Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev spaces and applications

We develop a constructive piecewise polynomial approximation theory in weighted Sobolev spaces with Muckenhoupt weights for any polynomial degree. The main ingredients to derive optimal error estimates for an averaged Taylor polynomial are a suitable weighted Poincare inequality, a cancellation property and a simple induction argument. We also construct a quasi-interpolation operator, built on local averages over stars, which is well defined for functions in $L^1$. We derive optimal error estimates for any polynomial degree on simplicial shape regular meshes. On rectangular meshes, these estimates are valid under the condition that neighboring elements have comparable size, which yields optimal anisotropic error estimates over $n$-rectangular domains. The interpolation theory extends to cases when the error and function regularity require different weights. We conclude with three applications: nonuniform elliptic boundary value problems, elliptic problems with singular sources, and fractional powers of elliptic operators

New computer-based search strategies for extreme functions of the Gomory--Johnson infinite group problem

We describe new computer-based search strategies for extreme functions for the Gomory--Johnson infinite group problem. They lead to the discovery of new extreme functions, whose existence settles several open questions.Comment: 54 pages, many figure

Copositive matrices with circulant zero pattern

Let n be an integer not smaller than 5 and let u1,...,un be nonnegative real n-vectors such that the indices of their positive elements form the sets 1,2,...,n-2,2,3,...,n-1,...,n,1,...,n-3, respectively. Here each index set is obtained from the previous one by a circular shift. The set of copositive forms which vanish on the vectors u1,...,un is a face of the copositive cone. We give an explicit semi-definite description of this face and of its subface consisting of positive semi-definite matrices, and study their properties. If the vectors u1,...,un and their positive multiples exhaust the zero set of an exceptional copositive form belonging to this face, then we call this form regular, otherwise degenerate. We show that degenerate forms are always extremal, and regular forms can be extremal only if n is odd. We construct explicit examples of extremal degenerate forms for any order n, and examples of extremal regular forms for any odd order n. The set of all degenerate forms, i.e., defined by different collections u1,...,un of zeros, is a submanifold of codimension 2n, the set of all regular forms a submanifold of codimension n

Theory and Applications of Robust Optimization

In this paper we survey the primary research, both theoretical and applied, in the area of Robust Optimization (RO). Our focus is on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying prominent theoretical results of RO, we also present some recent results linking RO to adaptable models for multi-stage decision-making problems. Finally, we highlight applications of RO across a wide spectrum of domains, including finance, statistics, learning, and various areas of engineering.Comment: 50 page