v2007.11.26 - Convex Optimization

4.2. FRAMEWORK 2414.2.3.1.1 Example. Minimum cardinality Boolean. [63] [27,4.3.4] [262](confer Example 4.5.1.0.1) Consider finding a minimum 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(577)where ‖x‖ 0 denotes cardinality of vector x (a.k.a, 0-norm; not a convexfunction).A minimum 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. [240] [157] [120] [121] 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 (577) instead the 1-norm or2-norm; id est, the two problems(577)minimize ‖x‖ 0xsubject to Ax = bx i ∈ {0, 1} ,i=1... n=minimize ‖x‖ 1xsubject to Ax = bx i ∈ {0, 1} ,(578)i=1... nare the same. The Boolean constraint makes the 1-norm problem nonconvex.Given data 4.11⎡⎤ ⎡ ⎤−1 1 8 1 1 01⎢1 1 1A = ⎣ −3 2 8 − 1 ⎥ ⎢ ⎥2 3 2 3 ⎦ , b = ⎣ ⎦ (579)−9 4 8141914 − 1 9the obvious and desired solution to the problem posed,x ⋆ = e 4 ∈ R 6 (580)has norm ‖x ⋆ ‖ 2 =1 and minimum cardinality; the minimum number ofnonzero entries in vector x . The Matlab backslash command x=A\b ,4.11 This particular matrix A is full-rank having three-dimensional nullspace (but thecolumns are not conically independent).1214