Concave function

About: Concave function is a research topic. Over the lifetime, 1415 publications have been published within this topic receiving 33278 citations.

Convex envelope results and strong formulations for a class of mixed‐integer programs

Abstract: In this article we present a novel technique for deriving the convex envelope of certain nonconvex fixed-charge functions of the type that arise in several related applications that have been considered in the literature. One common attribute of these problems is that they involve choosing levels for the undertaking of several activities. Two or more activities share a common resource, and a fixed charge is incurred when any of these activities is undertaken at a positive level. We consider nonconvex programming formulations for these problems in which the fixed charges are expressed in the form of concave functions. With the use of the developed convex envelope results, we show that the convex envelope relaxations of the nonconvex formulations lead to the linear programming relaxations of the strong IP/MIP formulations of these problems. Moreover, our technique for deriving convex envelopes offers a useful construct that could be exploited in other related contexts as well.

Goldstone models of modified gravity

Abstract: We investigate scalar-tensor theories where matter couples to the scalar field via a kinetically dependent conformal coupling. These models can be seen as the low-energy description of invariant field theories under a global Abelian symmetry. The scalar field is then identified with the Goldstone mode of the broken symmetry. It turns out that the properties of these models are very similar to the ones of ultralocal theories where the scalar-field value is directly determined by the local matter density. This leads to a complete screening of the fifth force in the Solar System and between compact objects, through the ultralocal screening mechanism. On the other hand, the fifth force can have large effects in extended structures with large-scale density gradients, such as galactic halos. Interestingly, it can either amplify or damp Newtonian gravity, depending on the model parameters. We also study the background cosmology and the linear cosmological perturbations. The background cosmology is hardly different from its $\Lambda$-CDM counterpart whilst cosmological perturbations crucially depend on whether the coupling function is convex or concave. For concave functions, growth is hindered by the repulsiveness of the fifth force whilst it is enhanced in the convex case. In both cases, the departures from the $\Lambda$-CDM cosmology increase on smaller scales and peak for galactic structures. For concave functions, the formation of structure is largely altered below some characteristic mass, as smaller structures are delayed and would form later through fragmentation, as in some warm dark matter scenarios. For convex models, small structures form more easily than in the $\Lambda$-CDM scenario.

A New Method for Interactive Multiobjective Optimization: A Boundary Point Ranking Method

Abstract: The method is based on a well-known method for unconstrained optimization developed by Nelder and Mead. It is assumed that the decision maker’s utility is not an explicitly known concave function of several objectives, which are linear functions of activities defined on a convex polyhedral set. It has thus been possible to modify the simplex procedure to handle constraints. Furthermore, in the interaction with the decision maker, the method only relies on a simple ranking procedure. The method seems to have many good properties. However, its advantages still have to be empirically verified.