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A Duality-Based Approach for Distributed Optimization with Coupling Constraints

IFAC-PapersOnLine 2017年第1期50卷 14326-14331页

作者： Notarnicola, Ivano Notarstefano, Giuseppe Department of Engineering Università del Salento Lecce Italy

In this paper we consider a distributed optimization scenario in which a set of agents has to solve a convex optimization problem with separable cost function, local constraint sets and a coupling inequality constraint. We propose a novel distributed algorithm based on a relaxation of the primal problem and an elegant exploration of duality theory. Despite its complex derivation based on several duality steps, the distributed algorithm has a very simple and intuitive structure. That is, each node solves a local version of the original problem relaxation, and updates suitable dual variables. We prove the algorithm correctness and show its effectiveness via numerical computations. © 2017

关键词： Distributed parameter control systems Constraint theory Convex optimization Cost functions Cyber Physical System Embedded systems Estimation Optimization algorithm correctness Convex optimization problems Distributed control Distributed optimization Inequality constraint Large scale network Large scale optimization Numerical computations

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algorithm Synthesis by Lazy Thinking: Examples and Implementation in Theorema

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Electronic Notes in Theoretical Computer Science 2004年 93卷 24-59页

作者： Bruno Buchberger Adrian Crăciun RISC - Research Institute for Symbolic Computation Johannes Kepler University Linz Austria

Recently, we proposed a systematic method for top-down synthesis and verification of lemmata and algorithms called “lazy thinking method” as a part of systematic mathematical theory exploration (mathematical knowledge management). The lazy thinking method is characterized: • by using a library of theorem and algorithm schemes • and by using the information contained in failing attempts to prove the schematic theorem or the correctness theorem for the algorithm scheme for inventing lemmata or requirements for subalgorithms , respectively. In this paper, we give a couple of examples for algorithm synthesis using the lazy thinking paradigm. These examples illustrate how the synthesized algorithm depends on the algorithm scheme used. Also, we give details about the implementation of the lazy thinking algorithm synthesis method in the frame of the Theorema system. In this implementation, the synthesis of the example algorithms can be carried out completely automatically, i.e. without any user interaction.

关键词： algorithm invention algorithm verification program synthesis algorithm correctness re-usable algorithms algorithm schemes learning from failure conjecture generation lazy thinking requirement engineering didactics of programming mathematical knowledge retrieval mathematical knowledge management sorting merging merge-sort Theorema

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Why is 1+1=2?

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JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 2002年第3期18卷 425-443页

作者： Yang, W Natl Chiao Tung Univ Dept Comp & Informat Sci Hsinchu 300 Taiwan

Too often we take the method of decimal addition to be the definition of addition. We clarify this misunderstanding by first giving a definition of addition and then proving that the decimal addition method is both sound and complete with respect to the given definition. By showing the soundness and completeness of decimal addition with respect to the addition of natural numbers, we actually propose a new approach to algorithm correctness. We also discuss the differences between our logic approach and the traditional axiomatic approach.

关键词： addition algorithm correctness arithmetic decimal addition decimal notation logic foundation of mathematics

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ON THE correctness OF A LINEAR-TIME VISIBILITY POLYGON algorithm

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 1990年第3-4期32卷 155-172页

作者： JOE, B Department of Computing Science University of Alberta Edmonton Alberta T6G 2HI Canada

We prove the correctness of the linear-time algorithm in Joe and Simpson [7] for computing the visibility polygon of a simple polygon from a viewpoint that is either interior to the polygon or in its blocked exterior. This proof establishes pre- and post-conditions for all steps of the algorithm. The algorithm and the proof are identical for interior and blocked exterior viewpoints. Other algorithms for computing the visibility polygon (usually for interior viewpoints only) are either nonlinear-time and harder to implement or can fail for polygons that wind sufficiently. © 1990, Taylor & Francis Group, LLC. All rights reserved.

关键词： algorithm correctness Computational geometry visibility

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MULTIVARIATE POLYNOMIAL FACTORIZATION

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JOURNAL OF THE ACM 1975年第2期22卷 291-308页

作者： MUSSER, DR UNIV WISCONSIN MADISON WI 53706 USA

This paper describes algorithms for factoring a polynomial in one or more variables, with integer coefficients, into factors which are irreducible over the integers These algorithms are based on the use of factorIzatmns over finite fields and “Hensel's lemma constructions” “Abstract algorithm” descripttons are used m the presentation of the underlying algebra:c theory Included s a new generahzatmn of Hensel's p-adlc constructmn which leads to a practical algorithm for factorlng multivariate polynommls The univarmte case algorithm is also specified m greater deta:l than in the prewous literature, with attentmn to a number of improvements wh:ch the author has developed based OR theoretical computing time analyses and experience with actual implementatmns. © 1975, ACM. All rights reserved.

关键词： abstract algorithms algorithm correctness Berlekamp's method factorlzation finite fields Hensel's lemma modular arithmet: c polynomials symbohc algebraic computation

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