Hard - Constrained InconsistentSignal Feasibility

| We consider the problem of synthesizing feasible signals in a Hilbert space in the presence of inconsistent convex constraints, some of which must imperatively be satissed. This problem is formalized as that of minimizing a convex objective measuring the amount of violation of the soft constraints over the intersection of the sets associated with the hard ones. The resulting convex optimization problem is analyzed and numerical solution schemes are presented along with convergence results. The proposed formalism and its algorithmic framework unify and extend existing approaches to inconsistent signal feasibility problems. An application to signal synthesis is demonstrated. Throughout, the signal space is a real Hilbert space H, with scalar product hh j i, norm k k, and distance d. The distance from a signal x 2 H to a nonempty set A H is deened as d(x; A) = inffkx ? yk j y 2 Ag. ? denotes the class of all lower semicontinuous proper convex functions from H into ]?1; +1] 9]. Given g 2 ? and 2 R, the closed and convex set lev g = fx 2 H j g(x) g is the lower level set of g at height and the nonempty convex set dom g = fx 2 H j g(x) < +1g its domain.