Using polarity, we give an outer polyhedral approximation for the epigraph of set functions. For a submodular function, prove that corresponding polar relaxation is exact; hence, it equivalent to Lovász extension. The approach provides alternative proof convex hull description function. Computational experiments show inequalities from approximations can be effective as cutting planes solving we...

Let F be a class of measurable functions f :S 7→ [0,1] defined on a probability space (S,A, P ). Given a sample (X1, . . . ,Xn) of i.i.d. random variables taking values in S with common distribution P , let Pn denote the empirical measure based on (X1, . . . ,Xn). We study an empirical risk minimization problem Pnf →min, f ∈ F . Given a solution f̂n of this problem, the goal is to obtain very ge...

For a non-negative integer k, a language is k-piecewise testable (k-PT) if it is a finite boolean combination of languages of the form Σa1Σ ∗ · · ·ΣanΣ for ai ∈ Σ and 0 ≤ n ≤ k. We study the following problem: Given a DFA recognizing a piecewise testable language, decide whether the language is k-PT. We provide a complexity bound on this problem and a detailed analysis for small k’s. The result...

Assuming axial symmetry, we introduce a surface of revolution approach to model the membrane at equilibrium. We consider a generating curve γ parameterized by arc length s lying in the x− z plane. The curve γ is expressed as γ(0, s1) → R γ(s) = (R(s), 0, z(s)) (S1.1) where s1 is the total arc-length. This generating curve leads to a global parametrization of the membrane expressed as X : (0, s1...