ABSTRACT
Until now, we discussed the convex programming problem (CP ) with the convex feasible set C given by (3.1), that is,
C = {x ∈ Rn : gi(x) ≤ 0, i = 1, 2 . . . ,m}, where gi : Rn → R, i = 1, 2, . . . ,m, are convex functions and its variations like (CP1) and (CCP ). But is the convexity of the functions forming the convex feasible set C important? For example, assume C as a subset of R2 given by
C = {(x1, x2) ∈ R2 : 1− x1x2 ≤ 0, x1 ≥ 0}. This set is convex even though g(x1, x2) = 1−x1x2 is a nonconvex function. As stated in Chapter 1, convex optimization basically means minimizing a convex function over a convex set with no emphasis on as to how the feasible set is obtained. Very recently (2010), Lasserre [74] published a very interesting paper discussing this aspect of convex feasibility for smooth convex optimization.
