No abstract

Given an n × m nonnegative matrix A = (a_{ij}) and positive integral vectors r in R^n and c in R^m having a common one-norm h, the (r, c)-scaling problem is to obtain positive diagonal matrices X and Y, if they exist, such that XAY has row and column sums equal to r and c, respectively. The entropy minimization problem corresponding to A is to find an n × m matrix z = (z_{ij}) having the same zero pattern as A, the sum of whose entries is a given number h, its row and column sums are within given integral vectors of lower and upper bounds, and such that the entropy function consisting of the sum of the terms z_{ij} ln(z_{ij}/a_{ij}) is minimized. When the lower and upper bounds coincide, matrix scaling and entropy minimization are closely related. In this paper we present several complexity bounds for the \epsilon-approximate (r, c)-scaling problem, polynomial in n, m, h, 1/\epsilon, and ln V , where V and v are the largest and the smallest positive entries of A, respectively. These bounds, although not polynomial in ln(1/\epsilon), not only provide alternative complexities for the polynomial time algorithms, but could result in better overall complexities. In particular, our theoretical analysis supports the practicality of the well-known RAS algorithm

Given S = {v1, . . . , vn} ⊂ R m and p ∈ R m , testing if p ∈ conv(S), the convex hull of S, is a fundamental problem in computational geometry and linear programming. First, we prove a Euclidean distance duality, distinct from classical separation theorems such as Farkas Lemma: p lies in conv(S) if and only if for each p ′ ∈ conv(S) there exists a pivot,Equivalently, p ∈ conv(S) if and only if there exists a witness, p ′ ∈ conv(S) whose Voronoi cell relative to p contains S. A witness separates p from conv(S) and approximate d(p, conv(S)) to within a factor of two. Next, we describe the Triangle Algorithm: given ǫ ∈ (0, 1), an iterate, p ′ ∈ conv(S), and, it stops. Otherwise, if there exists a pivot vj, it replace v with vj and p ′ with the projection of p onto the line p ′ vj . Repeating this process, the algorithm terminates in O(mn min{ǫ −2 , c −1 ln ǫ −1 }) arithmetic operations, where c is the visibility factor, a constant satisfying c ≥ ǫ 2 and sin(∠ppIn particular, the geometry of the input data may result in efficient complexities such as O(mn, t a natural number, or even O(mn ln ǫ −1 ). Additionally, (i) we prove a strict distance duality and a related minimax theorem, resulting in more effective pivots; (ii) describe O(mn ln ǫ −1 )-time algorithms that may compute a witness or a good approximate solution; (iii) prove generalized distance duality and describe a corresponding generalized Triangle Algorithm; (iv) prove a sensitivity theorem to analyze the complexity of solving LP feasibility via the Triangle Algorithm. The Triangle Algorithm is practical and competitive with the simplex method, sparse greedy approximation and first-order methods. Finally, we discuss future work on applications and generalizations.

No abstract

No abstract