# Journal of Advances in Applied Mathematics

### Scalarizations for Maximization with Respect to Polyhedral Cones

Download PDF (403.1 KB) PP. 151 - 163 Pub. Date: July 31, 2017

### Author(s)

**H.W Corley**^{*}

Center On Stochastic Modeling, Optimization, & Statistics (COSMOS), The University of Texas at Arlington, Arlington, Texas, United States

### Abstract

### Keywords

### References

[1] L. Hurwicz, “Programming in linear spaces,” Studies in linear and nonlinear programming, edited by K.J. Arrow, L. Hurwicz, and H. Uzawa, Stanford University Press, pp. 38-102, 1958.

[2] L.W. Neustadt, “A general theory of extremals,” Journal of computer and system sciences, vol. 3, pp. 57-92, 1969.

[3] K. Ritter, “Optimization theory in linear spaces III: Mathematical programming in partially ordered Banach spaces,” Mathematische annalen, vol. 184, pp. 133-154, 1970.

[4] B.D. Craven, “Nonlinear programming in locally convex spaces,” Journal of optimization theory and applications, vol. 10, pp. 197-210, 1972.

[5] P.L. Yu, “Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives,” Journal of optimization theory and applications, vol. 14, pp. 319–377, 1974.

[6] J. Borwein, “Proper efficient points for maximizations with respect to cones,” SIAM journal on control and optimization, vol. 15, pp. 57-63, 1977.

[7] N. Christopeit, “Necessary optimality conditions with applications to a variational problem,” SIAM journal on control and optimization, vol. 15, pp. 683-698, 1977.

[8] H.W. Corley, “On optimality conditions for maximizations with respect to cones,” Journal of optimization theory and applications, vol. 46, pp. 67-78, 1985.

[9] H.W. Corley, “Duality theory for maximizations with respect to cones,” Journal of mathematical analysis and applications, vol. 84, pp. 560-568, 1981.

[10] J. Jahn, “Scalarization in vector optimization,” Mathematical programming, vol. 29, pp. 203–218, 1984.

[11] R.M. Soland, “Multicriteria optimization: a general characterization of efficient solutions,” Decision sciences, vol. 10, pp. 26–38, 1979.

[12] P.L. Yu, Multiple-criteria decision making. Plenum Press, 1985.

[13] Y. Sawaragi, H. Nakavama, and T. Tanino, Theory of multiobjective optimization. Academic Press, 1985.

[14] Y. Collette and P. Siarry, Multiobjective optimization: principles and case studies. Springer, 2003.

[15] M. Ehrgott, Multicriteria optimization. Springer, 2005.

[16] J. Jahn, Vector optimization – theory, applications, and extensions. Springer, 2011.

[17] A. Auslender and M. Teboulle, Asymptotic cones and functions in optimization and variational inequalities, Springer, 2015.

[18] S.M. Grad, Vector optimization and monotone operators via convex duality: recent advances. Springer, 2015.

[19] C. Gutierrez, L. Huerga, and V. Novo, “Nonlinear scalarization in multiobjective optimization with a polyhedral ordering cone,” International transactions in operational research, doi:10.1111/itor.12398, 2017.

[20] C. Davis, “Theory of positive dependence,” American journal of mathematics, vol. 76, pp. 733-746, 1954.

[21] J.Dattorro, Convex optimization and Euclidean distance geometry. Lulu.com, 2010.

[22] A. Schrijver, Theory of linear and integer programming. John Wiley and Sons, 1986.

[23] A.M. Geoffrion, “Proper efficiency and the theory of vector maximization,” Journal of mathematical analysis and applications, vol. 22, pp. 618-630, 1968.

[24] M.I. Henig, “Proper efficiency with respect to cones,” Journal of optimization theory and applications, vol. 36, pp. 387–407, 1982.

[25] H.P. Benson, “Efficiency and proper efficiency in vector maximization with respect to cones,” Journal of mathematical analysis and applications, vol. 93, pp. 273-289, 1983.

[26] X. Zheng and X. Yang, “The structure of weak Pareto solution sets in piecewise linear multiobjective optimization in normed spaces,” Science in China series A:mathematics, vol. 51, pp. 1243–1256, 2008.

[27] R. Kasimbeyli, “A nonlinear cone separation theorem and scalarization in nonconvex vector optimization,” SIAM Journal of optimization, vol. 20, pp. 1591–1619, 2010.

[28] R Kasimbeyli, “A conic scalarization method in multi-objective optimization,” Journal of global optimization, vol. 56, pp. 279–297, 2013.

[29] K. Q. Zhao, Y.M. Xia, and Y.M. Yang, “Nonlinear scalarization characterization of E-efficiency in vector optimization,” Taiwan journal of mathematics, vol. 19, pp. 455–466, 2015.

[30] K.Q. Zhao, G.Y. Chen, and X.M. Yang, “Approximate proper efficiency in vector optimization,” Optimization, vol. 64, pp. 1777–1793, 2015.

[31] K.Q. Zhao and X.M. Yang, “E-Benson proper efficiency in vector optimization,” Optimization, vol. 64, pp. 739– 752, 2015.

[32] C-R Chen, X. Zuo, F. Lu, and S-J Li, “Vector equilibrium problems under improvement sets and linear scalarization with stability applications,” Optimization methods and software, vol. 31, pp. 1240-1257, 2016.

[33] H. Guo, and W.J. Zhang, “Conic scalarizations for approximate efficient solutions in nonconvex vector optimization problems,” Journal of the operations research society of China, pp. 1-12, 2016.

[34] K. Khaledian, E. Khorram, and M. Soleimani-damaneh, “Strongly proper efficient solutions: efficient solutions with bounded trade-offs,” Journal of Optimization theory and applications, vol. 168, pp. 864–883, 2016.

[35] S.Shahbeyk and M.Soleimani-damaneh, “Proper minimal points of nonconvex sets in Banach spaces in terms of the limiting normal cone,” Optimization, vol. 66, pp. 473-489, 2017.

[36] A. Engau, “Proper efficiency and tradeoffs in multiple criteria and stochastic optimization,” Mathematics of operations research, vol. 42, pp. 119-134, 2017.

[37] K.Q. Zhao, W.D.Rong, and X.M. Yang, “New nonlinear separation theorems and applications in vector optimization,” Scientia sinica mathematica, vol. 47, pp. 533-544, 2017.

[38] C. Gutierrez, B. Jimenez, and V. Novo, “Improvement sets and vector optimization,” European journal of operations research, vol. 223, pp. 304-311, 2017.

[39] E-D Rahmo and M. Studniarski, “A new global scalarization method for multiobjective optimization with an arbitrary ordering cone,” Applied mathematics, vol. 8, pp. 154-163, 2017.