Intermediate integer programming representations using value disjunctions

We introduce a general technique to create an extended formulation of a mixed-integer program. We classify the integer variables into blocks, each of which generates a finite set of vector values. The extended formulation is constructed by creating a new binary variable for each generated value. Initial experiments show that the extended formulation can have a more compact complete description than the original formulation. We prove that, using this reformulation technique, the facet description decomposes into one ``linking polyhedron'' per block and the ``aggregated polyhedron''. Each of these polyhedra can be analyzed separately. For the case of identical coefficients in a block, we provide a complete description of the linking polyhedron and a polynomial-time separation algorithm. Applied to the knapsack with a fixed number of distinct coefficients, this theorem provides a complete description in an extended space with a polynomial number of variables.Comment: 26 pages, 5 figure

Intermediate integer programming representations using value disjunctions

We introduce a general technique for creating an extended formulation of a mixed-integer program. We classify the integer variables into blocks, each of which generates a finite set of vector values. The extended formulation is constructed by creating a new binary variable for each generated value. Initial experiments show that the extended formulation can have a more compact complete description than the original formulation. We prove that, using this reformulation technique, the facet description decomposes into one “linking polyhedron” per block and the “aggregated polyhedron”. Each of these polyhedra can be analyzed separately. For the case of identical coefficients in a block, we provide a complete description of the linking polyhedron and a polynomial-time separation algorithm. Applied to the knapsack with a fixed number of distinct coefficients, this theorem provides a complete description in an extended space with a polynomial number of variables. On the basis of this theory, we propose a new branching scheme that analyzes the problem structure. It is designed to be applied in those subproblems of hard integer programs where LP-based techniques do not provide good branching decisions. Preliminary computational experiments show that it is successful for some benchmark problems of multi-knapsack type

Transformation of propositional calculus statements into integer and mixed integer programs: An approach towards automatic reformulation

A systematic procedure for transforming a set of logical statements or logical conditions imposed on a model into an Integer Linear Progamming (ILP) formulation Mixed Integer Programming (MIP) formulation is presented. An ILP stated as a system of linear constraints involving integer variables and an objective function, provides a powerful representation of decision problems through a tightly interrelated closed system of choices. It supports direct representation of logical (Boolean or prepositional calculus) expressions. Binary variables (hereafter called logical variables) are first introduced and methods of logically connecting these to other variables are then presented. Simple constraints can be combined to construct logical relationships and the methods of formulating these are discussed. A reformulation procedure which uses the extended reverse polish representation of a compound logical form is then described. These reformulation procedures are illustrated by two examples. A scheme of implementation.ithin an LP modelling system is outlined

Deadlock detection of Java Bytecode

This paper presents a technique for deadlock detection of Java programs. The technique uses typing rules for extracting infinite-state abstract models of the dependencies among the components of the Java intermediate language -- the Java bytecode. Models are subsequently analysed by means of an extension of a solver that we have defined for detecting deadlocks in process calculi. Our technique is complemented by a prototype verifier that also covers most of the Java features.Comment: Pre-proceedings paper presented at the 27th International Symposium on Logic-Based Program Synthesis and Transformation (LOPSTR 2017), Namur, Belgium, 10-12 October 2017 (arXiv:1708.07854

On Computational Small Steps and Big Steps: Refocusing for Outermost Reduction

We study the relationship between small-step semantics, big-step semantics and abstract machines, for programming languages that employ an outermost reduction strategy, i.e., languages where reductions near the root of the abstract syntax tree are performed before reductions near the leaves.In particular, we investigate how Biernacka and Danvy's syntactic correspondence and Reynolds's functional correspondence can be applied to inter-derive semantic specifications for such languages.The main contribution of this dissertation is three-fold:First, we identify that backward overlapping reduction rules in the small-step semantics cause the refocusing step of the syntactic correspondence to be inapplicable.Second, we propose two solutions to overcome this in-applicability: backtracking and rule generalization.Third, we show how these solutions affect the other transformations of the two correspondences.Other contributions include the application of the syntactic and functional correspondences to Boolean normalization.In particular, we show how to systematically derive a spectrum of normalization functions for negational and conjunctive normalization

Counting inequivalent monotone Boolean functions

Monotone Boolean functions (MBFs) are Boolean functions $f: {0,1}^n \rightarrow {0,1}$ satisfying the monotonicity condition $x \leq y \Rightarrow f(x) \leq f(y)$ for any $x,y \in {0,1}^n$. The number of MBFs in n variables is known as the $n$th Dedekind number. It is a longstanding computational challenge to determine these numbers exactly - these values are only known for $n$ at most 8. Two monotone Boolean functions are inequivalent if one can be obtained from the other by renaming the variables. The number of inequivalent MBFs in $n$ variables was known only for up to $n = 6$. In this paper we propose a strategy to count inequivalent MBF's by breaking the calculation into parts based on the profiles of these functions. As a result we are able to compute the number of inequivalent MBFs in 7 variables. The number obtained is 490013148

Synthesis and design of integrated reaction-separation systems with complex configurations and rigorous models

Chemical engineering, and specially process design, synthesis and intensification, are well positioned to support both society and industry in overcoming present global challenges of environment degradation, energy supply, water scarcity and food supply. These challenges have been translated into industrial problems that involve the design of chemical processes with decreased water and energy consumption, and improved efficiencies. In this context the present study focuses on the simultaneous synthesis and design of reaction-separation systems including complex configuration distillation columns and using rigorous models. The study is considered a further step in this research area, as previous works have usually focused on the synthesis of sub-networks and have used shortcut models. Additionally, among complex configuration, thermally coupled distillation columns are reported to present significant savings in terms of the total annualised cost of the system. Among the available approaches to synthesis and design, a superstructure optimisation approach is used. The procedure involves the construction of a superstructure that includes a reaction superstructure, taken from Ma et al. (Ma et al. 2019) and a separation superstructure, proposed by Sargent and Gaminibandara (Sargent and K. Gaminibandara 1976). The modelling is performed using generalised disjunctive programming (GDP) to produce a logic-based model. This model is then reformulated into a mixed-integer nonlinear programming (MINLP) optimisation problem, where the objective is to minimise the total annualised cost of the process. For the reformulation convex hull and bypass efficiency methods are used. A modified version of the solving strategy presented by Ma et al. (Ma et al. 2019) is used, which involves using the solver SBB in General Algebraic Modelling System (GAMS). The proposed framework is applied to a case study previously addressed by Zhang et al. (Zhang et al. 2018) and Ma et al. (Ma et al. 2019). Economic models and assumptions made in those studies are maintained in order to evaluate the benefits of including complex configuration columns in the design possibilities. Results present a flowsheet with one PFR reactor and complex configuration distillation columns that are partially thermally coupled. The total annualised cost of the process is 5.85x105 $/yr, which is 6.3% and 4.7% less than the value achieved by Zhang et al. (Zhang et al. 2018)and Ma et al., respectively. Results show that it is both possible and beneficial to consider complex configuration distillation columns, including thermally coupled ones, in the simultaneous synthesis and design of reaction-separation systems using rigorous models.Chevening AwardsAgencia Nacional de Investigación e Innovació

Predicting globally-coherent temporal structures from texts via endpoint inference and graph decomposition

International audienceAn elegant approach to learning temporal order- ings from texts is to formulate this problem as a constraint optimization problem, which can be then given an exact solution using Integer Linear Programming. This works well for cases where the number of possible relations between temporal entities is restricted to the mere precedence rela- tion [Bramsen et al., 2006; Chambers and Jurafsky, 2008], but becomes impractical when considering all possible interval relations. This paper proposes two innovations, inspired from work on temporal reasoning, that control this combinatorial blow-up, therefore rendering an exact ILP inference viable in the general case. First, we translate our network of constraints from temporal intervals to their end- points, to handle a drastically smaller set of con- straints, while preserving the same temporal infor- mation. Second, we show that additional efficiency is gained by enforcing coherence on particular sub- sets of the entire temporal graphs. We evaluate these innovations through various experiments on TimeBank 1.2, and compare our ILP formulations with various baselines and oracle systems