# Article

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Keywords:
multi-objective program; efficient (Pareto) solution; properly efficient solution; LFS function; convex program; $l_{1}$ norm; $l_{\infty }$ norm; simultaneous optimization
Summary:
We find conditions, in multi-objective convex programming with nonsmooth functions, when the sets of efficient (Pareto) and properly efficient solutions coincide. This occurs, in particular, when all functions have locally flat surfaces (LFS). In the absence of the LFS property the two sets are generally different and the characterizations of efficient solutions assume an asymptotic form for problems with three or more variables. The results are applied to a problem in highway construction, where the quantity of dirt to be removed and the uniform smoothness of the shape of a terrain are optimized simultaneously.
References:
[1] S. M. Allende, C. N. Bouza: A parametric mathematical programming approach to the estimation of the coefficients of the linear regression model. Parametric Optimization and Related Topics III, J. Guddat et al. (eds.), Akademie Verlag, Berlin, 1993, pp. 9–20. MR 1241214
[2] A. Ben-Israel, A. Ben-Tal, S. Zlobec: Optimality in Nonlinear Programming: A Feasible Directions Approach. Wiley Interscience, New York, 1981. MR 0607673
[3] M. D. Canon, C. D. Cullum, E. Polak: Theory of Optimal Control and Mathematical Programming. McGraw-Hill, New York, 1970. MR 0397497
[4] A. Charnes: Correspondence. 1989.
[5] A. Charnes, W. W. Cooper: Management Models and Industrial Applications of Linear Programming, Vol. I. Wiley, New York, 1961. MR 0157774
[6] S. J. Citron: Elements of Optimal Control. Holt, Rinehard and Winston, New York, 1969. Zbl 0221.49002
[7] R. W. Farebrother: The historical development of $L_{1}$ and $L_{\infty }$ estimation procedures. Statistical Data Analysis Based on $L_{1}$-norm and Related Methods, Y. Dodge (ed.), North Holland, Amsterdam, 1987, pp. 37–63. MR 0949218
[8] M. D. Intriligator: Mathematical Optimization and Economic Theory. Prentice Hall, Englewood Cliffs, New Jersey, 1972. MR 0353945
[9] S. Karlin: Mathematical Methods in Theory of Games, Programming and Economics, Vol. I. Addison-Wesley, Reading, Massachussetts, 1959. MR 0111634
[10] O. J. Karst: Linear curve fitting using least deviations. Journal of the American Statistical Association 53 (1958), 118–132. MR 0134453 | Zbl 0080.13402
[11] Lj. Martić: Bicriterial programming in regression analysis. Proceedings of KOI $^{\prime }$91, Lj. Martić and L. Neralić (eds.), Faculty of Economics, Zagreb, 1991, pp. 37–45. (Croatian)
[12] Lj. Martić: A simple regression by $l_{1}$ and $L_{\infty }$ criteria. Proceedings of KOI  $^{\prime }$92, V. Bahovec, Lj. Martić and L. Neralić (eds.), Croatian Operational Research Society, Rovinj, 1992, pp. 17–32. (Croatian)
[13] M. E. Salukvadze: Vector-Valued Optimization Problems in Control Theory. Academic Press, New York, 1969. MR 0563922
[14] F. Sharifi Mokhtarian: Mathematical Programming with LFS Functions. M. Sc. Thesis, McGill University, Montreal, Quebec, 1992.
[15] F. Sharifi Mokhtarian, S. Zlobec: Mathematical Programming with LFS Functions. Utilitas Mathematica 45 (1994), 3–15. MR 1284014
[16] M. van Rooyen, X. Zhou, S. Zlobec: A saddle-point characterization of Pareto optima. Mathematical Programming 67 (1994), 77–88. MR 1300819
[17] T. L. Vincent, W. J. Grantham: Optimality in Parametric Systems. Wiley Interscience, New York, 1981. MR 0628316
[18] X. Zhou, F. Sharifi Mokhtarian, S. Zlobec: A simple constraint qualification in convex programming. Mathematical Programming 61 (1993), 385–397. MR 1242469
[20] S. Zlobec: Two characterizations of Pareto minima in convex multicriteria optimization. Aplikace Matematiky 29 (1984), 342–349. MR 0772269 | Zbl 0549.90085
[19] S. Zlobec: Characterizations of optimality in nonconvex programming. The Fourteenth Symposium on Mathematical Programming with Data Perturbations, The George Washington University, Washington, D. C., May 23, 1992.

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