Abstract
In this chapter we present an algorithm of quasi-linear complexity, based on the calculation of the infimal convolution of convex quadratic functions, that leads to the determination of the analytical optimal solution of the Continuous Quadratic Knapsack problem. The algorithm both exactly and simultaneously solves a separable uniparametric family of quadratic programming problems resulting from varying the equality constraint. We prove that the analytical solution of the problem is piecewise quadratic, continuous and, under certain conditions, belongs to the class C 1. Moreover we analyze the complexity of the algorithm presented and prove that the complexity is quasi-linear in order. We demonstrate that our algorithm is able to deal with large-scale quadratic programming problems of this type. We present a very important application: the classical Problem of Economic Dispatch. Finally, we release the source code for our algorithm in the computer language Mathematica.
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References
Audet, C., Hansen, P., Le Digabel, S.: Exact solution of three nonconvex quadratic programming problems. In: Frontiers in Global Optimization. Nonconvex Optimization and Applications, vol. 20, pp. 25–45. Kluwer Acad. Publ., Dordrecht (2004)
Bayón, L., Grau, J.M., Suárez, P.M.: A New Formulation of the Equivalent Thermal in Optimization of Hydrothermal Systems. Math. Probl. Eng. 8(3), 181–196 (2002)
Bayón, L., Grau, J.M., Ruiz, M.M., Suárez, P.M.: New developments on equivalent thermal in hydrothermal optimization: an algorithm of approximation. J. Comput. Appl. Math. 175(1), 63–75 (2005)
Bayón, L., Grau, J.M., Ruiz, M.M., Suárez, P.M.: An analytic solution for some separable convex quadratic programming problems with equality and inequality constraints. Journal of Mathematical Inequalities 4(3), 453–465 (2010)
Bayón, L., Grau, J.M., Ruiz, M.M., Suárez, P.M.: A quasi-linear algorithm for calculating the infimal convolution of convex quadratic functions. In: Vigo-Aguiar, J. (ed.) Proceedings of the 2010 International Conference on Computational and Mathematical Methods in Science and Engineering, vol. I, pp. 169–172 (2010)
Cosares, S., Hochbaum, D.S.: Strongly polynomial algorithms for the quadratic transportation problem with a fixed number of sources. Math. Oper. Res. 19(1), 94–111 (1994)
Dostal, Z.: Inexact semimonotonic augmented Lagrangians with optimal feasibility convergence for convex bound and equality constrained quadratic programming. SIAM J. Numer. Anal. 43(1), 96–115 (2005)
Dostal, Z.: An optimal algorithm for bound and equality constrained quadratic programming problems with bounded spectrum. Computing 78(4), 311–328 (2006)
Goh, C.J., Yang, X.Q.: Analytic efficient solution set for multi-criteria quadratic programs. European J. Oper. Res. 92(1), 166–181 (1996)
Gould, N.I.M., Toint, P.L.: A Quadratic Programming Bibliography (2001), http://www.optimization-online.org/DB_HTML/2001/02/285.html
Gould, N.I.M., Toint, P.L.: A Quadratic Programming Page, http://www.numerical.rl.ac.uk/qp/qp.html
Kim, S., Kojima, M.: Exact Solutions of Some Nonconvex Quadratic Optimization Problems via SDP and SOCP Relaxations. Comput. Optim. Appl. 26(2), 143–154 (2003)
Megiddo, N., Tamir, A.: Linear time algorithms for some separable quadratic programming problems. Oper. Res. Lett. 13, 203–211 (1993)
Mittelmann, H.D.: Decision Tree for Optimization Software, http://plato.asu.edu/guide.html
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Stromberg, T.: The operation of infimal convolution. Diss. Math. 352 (1996)
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Bayón, L., Grau, J.M., Ruiz, M.M., Suárez, P.M. (2013). An Exact Algorithm for the Continuous Quadratic Knapsack Problem via Infimal Convolution. In: Zelinka, I., Snášel, V., Abraham, A. (eds) Handbook of Optimization. Intelligent Systems Reference Library, vol 38. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30504-7_5
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DOI: https://doi.org/10.1007/978-3-642-30504-7_5
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