Terms Of Disjunction Research Articles

On handling indicator constraints in mixed integer programming

Mixed integer programming (MIP) is commonly used to model indicator constraints, i.e., constraints that either hold or are relaxed depending on the value of a binary variable. Unfortunately, those models tend to lead to weak continuous relaxations and turn out to be unsolvable in practice; this is what happens, for e.g., in the case of Classification problems with Ramp Loss functions that represent an important application in this context. In this paper we show the computational evidence that a relevant class of these Classification instances can be solved far more efficiently if a nonlinear, nonconvex reformulation of the indicator constraints is used instead of the linear one. Inspired by this empirical and surprising observation, we show that aggressive bound tightening is the crucial ingredient for solving this class of instances, and we devise a pair of computationally effective algorithmic approaches that exploit it within MIP. One of these methods is currently part of the arsenal of IBM-Cplex since version 12.6.1. More generally, we argue that aggressive bound tightening is often overlooked in MIP, while it represents a significant building block for enhancing MIP technology when indicator constraints and disjunctive terms are present.

Implicational (semilinear) logics II: additional connectives and characterizations of semilinearity

This is the continuation of the paper (Cintula and Noguera in Arch Math Log 49(4):417---446, 2010). We continue the abstract study of non-classical logics based on the kind of generalized implication connectives they possess and we focus on semilinear logics, i.e. those that are complete with respect to the class of models where the implication defines a linear order. We obtain general characterizations of semilinearity in terms of the intersection-prime extension property, the syntactical semilinearity metarule and the class of finitely subdirectly irreducible models. Moreover, we consider extensions of the language with lattice connectives and generalized disjunctions, study their interplay with implication and obtain axiomatizations and further descriptions of semilinear logics in terms of disjunctions and the proof by cases property.

Rethinking the Relation between Science and Religion: Some Epistemological and Political Implications

Interview of In contemporary western societies we have become used to thinking of the relation between “science” and “religion” (or between “faith” and “reason”) in disjunctive terms, assuming a necessary opposition and/or the overcoming of one of them by the other (science as an understanding of the world necessarily opposed to religious beliefs and practices, one which tends historically to overcome the latter in the progress of civilization). An example of this pervasive assumption is the...

A high-performance algorithm for calculating conjunctive and disjunctive terms in polynomial representations of Boolean functions

The computer calculation of values of Boolean functions and systems of Boolean functions represented by normal polynomial forms starts from calculating values of terms of this polynomial followed by operations performed over the results. For the classical calculation method, this process is sequential. In this work, an algorithm is given that uses no more than one logical operation and one comparison to calculate the values of disjunctive and conjunctive terms of polynomial forms.

Una revisión del estado del arte en optimización

En este artículo se hace una revisión de las técnicas de optimización más importantes: programación lineal y no lineal con y sin variables discretas, así como de los algoritmos disponibles hasta la fecha para resolver dichos problemas. Además de los problemas clásicos se hace una extensión para incluir problemas planteados mediante disyunciones y relaciones lógicas, optimización global determinista y estocástica, optimización en problemas sin estructura definida y una visión global de los sistemas de modelado algebraico.

EvIA - Evidential Interval Algebra and Heuristic Backtrack-Free Algorithm

The Interval Algebra (IA) framework for temporal reasoning encodes indefinite knowledge in terms of disjunctions of relations. Many problems arising in practice can have evidences from past or from other external sources to indicate that some relations in a disjunction may be more probable than others. IA framework is inadequate to encode this information. The aim of the present study is two fold. First, to extend IA framework by associating numeric weights to the relations for capturing additional information and provide a reasoning methodology for the extended framework. Second, to apply the extended framework for developing a heuristic algorithm which finds a solution of the conventional IA network problem without backtrack. We make use of well-known evidential reasoning techniques to develop the new framework, Evidential Interval Algebra (EvIA). EvIA is an augmentation of interval algebra with evidential techniques. The constraint, constraint operators namely converse, composition and intersection, and path consistency algorithm of interval algebra are overlayed by evidential function and evidential operations to get enhanced expressiveness and efficient reasoning capability. The efficiency of the EvIA framework is demonstrated in the form of a heuristic which finds a solution of the interval algebra network without backtrack. Experimental results of the heuristic algorithm reveal that the algorithm is sound and for some specific types of the problems, the success of finding a solution is more than 90 percent. The results also show that the algorithm is efficient in terms of runtime when compared with a backtrack search algorithm.

Short Single Axioms for Boolean Algebra

We present short single equational axioms for Boolean algebra in terms of disjunction and negation and in terms of the Sheffer stroke. Previously known single axioms for these theories are much longer than the ones we present. We show that there is no shorter axiom in terms of the Sheffer stroke. Automated deduction techniques were used in several parts of the work.

Conditional Modeling. 2. Solving Using Complementarity and Boundary-Crossing Formulations

In this second paper on conditional modeling, we investigate the solving of the special subclass of conditional models which display continuous values for the problem variables as one crosses from one region to the next for such models. First, we present a new complementarity formulation for representing algebraic systems of disjunctive equations, and then we discuss and demonstrate the use of a boundary-crossing algorithm proposed by Zaher (Ph.D. Thesis, Carnegie Mellon University, Pittsburgh, PA, 1995). Our complementarity formulation not only establishes the complementarity condition among equations belonging to different disjunctive terms but also enforces simultaneous satisfaction of all the equations appearing within the same disjunctive term. This approach performed reliably on several example problems where the number of equations in each disjunctive term is small. Zaher also proposed the boundary-crossing algorithm as an alternative to using MINLP techniques for solving the special subclass of continuous conditional models. This algorithm involves the execution of several well-differentiated activities including logical analysis and continuous reconfiguration of the equations constituting the problem, calculation of Newton-like steps, and calculation of subgradient steps. We describe the practical implementation of the boundary-crossing algorithm as a conditional-modeling solution tool. In such an implementation, we have integrated the entities created and/or used to perform each of the activities in an object-oriented solving engine: the conditional-modeling solver CMSlv. We solve several examples of conditional models in chemical engineering by using both the approaches for solving, and we identify the advantages and disadvantages of such approaches.

Transfinite expansions of arithmetic formulas

A transfinite iteration of carrying the classical existence ⌉∨ϰ⌉ (denoted by\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{\exists } x\)) forward in the form of constructive realizability (∃ϰ) by the scheme is examined. This idea was expressed by Shanin who used it to define the majorant of arithmetic formulas. For any negative formula F the expansion Fα of (transfinite) rank α is defined as the α-fold carrying forward of the quantifiers ∃ and ∀ by the scheme described above and of the retaining at the end only quantifier-free disjunctive terms. It is proved that the constructive truth of Fα is equivalent to the existence of a recursive infinite derivation of formula F (by the classical rules) of height α. This permits the proof of Fα being a disjunctive iteration of classical arithmetic. A modification of expansions Fα, equivalent to majorants in the sense of Shanin, is indicated.