Computational aspects of infeasibility analysis in mixed integer programming

The analysis of infeasible subproblems plays an important role in solving mixed integer programs (MIPs) and is implemented in most major MIP solvers. There are two fundamentally different concepts to generate valid global constraints from infeasible subproblems: conflict graph analysis and dual proof analysis. While conflict graph analysis detects sets of contradicting variable bounds in an implication graph, dual proof analysis derives valid linear constraints from the proof of the dual LP’s unboundedness. The main contribution of this paper is twofold. Firstly, we present three enhancements of dual proof analysis: presolving via variable cancellation, strengthening by applying mixed integer rounding functions, and a filtering mechanism. Further, we provide a comprehensive computational study evaluating the impact of every presented component regarding dual proof analysis. Secondly, this paper presents the first combined approach that uses both conflict graph and dual proof analysis simultaneously within a single MIP solution process. All experiments are carried out on general MIP instances from the standard public test set Miplib 2017; the presented algorithms have been implemented within the non-commercial MIP solver and the commercial MIP solver .

From mathematical axioms to mathematical rules of proof: recent developments in proof analysis

A short text in the hand of David Hilbert, discovered in Göttingen a century after it was written, shows that Hilbert had considered adding a 24th problem to his famous list of mathematical problems of the year 1900. The problem he had in mind was to find criteria for the simplicity of proofs and to develop a general theory of methods of proof in mathematics. In this paper, it is discussed to what extent proof theory has achieved the second of these aims. This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert's 24th problem’.

Lakatos, Imre (1922–74)

Imre Lakatos made important contributions to the philosophy of mathematics and of science. His ‘Proofs and Refutations’ (1963–4) develops a novel account of mathematical discovery. It shows that counterexamples (‘refutations’) play an important role in mathematics as well as in science and argues that both proofs and theorems are gradually improved by searching for counterexamples and by systematic ‘proof analysis’. His ‘methodology of scientific research programmes’ (which he presented as a ‘synthesis’ of the accounts of science given by Popper and by Kuhn) is based on the idea that science is best analysed, not in terms of single theories, but in terms of broader units called research programmes. Such programmes issue in particular theories, but in a way again governed by clear-cut heuristic principles. Lakatos claimed that his account supplies the sharp criteria of ‘progress’ and ‘degeneration’ missing from Kuhn’s account, and hence captures the ‘rationality’ of scientific development. Lakatos also articulated a ‘meta-methodology’ for appraising rival methodologies of science in terms of the ‘rational reconstructions’ of history they provide.

Knowledge, Luck, and Virtue

The Gettier Problem is conceived in a specific fashion as the problem of offering an informative (but not necessarily reductive) Gettier-proof analysis of knowledge. A solution is offered to this problem via anti-luck virtue epistemology. This is an account of knowledge which incorporates both an anti-luck condition and a virtue condition, and which is thereby able to avoid problems that face some of the main competing accounts of knowledge, particularly those offered by proponents of robust virtue epistemology. In particular, it is able to accommodate the epistemic dependence of knowledge on external factors, where this has both a positive and a negative aspect. Relatedly, it can also avoid the problem posed by epistemic twin earth cases. Anti-luck virtue epistemology is then motivated and defended in light of a range of objections, in order to demonstrate its potential as a resolution to the Gettier Problem, so conceived.

ELIMINATING DISJUNCTIONS BY DISJUNCTION ELIMINATION

Completeness and other forms of Zorn’s Lemma are sometimes invoked for semantic proofs of conservation in relatively elementary mathematical contexts in which the corresponding syntactical conservation would suffice. We now show how a fairly general syntactical conservation theorem that covers plenty of the semantic approaches follows from an utmost versatile criterion for conservation given by Scott in 1974.To this end we work with multi-conclusion entailment relations as extending single-conclusion entailment relations. In a nutshell, the additional axioms with disjunctions in positive position can be eliminated by reducing them to the corresponding disjunction elimination rules, which in turn prove admissible in all known mathematical instances. In deduction terms this means to fold up branchings of proof trees by way of properties of the relevant mathematical structures.Applications include the syntactical counterparts of the theorems or lemmas known under the names of Artin–Schreier, Krull–Lindenbaum, and Szpilrajn. Related work has been done before on individual instances, e.g., in locale theory, dynamical algebra, formal topology and proof analysis.