Abstract
Convexity is an important property in nonlinear optimization since it allows to apply efficient local methods for finding global solutions. We propose to apply symbolic methods to prove or disprove convexity of rational functions over a polyhedral domain. Our algorithms reduce convexity questions to real quantifier elimination problems. Our methods are implemented and publicly available in the open source computer algebra system Reduce. Our long term goal is to integrate Reduce as a “workhorse” for symbolic computations into a numerical solver.
This work was supported by the DFG Research Center Matheon Mathematics for key technologies in Berlin, http://www.matheon.de
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References
Ballerstein, M., Michaels, D., Seidel-Morgenstern, A., Weismantel, R.: A theoretical study of continuous counter-current chromatography for adsorption isotherms with inflection points. Computers & Chemical Engineering 34(4), 447–459 (2010)
Grossmann, I.E. (ed.): Global Optimization in Engineering Design. Kluwer Academic Publishers, Dordrecht (1996)
Grossmann, I.E., Kravanja, Z.: Mixed-integer nonlinear programming: A survey of algorithms and applications. In: Conn, A., Biegler, L., Coleman, T., Santosa, F. (eds.) Large-Scale Optimization with Applications, Part II: Optimal Design and Control. Springer, Heidelberg (1997)
Jüdes, M., Tsatsaronis, G., Vigerske, S.: Optimization of the design and partial-load operation of power plants using mixed-integer nonlinear programming. In: Kallrath, J., Pardalos, P., Rebennack, S., Scheidt, M. (eds.) Optimization in the Energy Industry. Springer, Heidelberg (2009)
Tawarmalani, M., Sahinidis, N.V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications. Kluwer Academic Publishers, Dordrecht (2002)
Nocedal, J., Wright, S.: Numerical Optimization. Springer, Heidelberg (2000)
Adjiman, C.S., Floudas, C.A.: Rigorous convex underestimators for general twice-differentiable problems. Journal of Global Optimization 9, 23–40 (1997)
Fourer, R., Maheshwari, C., Neumaier, A., Orban, D., Schichl, H.: Convexity and concavity detection in computational graphs: Tree walks for convexity assessment. INFORMS Journal on Computing 22(1), 26–43 (2009)
Mönnigmann, M.: Efficient calculation of bounds on spectra of Hessian matrices. SIAM Journal on Scientific Computing 30(5), 2340–2357 (2008)
Nenov, I.P., Fylstra, D.H., Kolev, L.V.: Convexity determination in the Microsoft Excel solver using automatic differentiation techniques. Technical report, Frontline Systems Inc. (2004)
Sturm, T., Weber, A.: Investigating generic methods to solve Hopf bifurcation problems in algebraic biology. In: Horimoto, K., Regensburger, G., Rosenkranz, M., Yoshida, H. (eds.) AB 2008. LNCS, vol. 5147, pp. 200–215. Springer, Heidelberg (2008)
Sturm, T., Weber, A., Abdel-Rahman, E.O., El Kahoui, M.: Investigating algebraic and logical algorithms to solve Hopf bifurcation problems in algebraic biology. Mathematics in Computer Science 2(3), 493–515 (2009)
Tarski, A.: A decision method for elementary algebra and geometry. Prepared for publication by J.C.C. McKinsey. RAND Report R109, August 1 (1948) (revised May 1951); Second Edition, RAND, Santa Monica, CA (1957)
Basu, S., Pollack, R., Roy, M.F.: On the combinatorial and algebraic complexity of quantifier elimination. Journal of the ACM 43(6), 1002–1045 (1996)
Weispfenning, V.: The complexity of linear problems in fields. Journal of Symbolic Computation 5(1&2), 3–27 (1988)
Weispfenning, V.: Quantifier elimination for real algebra—the quadratic case and beyond. Applicable Algebra in Engineering Communication and Computing 8(2), 85–101 (1997)
Collins, G.E., Hong, H.: Partial cylindrical algebraic decomposition for quantifier elimination. Journal of Symbolic Computation 12(3), 299–328 (1991)
Dolzmann, A., Sturm, T.: Redlog: Computer algebra meets computer logic. ACM SIGSAM Bulletin 31(2), 2–9 (1997)
Davenport, J.H., Heintz, J.: Real quantifier elimination is doubly exponential. Journal of Symbolic Computation 5(1-2), 29–35 (1988)
Dolan, E.D., Moré, J.J., Munson, T.S.: Benchmarking optimization software with COPS 3.0. Technical Report ANL/MCS-273, Mathematics and Computer Science Division, Argonne National Laboratory (2004), http://www.mcs.anl.gov/~more/cops
Bonami, P., Kilinç, M., Linderoth, J.: Algorithms and software for convex mixed integer nonlinear programs (2009), Optimization Online, http://www.optimization-online.org/DB_HTML/2009/10/2429.html
Bussieck, M.R., Drud, A.S., Meeraus, A.: MINLPLib—A Collection of Test Models for Mixed-Integer Nonlinear Programming. INFORMS Journal on Computing 15(1), 114–119 (2003), http://www.gamsworld.org/minlp/minlplib.htm
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Neun, W., Sturm, T., Vigerske, S. (2010). Supporting Global Numerical Optimization of Rational Functions by Generic Symbolic Convexity Tests . In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2010. Lecture Notes in Computer Science, vol 6244. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15274-0_19
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DOI: https://doi.org/10.1007/978-3-642-15274-0_19
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