v2010.10.26 - Convex Optimization

292 CHAPTER 4. SEMIDEFINITE PROGRAMMING4.2.3.1.1 Example. Minimal cardinality Boolean. [93] [34,4.3.4] [345](confer Example 4.5.1.5.1) Consider finding a minimal cardinality Booleansolution x to the classic linear algebra problem Ax = b given noiseless dataA∈ R m×n and b∈ R m ;minimize ‖x‖ 0xsubject to Ax = bx i ∈ {0, 1} ,i=1... n(680)where ‖x‖ 0 denotes cardinality of vector x (a.k.a, 0-norm; not a convexfunction).A minimal cardinality solution answers the question: “Which fewestlinear combination of columns in A constructs vector b ?” Cardinalityproblems have extraordinarily wide appeal, arising in many fields of scienceand across many disciplines. [318] [214] [171] [170] Yet designing an efficientalgorithm to optimize cardinality has proved difficult. In this example, wealso constrain the variable to be Boolean. The Boolean constraint forcesan identical solution were the norm in problem (680) instead the 1-norm or2-norm; id est, the two problems(680)minimize ‖x‖ 0xsubject to Ax = bx i ∈ {0, 1} ,i=1... n=minimize ‖x‖ 1xsubject to Ax = bx i ∈ {0, 1} ,(681)i=1... nare the same. The Boolean constraint makes the 1-norm problem nonconvex.Given dataA =⎡−1 1 8 1 1 0⎢1 1 1⎣ −3 2 82 3−9 4 81419− 1 2 31− 1 4 9⎤⎥⎦ , b =the obvious and desired solution to the problem posed,⎡⎢⎣11214⎤⎥⎦ (682)x ⋆ = e 4 ∈ R 6 (683)