A recipe for finding good solutions to MINLPs

Finding good (or even just feasible) solutions for Mixed-Integer Nonlinear Programming problems independently of the specific problem structure is a very hard but practically important task, especially when the objective and/or the constraints are nonconvex. With this goal in mind, we present a general-purpose heuristic based on Variable Neighborhood Search, Local Branching, a local Nonlinear Programming algorithm and Branch-and-Bound. We test the proposed approach on MINLPLib, comparing with several existing heuristic and exact methods. An implementation of the proposed heuristic is freely available and can employ all NLP/MINLP solvers with an AMPL interface as the main search tools.     

Practical strategies for generating rank-1 split cuts in mixed-integer linear programming

In this paper we propose practical strategies for generating split cuts, by considering integer linear combinations of the rows of the optimal simplex tableau, and deriving the corresponding Gomory mixed-integer cuts; potentially, we can generate a huge number of cuts. A key idea is to select subsets of variables, and cut deeply in the space of these variables. We show that variables with small reduced cost are good candidates for this purpose, yielding cuts that close a larger integrality gap. An extensive computational evaluation of these cuts points to the following two conclusions. The first is that our rank-1 cuts improve significantly on existing split cut generators (Gomory cuts from single tableau rows, MIR, Reduce-and-Split, Lift-and-Project, Flow and Knapsack cover): on MIPLIB instances, these generators close 24% of the integrality gap on average; adding our cuts yields an additional 5%. The second conclusion is that, when incorporated in a Branch-and-Cut framework, these new cuts can improve computing time on difficult instances.     

A hybrid shuffled complex evolution approach based on differential evolution for unconstrained optimization

Numerous optimization methods have been proposed for the solution of the unconstrained optimization problems, such as mathematical programming methods, stochastic global optimization approaches, and metaheuristics. In this paper, a metaheuristic algorithm called Modified Shuffled Complex Evolution (MSCE) is proposed, where an adaptation of the Downhill Simplex search strategy combined with the differential evolution method is proposed. The efficiency of the new method is analyzed in terms of the mean performance and computational time, in comparison with the genetic algorithm using floating-point representation (GAF) and the classical shuffled complex evolution (SCE-UA) algorithm using six benchmark optimization functions. Simulation results and the comparisons with SCE-UA and GAF indicate that the MSCE improves the search performance on the five benchmark functions of six tested functions.     

Deciding the existence of uniform interpolants over transitive models

We consider the problem of the existence of uniform interpolants in the modal logic K4. We first prove that all ${\square}$ -free formulas have uniform interpolants in this logic. In the general case, we shall prove that given a modal formula ${\phi}$ and a sublanguage L of the language of the formula, we can decide whether ${\phi}$ has a uniform interpolant with respect to L in K4. The ${\square}$ -free case is proved using a reduction to the G?del L?b Logic GL, while in the general case we prove that the question of whether a modal formula has uniform interpolants over transitive frames can be reduced to a decidable expressivity problem on the???-calculus.     

On a positive set theory with inequality

We introduce a quite natural Frege‐style set theory, which we call Strong‐Frege‐2 $(\mathsf {SF}_2)$, a sort of simplification of the theory considered in 13 (under the name Strong‐Frege‐3) and 1 (under the name F2). We give a model of a weaker variant of $\mathsf {SF}_2$, called $\mathsf {SF}_2\mathsf {AC}$, where atoms and coatoms are allowed. To construct the model we use an enumeration “almost without repetitions” of the Π¹₁ sets of natural numbers; such an enumeration can be obtained via a classical priority argument much in the style of 5 and 15 . © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

On Fractional Evolution Inclusion Coupled with a Time and State Dependent Maximal Monotone Operator

Set-Valued and Variational Analysis - The paper deals with second-order evolution problems driven by time and state dependent maximal monotone operators with non-Lipschitz perturbations. Systems...     

On the dimensions of automorphism groups of 8-dimensional ternary fields I

Let be an eight-dimensional, connected, locally compact ternary field and let denote a connected closed subgroup of its automorphism group which is taken with the compact-open topology. It is proved that is either isomorphic to the compact exceptional Lie group G₂, or the (covering) dimension of is at most 11. This bound can be decreased to 10, if the ternary fixed fieldF of is connected.     

An uncertainty principle for the basic Bessel transform

The aim of this paper is to prove an uncertainty principle for the basic Bessel transform of order . In order to obtain a sharp uncertainty principle, we introduce and study a generalized q-Bessel-Dunkl transform which is based on the q-eigenfunctions of the q-Dunkl operator newly given by:

Prolongement d'un courant négatif plurisousharmonique avec condition sur les tranches

In this Note, we prove a theorem on the extension of a negative (or positive) plurisubharmonic current T (i.e. such that $d{d}^{c}T?0$) with condition on the slices with respect to some coordinates. This theorem generalizes a result proved by El Mir–Ben Messaoud relative to d-closed positive currents with a condition on slices. The method consists first of proving a Chern–Levine–Nirenberg inequality for a positive (or negative) psh current, which is a generalization of results obtained by Bedford–Taylor, Demailly and Sibony for d-closed positive currents. Also we prove an Oka type inequality for positive psh currents, thereby generalizing former results by Ben Messaoud–El Mir concerning positive currents with a negative $d{d}^c$. To cite this article: M. Toujani, H. Ben Messaoud, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Smooth Stable Planes

This paper deals with smooth stable planes which generalize the notion of differentiable (affine or projective) planes [7]. It is intended to be the first one of a series of papers on smooth incidence geometry based on the Habilitationsschrift of the author. It contains the basic definitions and results which are needed to build up a foundation for a systematic study of smooth planes. We define smooth stable planes, and we prove that point rows and line pencils are closed submanifolds of the point set and line set, respectively (Theorem (1.6)). Moreover, the flag space is a closed submanifold of the product manifold $P\times {\cal L}$ (Theorem (1.14)), and the smooth structure on the set P of points and on the set ${\cal L}$ of lines is uniquely determined by the smooth structure of one single line pencil. In the second section it is shown that for any point p \te P the tangent space T_pP carries the structure of a locally compact affine translation plane ${\cal A}_p$ , see Theorem (2.5). Dually, we prove in Section 3 that for any line $L \in {\cal L}$ the tangent space ${\rm T}_L{\cal L}$ together with the set ${\cal \rm S}_L=\lbrace {\rm T}_{L}{\cal L}_p\mid p \in L\rbrace$ gives rise to some shear plane. It turned out that the translation planes ${\cal A}_p$ are one of the most important tools in the investigation of smooth incidence geometries. The linearization theorems (3.9), (3.11), and (4.4) can be viewed as the main results of this paper. In the closing section we investigate some homogeneity properties of smooth projective planes.