Convexity and Concavity Detection in Computational Graphs: Tree Walks for Convexity Assessment

Authors:
Robert Fourer;Chandrakant Maheshwari;Arnold Neumaier;Dominique Orban;Hermann Schichl
Affiliations:
Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois 60208;Department of Mechanical Engineering, India Institute of Technology Guwahati, Guwahati 781039, India;Department of Mathematics, University of Vienna, A-1090 Vienna, Austria;GERAD and Département de Mathématiques et Génie Industriel, École Polytechnique de Montréal, Montréal, Québec H3C 3A7, Canada;Department of Mathematics, University of Vienna, A-1090 Vienna, Austria
Venue:
INFORMS Journal on Computing
Year:
2010

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Abstract

We examine symbolic tools associated with two modeling systems for mathematical programming, which can be used to automatically detect the presence or absence of convexity and concavity in the objective and constraint functions, as well as convexity of the feasible set in some cases. The coconut solver system [Schichl, H. 2004a. COCONUT: COntinuous CONstraints---Updating the technology] focuses on nonlinear global continuous optimization and possesses its own modeling language and data structures. The Dr. Ampl meta-solver [Fourer, R., D. Orban. 2007. Dr. Ampl---A meta solver for optimization. Technical Report G-2007-10, GERAD, Montréal] aims to analyze nonlinear differentiable optimization models and hooks into the ampl Solver Library [Gay, D. M. 2002. Hooking your solver to AMPL]. Our symbolic convexity analysis may be supplemented, when it returns inconclusive results, with a numerical phase that may detect nonconvexity. We report numerical results using these tools on sets of test problems for both global and local optimization.