page 3 and 4: Piecewise-linear function f : R n
Page 5 and 6: Minimizing a sum of piecewise-linea
Page 7 and 8: l ∞ -Norm (Cheybshev) approximati
Page 9 and 10: l 1 -Norm approximation minimize
Page 11 and 12: Comparison with least-squares solut
Page 13 and 14: Sparse signal recovery via l 1 -nor
Page 15 and 16: Exact recovery when are the followi
Page 17 and 18: Proof of nullspace condition notati
Page 19 and 20: necessity: suppose A does not satis
Page 21: Approximate linear separation of no

Lecture 2 Piecewise-linear optimization

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