# An interior-point method for the single-facility location problem with mixed norms using a conic formulation

- 70 Downloads
- 1 Citations

## 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.

## Keywords

Nonsymmetric conic optimization Conic reformulation Sum of norm minimization Single-facility location problems Interior-point methods## 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): 243–262MATHCrossRefMathSciNetGoogle Scholar
- Ben-Tal A, Nemirovski A (2001) Lectures on modern convex optimization. Analysis, Algorithms, and Engineering Applications, SIAM, PhiladelphiaMATHGoogle Scholar
- Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, LondonMATHGoogle Scholar
- Carrizosa E, Fliege J (2002) Generalized goal programming: Polynomial methods and applications. Math Programm 93(2): 281–303MATHCrossRefMathSciNetGoogle Scholar
- 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): 1–21CrossRefGoogle Scholar
- 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: 75–88CrossRefMathSciNetGoogle Scholar
- Fliege J (2000) Solving convex location problems with gauges in polynomial time. Stud Locat Anal 14: 153–172MATHMathSciNetGoogle Scholar
- Fliege J, Nickel S (2000) An interior point method for multifacility location problems with forbidden regions. Stud Locat Anal 14: 23–46MATHMathSciNetGoogle Scholar
- Glineur F (2001a) Proving strong duality for geometric optimization using a conic formulation. Ann Oper Res 105: 155–184MATHCrossRefMathSciNetGoogle Scholar
- Glineur F (2001b) Topics in convex optimization: interior-point methods, conic duality and approximations. Ph.D. thesis, Faculté Polytechnique de Mons, Mons, Belgium, January 2001Google Scholar
- Glineur F, Terlaky T (2004) Conic formulation for
*l*_{p}-norm optimization. J Optim Theory Appl 122(2): 285–307MATHCrossRefMathSciNetGoogle Scholar - Griewank A (2000) Evaluating derivatives: principles and techniques of algorithmic differentiation. Frontiers in applied mathematics, vol 19. SIAM, PhiladelphiaGoogle Scholar
- Karmarkar N (1984) A new polynomial-time algorithm for linear programming. Combinatorica 4: 373–395MATHCrossRefMathSciNetGoogle Scholar
- Love RF, Morris JG, Wesolowsky GO (1988) Facilities location: models & methods. North Holland, AmsterdamMATHGoogle Scholar
- Nash S, Sofer A (1998) On the complexity of a practical interior-point method. SIAM J Optim 8(3): 833–849MATHCrossRefMathSciNetGoogle Scholar
- Nesterov Y (1997) Interior-point methods: An old and new approach to nonlinear programming. Math Programm 79: 285–297MathSciNetGoogle Scholar
- Nesterov Y (2006) Towards nonsymmetric conic optimization. CORE Discussion Paper, 28Google Scholar
- Nesterov Y, Nemirovski A (1994) Interior-point polynomial algorithms in convex programming. SIAM, PhiladelphiaMATHGoogle Scholar
- Nesterov Y, Todd MJ (1997) Self-scaled barriers and interior-point methods for convex programming. Math Oper Res 22: 1–42MATHCrossRefMathSciNetGoogle Scholar
- Renegar J (2001) A mathematical view of interior-point methods in convex optimization. SIAM, PhiladelphiaMATHGoogle Scholar
- Weiszfeld E (1937) Sur le point par lequel le somme des distances de n points donnés est minimum. Tohoku Math J 4: 355–386Google Scholar
- Xue G, Ye Y (1997) An efficient algorithm for minimizing a sum of euclidean norms with applications. SIAM J Optim 7(4): 1017–1036MATHCrossRefMathSciNetGoogle Scholar
- Xue G, Ye Y (1998) An efficient algorithm for minimizing a sum of p-norms. SIAM J Optim 10(2): 551–579CrossRefMathSciNetGoogle Scholar