Mathematical programs with a two-dimensional reverse convex constraint

We consider the problem min{f(χ) : χ ∈ G, T(χ) ∉ int D}, where f is a lower semicontinuous function, G a compact, nonempty set in IRn, D a closed convex set in JR² with nonempty interior, and T a continuous mapping from IRn to IR². The constraint T(χ) ∉. int D is areverse convex constraint, so the feasible domain may be disconnected even when f, T are affine and G is a polytope. We show that this problem can be reduced to a quasiconcave minimization problem over a compact convex set in IR², and hence can be solved effectively provided f, T are convex and G is convex or discrete. In particular, we discuss areverse convex constraint of the form (c, χ) . (d, χ) ≤ 1. We also compare the approach in this paper with the parametric approach

Neutron scattering study of the field-dependent ground state and the spin dynamics in S=1/2 NH4CuCl3

Elastic and inelastic neutron scattering experiments have been performed on the dimer spin system NH4CuCl3, which shows plateaus in the magnetization curve at m=1/4 and m=3/4 of the saturation value. Two structural phase transitions at T1≈156  K and at T2=70  K lead to a doubling of the crystallographic unit cell along the b direction and as a consequence a segregation into different dimer subsystems. Long-range magnetic ordering is reported below TN=1.3  K. The magnetic field dependence of the excitation spectrum identifies successive quantum phase transitions of the dimer subsystems as the driving mechanism for the unconventional magnetization process in agreement with a recent theoretical model

Certainty Closure: Reliable Constraint Reasoning with Incomplete or Erroneous Data

Constraint Programming (CP) has proved an effective paradigm to model and solve difficult combinatorial satisfaction and optimisation problems from disparate domains. Many such problems arising from the commercial world are permeated by data uncertainty. Existing CP approaches that accommodate uncertainty are less suited to uncertainty arising due to incomplete and erroneous data, because they do not build reliable models and solutions guaranteed to address the user's genuine problem as she perceives it. Other fields such as reliable computation offer combinations of models and associated methods to handle these types of uncertain data, but lack an expressive framework characterising the resolution methodology independently of the model. We present a unifying framework that extends the CP formalism in both model and solutions, to tackle ill-defined combinatorial problems with incomplete or erroneous data. The certainty closure framework brings together modelling and solving methodologies from different fields into the CP paradigm to provide reliable and efficient approches for uncertain constraint problems. We demonstrate the applicability of the framework on a case study in network diagnosis. We define resolution forms that give generic templates, and their associated operational semantics, to derive practical solution methods for reliable solutions.Comment: Revised versio

Lagrange Duality in Set Optimization

Based on the complete-lattice approach, a new Lagrangian duality theory for set-valued optimization problems is presented. In contrast to previous approaches, set-valued versions for the known scalar formulas involving infimum and supremum are obtained. In particular, a strong duality theorem, which includes the existence of the dual solution, is given under very weak assumptions: The ordering cone may have an empty interior or may not be pointed. "Saddle sets" replace the usual notion of saddle points for the Lagrangian, and this concept is proven to be sufficient to show the equivalence between the existence of primal/dual solutions and strong duality on the one hand and the existence of a saddle set for the Lagrangian on the other hand

Set optimization - a rather short introduction

Recent developments in set optimization are surveyed and extended including various set relations as well as fundamental constructions of a convex analysis for set- and vector-valued functions, and duality for set optimization problems. Extensive sections with bibliographical comments summarize the state of the art. Applications to vector optimization and financial risk measures are discussed along with algorithmic approaches to set optimization problems

Mathematical programs with a two-dimensional reverse convex constraint

We consider the problem min{f(χ) : χ ∈ G, T(χ) ∉ int D}, where f is a lower semicontinuous function, G a compact, nonempty set in IRn, D a closed convex set in JR² with nonempty interior, and T a continuous mapping from IRn to IR². The constraint T(χ) ∉. int D is areverse convex constraint, so the feasible domain may be disconnected even when f, T are affine and G is a polytope. We show that this problem can be reduced to a quasiconcave minimization problem over a compact convex set in IR², and hence can be solved effectively provided f, T are convex and G is convex or discrete. In particular, we discuss areverse convex constraint of the form (c, χ) . (d, χ) ≤ 1. We also compare the approach in this paper with the parametric approach

High-resolution isotope analysis of tree rings

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