Formal Proofs for Nonlinear Optimization

Formal Proofs for Global Optimization - Templates and Sums of Squares. (Preuves formelles pour l'optimisation globale - Méthodes de gabarits et sommes de carrés)

Recent Progress in Proof Mining in Nonlinear Analysis

During the last two decades the program of ‘proof mining’ emerged which uses tools from mathematical logic (so-called proof interpretations) to systematically extract explicit quantitative information (e.g. rates of convergence) from prima facie nonconstructive proofs (e.g. convergence proofs). This has been applied particularly successful in the context of nonlinear analysis: fixed point theor...

An ILP-based Proof System for the Crossing Number Problem

Formally, approaches based on mathematical programming are able to find provably optimal solutions. However, the demands on a verifiable formal proof are typically much higher than the guarantees we can sensibly attribute to implementations of mathematical programs. We consider this in the context of the crossing number problem, one of the most prominent problems in topological graph theory. Th...

NLCertify: A Tool for Formal Nonlinear Optimization

NLCertify is a software package for handling formal certification of nonlinear inequalities involving transcendental multivariate functions. The tool exploits sparse semialgebraic optimization techniques with approximation methods for transcendental functions, as well as formal features. Given a box and a transcendental multivariate function as input, NLCertify provides OCaml libraries that pro...

Representing Proof Transformations for Program Optimization

In the proofs as programs methodology a program is derived from a formal constructive proof. Because of the close relationship between proof and program structure, transformations can be applied to proofs rather than to programs in order to improve performance. We describe a method for implementing transformations of formal proofs and show that it is applicable to the optimization of extracted ...