The problem of optimizing a real function over the efficient set of a multiple objective programming problem arises in a variety of applications. In this article, we propose an outer approximation algorithm for maximizing a function $h(x) = \varphi(f(x))$ over the efficient set $X_E$ of the bi-criteria convex programming problem $ {\rm Vmin} \{f(x)=(f_1(x), f_2(x))^T | x \in X \}$, where $\varphi$ is an increasing function on $f(X)$. The convergence of the algorithm is established. To illustrate the new algorithm, we apply it to the solution of the sample problem. Preliminary computational results with the proposed algorithm are reported.
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[1] L.T.H. An, P. D. Tao and L. D. Muu, Numerical solution for optimization over the efficient set by d.c. optimization algorithm, Oper. Res. Lett. 19 (1996) 117-128. [2] H.P. Benson and S. Sayin, Optimization over the efficient set: Four Special Cases, J. Optim. Theory Appl. 80 (1994) 3-18. [3] H. P. Benson and D. Lee, Outcome-based algorithm for optimizing over the efficient set of a bicriteria linear programming problem, J. Optim. Theory Appl. 88 (1996) 77-105. [4] H. P. Benson, Simplicial branch-and-reduce algorithm for convex programs with a multiplicative constraint, J. Optim. Theory Appl. 145 (2010) 213-233. [5] J.P. Dauer and T.A. Fosnaugh, Optimization over the efficient set, J. Global Optim. 7 (1995) 261-277. [6] B. Jaumard, C. Meyer and H. Tuy, Generalized convex multiplicative programming via quasiconcave minimization, J. Global Optim. 10 (1997) 229-256. [7] R. Horst, N.V. Thoai, Y. Yamamoto, D. Zenke, On optimization over the efficient set in linear multicriteria programming, J. Optim. Theory Appl. 134 (2007) 433-443. [8] N.T.B. Kim, An algorithm for optimizing over the efficient set, Vietnam J. Math. 28 (2000) 329-340. [9] N.T.B. Kim and L.D. Muu, On the projection of the efficient set and potential application, Optimization 51 (2002) 401-421. [10] D. T. Luc, Theory of vector optimization, Springer-Verlag, Berlin Germany, 1989. [11] J. Philip, Algorithms for the vector maximization problem, Math. Program. 2 (1972) 207-229. [12] H. X. Phu, On efficient sets in R2, Vietnam J. Math. 33 (2005) 463-468. [13] N. V. Thoai, Decomposition branch and bound algorithm for optimization problems over efficient set, J. Ind. Manag. Optim. 4 (2008) 647-660. [14] N. V. Thoai, Reverse convex programming approach in the space of extreme criteria for optimization over effcient sets, J. Optim. Theory and Appl. 147 (2010) 263-277 . [15] Y. Yamamoto, Optimization over the efficient set: overview, J. Global Optim. 22 (2002) 285-317. [16] P. L. Yu, Multiple-criteria decision making, Plenum Press, New York and London, 1985.
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