Disjunctive Rado numbers

If L₁ and L₂ are linear equations, then the disjunctive Rado number of the set {L₁,L₂} is the least integer n, provided that it exists, such that for every 2-coloring of the set {1,2,…,n} there exists a monochromatic solution to either L₁ or L₂. If such an integer n does not exist, then the disjunctive Rado number is infinite. In this paper, it is shown that for all integers and b1, the disjunctive Rado number for the equations x₁+a=x₂ and x₁+b=x₂ is a+b+1-gcd(a,b) if is odd and the disjunctive Rado number for these equations is infinite otherwise. It is also shown that for all integers a>1 and b>1, the disjunctive Rado number for the equations ax₁=x₂ and bx₁=x₂ is c^s+t-1 if there exist natural numbers c,s, and t such that a=c^s and b=c^t and s+t is an odd integer and c is the largest such integer, and the disjunctive Rado number for these equations is infinite otherwise.     

Disjunctive networks and update schedules

In this paper, we present a study of the dynamics of disjunctive networks under all block-sequential update schedules. We also present an extension of this study to more general fair periodic update schedules, that is, periodic update schedules that do not update some elements much more often than some others. Our main aim is to classify disjunctive networks according to the robustness of their dynamics with respect to changes of their update schedules. To study this robustness, we focus on one property, that of being able to cycle dynamically.     

Disjunctive programming and relaxations of polyhedra

Given a polyhedron \(L\) with \(h\) facets, whose interior contains no integral points, and a polyhedron \(P\), recent work in integer programming has focused on characterizing the convex hull of \(P\) minus the interior of \(L\). We show that to obtain such a characterization it suffices to consider all relaxations of \(P\) defined by at most \(n(h-1)\) among the inequalities defining \(P\). This extends a result by Andersen, Cornuéjols, and Li.     

主范式的运算性质

研究了极大项、极小项的运算性质 ,利用这些性质给出了求 A,A∨ B,A∧ B,A→ B,A B的主范式的公式 ,由此可用程序化的方法求任意公式的主范式 .     

Disjunctive cuts in Mixed-Integer Conic Optimization

Mathematical Programming - This paper studies disjunctive cutting planes in Mixed-Integer Conic Programming. Building on conic duality, we formulate a cut-generating conic program for separating...     

Restorations of punctured languages and similarity of languages

Punctured languages are languages whose words are partial words in the sense that the letters at some positions are unknown. We investigate to which extent restoration of punctured languages is possible if the number of unknown positions or the proportion of unknown positions per word, respectively, is bounded, and we study their relationships for different boundings. The considered restoration classes coincide with similarity classes according to some kind of similarity for languages. Thus all results we can also formulate in the language of similarity. We show some hierarchies of similarity classes for each class from the Chomsky hierarchy and prove the existence of linear languages which are not δ ‐similar to any regular language for any δ < ½. For δ ≥ ½ this is unknown but it could only be possible in the case of non‐slender linear languages. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Disjunctive Optimization: Critical Point Theory

In this paper, we introduce the concepts of (nondegenerate) stationary points and stationary index for disjunctive optimization problems. Two basic theorems from Morse theory, which imply the validity of the (standard) Morse relations, are proved. The first one is a deformation theorem which applies outside the stationary point set. The second one is a cell-attachment theorem which applies at nondegenerate stationary points. The dimension of the cell to be attached equals the stationary index. Here, the stationary index depends on both the restricted Hessian of the Lagrangian and the set of active inequality constraints. In standard optimization problems, the latter contribution vanishes.     

Disjunctive cuts for continuous linear bilevel programming

This work shows how disjunctive cuts can be generated for a bilevel linear programming problem (BLP) with continuous variables. First, a brief summary on disjunctive programming and bilevel programming is presented. Then duality theory is used to reformulate BLP as a disjunctive program and, from there, disjunctive programming results are applied to derive valid cuts. These cuts tighten the domain of the linear relaxation of BLP. An example is given to illustrate this idea, and a discussion follows on how these cuts may be incorporated in an algorithm for solving BLP.     

Neural languages

In order to provide the theoretical framework necessary to study the neural mechanisms underlying languages, we present here a mathematical formalization of some neural behaviors. In such a context: (i) the neuron is defined as a coupling of automata, dealing with the transduction and codifying processes; (ii) the coupling between neurons is measured by a membership relation defined as the ratio between the transmitted and the system entropies; (iii) the attention given to the messages arriving at the system is considered as the difference between the couplings of the excitatory and inhibitory pools of neurons; (iv) the graphs in the neural systems are described by means of these couplings and, finally (v) the semantic productions are described by the fuzzy formal languages accepted by the automata formalized on these graphs. In such a way, both the verbal and neural semantics can now be correlated and experimentally investigated.     

Solving Problems with Min-Type Functions by Disjunctive Programming

We consider applications of disjunctive programming to global optimization and problems with equilibrium constraints. We propose a modification of the algorithm of F. Beaumont for disjunctive programming problems and show its numerical efficiency.     

On the Complexity of Dualization of Monotone Disjunctive Normal Forms

We show that the duality of a pair of monotone disjunctive normal forms of sizencan be tested inn^{o(log n)}time.     

System function languages

In this paper we define the concept of a system function language which is a language generated by a system function. We identify system function languages with recursively enumerable sets which are non-simple and co-infinite. We then define restricted system function languages and identify them with recursive sets which are co-infinite. Finally we state and prove some independence and dependence relationships between system function languages and some of the more well-known decision problems. MSC: 03D05, 03D20, 03D25.     

Optimization of Discrete-Continuous Dynamic Systems Based on Disjunctive Programming

In this contribution, a novel approach for the modeling and optimization of discrete-continuous dynamic systems based on a disjunctive problem formulation is proposed. It will be shown that a disjunctive model representation, which constitutes an alternative to mixed-integer model formulations, provides a very flexible and intuitive way to formulate discrete-continuous dynamic optimization problems. Moreover, the structure and properties of the disjunctive process models can be exploited for an efficient and robust numerical solution by applying generalized disjunctive programming techniques. The proposed modeling and optimization approach will be illustrated by means of an optimal control problem that embeds a linear discretecontinuous dynamic system. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Parametric Disjunctive Programming: One-Sided Differentiability of the Value Function

We consider a countable family of one-parameter convex programs and give sufficient conditions for the one-sided differentiability of its optimal value function. The analysis is based on the Borwein dual problem for a family of convex programs (a convex disjunctive program). We give conditions that assure stability of the situation of perfect duality in the Borwein theory.For the reader's convenience, we start with a review of duality results for families of convex programs. A parametric family of dual problems is introduced that contains the dual problems of Balas and Borwein as special cases. In addition, a vector optimization problem is defined as a dual problem. This generalizes a result by Helbig about families of linear programs.     

On-line user languages

The principal differences between on-line and off-line user languages are discussed in some detail. The characteristic features of the OPL programming language are mentioned together with experiences of Project MAC, pointing at increased efficiency in the use of computers.Work reported herein was supported by Project MAC, and M. I. T. research program sponsored by the Advanced Research Projects Agency, Department of Defense, under Office of Naval Research contract NONR-4102(01). Reproduction in whole or in part is permitted for any purpose of the United States Government.