Abstract
We consider the single-facility location problem with mixed norms, i.e. the problem of minimizing the sum of the distances from a point to a set of fixed points in \({\mathbb{R}^n}\) , where each distance can be measured according to a different p-norm. We show how this problem can be expressed into a structured conic format by decomposing the nonlinear components of the objective into a series of constraints involving three-dimensional cones. Using the availability of a self-concordant barrier for these cones, we present a polynomial-time algorithm (a long-step path-following interior-point scheme) to solve the problem up to any given accuracy. Finally, we report computational results for this algorithm and compare with standard nonlinear optimization solvers applied to this problem.
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References
Andersen KD, Christiansen E, Conn AR, Overton ML (2000) An efficient primal–dual interior-point method for minimizing a sum of euclidean norms. SIAM J Sci Comput 22(1): 243–262
Ben-Tal A, Nemirovski A (2001) Lectures on modern convex optimization. Analysis, Algorithms, and Engineering Applications, SIAM, Philadelphia
Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, London
Carrizosa E, Fliege J (2002) Generalized goal programming: Polynomial methods and applications. Math Programm 93(2): 281–303
del Mar Hershenson M, Boyd SP, Lee TH (2001) Optimal design of a cmos op-amp via geometric programming. IEEE Trans Comput Aided Des Integr Circuits Syst 20(1): 1–21
den Hertog D, Jarre F, Roos C, Terlaky T (1995) A sufficient condition for self-concordance with application to some classes of structured convex programming problems. Math Programm 69: 75–88
Fliege J (2000) Solving convex location problems with gauges in polynomial time. Stud Locat Anal 14: 153–172
Fliege J, Nickel S (2000) An interior point method for multifacility location problems with forbidden regions. Stud Locat Anal 14: 23–46
Glineur F (2001a) Proving strong duality for geometric optimization using a conic formulation. Ann Oper Res 105: 155–184
Glineur F (2001b) Topics in convex optimization: interior-point methods, conic duality and approximations. Ph.D. thesis, Faculté Polytechnique de Mons, Mons, Belgium, January 2001
Glineur F, Terlaky T (2004) Conic formulation for l p -norm optimization. J Optim Theory Appl 122(2): 285–307
Griewank A (2000) Evaluating derivatives: principles and techniques of algorithmic differentiation. Frontiers in applied mathematics, vol 19. SIAM, Philadelphia
Karmarkar N (1984) A new polynomial-time algorithm for linear programming. Combinatorica 4: 373–395
Love RF, Morris JG, Wesolowsky GO (1988) Facilities location: models & methods. North Holland, Amsterdam
Nash S, Sofer A (1998) On the complexity of a practical interior-point method. SIAM J Optim 8(3): 833–849
Nesterov Y (1997) Interior-point methods: An old and new approach to nonlinear programming. Math Programm 79: 285–297
Nesterov Y (2006) Towards nonsymmetric conic optimization. CORE Discussion Paper, 28
Nesterov Y, Nemirovski A (1994) Interior-point polynomial algorithms in convex programming. SIAM, Philadelphia
Nesterov Y, Todd MJ (1997) Self-scaled barriers and interior-point methods for convex programming. Math Oper Res 22: 1–42
Renegar J (2001) A mathematical view of interior-point methods in convex optimization. SIAM, Philadelphia
Weiszfeld E (1937) Sur le point par lequel le somme des distances de n points donnés est minimum. Tohoku Math J 4: 355–386
Xue G, Ye Y (1997) An efficient algorithm for minimizing a sum of euclidean norms with applications. SIAM J Optim 7(4): 1017–1036
Xue G, Ye Y (1998) An efficient algorithm for minimizing a sum of p-norms. SIAM J Optim 10(2): 551–579
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This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. The first author acknowledges support by an FSR grant from Université Catholique de Louvain and a FRIA grant from Fonds de la Recherche Scientifique-FNRS. The scientific responsibility rests with its authors.
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Chares, R., Glineur, F. An interior-point method for the single-facility location problem with mixed norms using a conic formulation. Math Meth Oper Res 68, 383–405 (2008). https://doi.org/10.1007/s00186-008-0225-x
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DOI: https://doi.org/10.1007/s00186-008-0225-x