Abstract
The Constraint Consensus method moves quickly from an initial infeasible point to a point that is close to feasibility for a set of nonlinear constraints. It is a useful first step prior to launching an expensive local solver, improving the probability that the local solver will find a solution and the speed with which it is found. The two main ingredients are the method for calculating the feasibility vector for each violated constraint (the estimated vector to the closest point that satisfies the constraint), and the method of combining the feasibility vectors into a single consensus vector that updates the current point. We propose several improvements: (i) a simple new method for calculating the consensus vector, (ii) a predictor-corrector approach to adjusting the consensus vector, (iii) an improved way of selecting the output point, and (iv) ways of selecting subsets of the constraints to operate on at a given iteration. These techniques greatly improve the performance of barrier method local solvers. Quadratic feasibility vectors are also investigated. Empirical results are given for a large set of nonlinear and nonconvex models.
Similar content being viewed by others
References
Chinneck, J.W.: The constraint consensus method for finding approximately feasible points in nonlinear programs. INFORMS J. Comput. 16(3), 255–265 (2004)
Ibrahim, W., Chinneck, J.W.: Improving solver success in reaching feasibility for sets of nonlinear constraints. Comput. Oper. Res. 35(5), 1394–1411 (2008)
Chinneck, J.W.: Feasibility and Infeasibility in Optimization: Algorithms and Computational Methods, International Series in Operations Research and Management Sciences, vol. 118. Springer, Berlin (2008)
Mittelmann, H.D.: Benchmarks for optimization software: AMPL-NLP benchmark. Accessed Online: October 29 (2010). http://plato.asu.edu/ftp/ampl-nlp.html
Censor, Y., Zenios, S.: Parallel Optimization: Theory, Algorithms, and Applications. Oxford Univ. Press, New York (1997)
Gubin, L.G., Polyak, B.T., Raik, E.V.: The method of projections for finding the common point of convex sets. U.S.S.R. Comput. Math. Math. Phys. 7, 1–24 (1967)
Cimmino: Calcolo approssimato per soluzioni dei sistemi di equazioni lineari. La Ricerca Sci. XVI, Ser. II, IX 1, 326–333
Aharoni, R., Censor, Y.: Block-iterative projection methods for parallel computation of solutions to convex feasibility problems. Linear Algebra Appl. 120, 165–175 (1989)
Censor, Y., Lent, A.: Cyclic subgradient projections. Math. Program. 24, 233–235 (1982)
Kaczmarz, S.: Angenherte Auflsung von Systemen Linearer Gleichungen. Bull. Int. Acad. Pol. Sci. Lett., Ser. A, 35, 335–357 (1937)
McCormick, S.F.: An iterative procedure for the solution of constrained nonlinear equations with application to optimization problems. Numer. Math. 23(5), 271–385 (1975)
Meyn, K.H.: Solution of underdetermined nonlinear equations by stationary iteration methods. Numer. Math. 42(2), 161–172 (1983)
Martinez, J.M.: Solving systems of nonlinear equations by means of an accelerated successive orthogonal projections method. J. Comput. Appl. Math. 16(2), 169–179 (1986)
Censor, Y., Gordon, D., Gordon, R.: Component averaging: an efficient iterative parallel algorithm for large and sparse unstructured problems. Parallel Comput. 27(6), 777–808 (2001)
Tjalling, Y.J.: Historical development of the Newton-Raphson method. SIAM Rev. 37(4), 531–551 (1995)
Combettes, P.L.: The convex feasibility problem in image recovery. In: Hawkes, P. (ed.) Advances in Imaging and Electron Physics, vol. 95, pp. 155–270 (1996)
Mehrotra, S.: On the implementation of a primal–dual interior point method. SIAM J. Optim. 2(4), 575–601 (1992)
MacLeod, M.: Multistart constraint consensus for seeking feasibility in nonlinear programs. MASc Thesis, Systems and Computer Engineering, Carleton University, Ottawa, Canada (2006)
Wachter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)
Shcherbina, O., Neumaier, A., Sam-Haroud, D., Vu, X.H., Nguyen, T.V.: Benchmarking global optimization and constraint satisfaction codes. In: Bliek, C. et al. (eds.) COCOS 2002, LNCS, vol. 2861, pp. 211–222 (2003)
Fourer, R., Gay, D.M.M., Kernighan, B.W.: AMPL: A Modeling Language for Mathematical Programming. Duxbury Press, N. Scituate (2002)
Wachter, A., Laird, C., Kawajir, Y.: Introduction to Ipopt: A tutorial for downloading, installing, and using Ipopt. Document Revision 1830, Accessed Online: July 25 (2011). https://projects.coin-or.org/Ipopt
Smith, L.R.: Improved placement of local solver launch points for large-scale global optimization. Ph.D. Thesis, Systems and Computer Engineering Department, Carleton University, Ottawa, Ontario, Canada (2011)
Dolan, E.D., More, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)
Author information
Authors and Affiliations
Corresponding author
Electronic Supplementary Material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Smith, L., Chinneck, J. & Aitken, V. Improved constraint consensus methods for seeking feasibility in nonlinear programs. Comput Optim Appl 54, 555–578 (2013). https://doi.org/10.1007/s10589-012-9473-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-012-9473-z