K e y w o r d s V a r i a t i o n a l inequalities, Auxiliary principle, Iterative methods, Convergence. 1. I N T R O D U C T I O N Variational inequalities theory, which was introduced by Stampacchia [1] in 1964, has emerged as an interesting and fascinating branch of applicable mathematics with a wide range of applications in industry, physical, regional, social, pure, and applied sciences. T...

The Brunn-Minkowski inequality theory plays an important role in a number of mathematical disciplines such as measure theory, crystallography, optimal control theory, functional analysis, and geometric convexity. It has many useful applications in combinatorics, stochastic geometry, and mathematical economics. In recent years, several authors including Ball [1, 2, 3], Bourgain and Lindenstrauss...

i agree entirely with nir eyal’s perspective that denying treatment to obese patients is morally wrong. however, the reasons for this belief differ in some ways from eyal’s analysis. in this commentary, i will try to explain the similarities and differences in our perspectives. my primary claim is that the denial of treatment to obese patients is wrong principally because (i) it eschews a whole...

We consider the standard semidefinite programming (SDP) relaxation for the vertex cover problem to which all hypermetric inequalities supported on at most k vertices are added and show that the integrality gap for such SDPs remains 2− o(1) even for k = O( √ logn/ log log n). This extends results by Kleinberg-Goemans, Charikar and Hatami et al. who considered vertex cover SDPs tightened using th...

We survey recent research on mixed-integer rounding (MIR) inequalities and a generalization, namely the two-step MIR inequalities defined by Dash and Günlük (2006). We discuss the master cyclic group polyhedron of Gomory (1969) and discuss how other subadditive inequalities, similar to MIR inequalities, can be derived from this polyhedron. Recent numerical experiments have shed much light on th...

Many classes of valid and facet-inducing inequalities are known for the family of polytopes associated with the Symmetric Travelling Salesman Problem (STSP), including subtour elimination, 2-matching and comb inequalities. For a given class of inequalities, an exact separation algorithm is a procedure which, given an LP relaxation vector x∗, 7nds one or more inequalities in the class which are ...

We investigate efficient enumeration algorithms for conjunctive queries for databases over binary relations that satisfy the X property. Tree-like relations such as XPath axes or grids are natural examples of such relations. We first show that the result of an n-ary conjunctive query Q over an X structure S can be enumerated with a delay in O(n·|S|·|Q|) between two consecutive n-tuples. Then, w...

We show that there exists a family P of Knapsack polytopes such that for each P ∈ P and each ε > 0, any ε-approximated formulation of P in the original space Rn requires a number of inequalities that is super-polynomial in n. This answers a question by Bienstock and McClosky (2012). We also prove that, for any down-monotone polytope, an ε-approximated formulation in the original space can be ob...

We give an upper bound on the number of vertices of PI, the integer hull of a polyhedron P, in terms of the dimension n of the space, the number m of inequalities required to describe P, and the size ~ of these inequalities. For fixed n the bound is O(mn~n-1). We also describe an algorithm which determines the number of integer points in a polyhedron to within a multiplicative factor of 1 qE in...