Abstract
This paper examines nonsmooth constrained multi-objective optimization problems where the objective function and the constraints are compositions of convex functions, and locally Lipschitz and Gâteaux differentiable functions. Lagrangian necessary conditions, and new sufficient optimality conditions for efficient and properly efficient solutions are presented. Multi-objective duality results are given for convex composite problems which are not necessarily convex programming problems. Applications of the results to new and some special classes of nonlinear programming problems are discussed. A scalarization result and a characterization of the set of all properly efficient solutions for convex composite problems are also discussed under appropriate conditions.
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References
A. Ben-Tal and J. Zowe, “Necessary and sufficient optimality conditions for a class of nonsmooth minimization problems,”Mathematical Programming 24 (1982) 70–91.
J.V. Burke, “Decent methods for composite nondifferentiable optimization,”Mathematical Programming 33 (1985) 260–279.
J.V. Burke, “Second order necessary and sufficient conditions for composite NDO,”Mathematical Programming 38 (1987) 287–302.
K.L. Chew and E.U. Choo, “Pseudolinearity and efficiency,”Mathematical Programming 28 (1984) 226–239.
E.U. Choo, “Proper efficiency and linear fractional vector maximum problem,”Operations Research 32 (1984) 216–220.
F.H. Clarke,Optimization and Nonsmooth Analysis (Wiley, New York, 1983).
B.D. Craven,Mathematical Programming and Control Theory (Chapman and Hall, London, 1978).
R. Fletcher,Practical Methods of Optimization (Wiley, New York, 1987).
R. Fletcher, “A model algorithm for composite nondifferentiable optimization problems,”Mathematical Programming Study 17 (1982) 67–76.
A.M. Geoffrion, “Proper efficiency and the theory of vector maximization,”Journal of Mathematical Analysis and Applications 22 (1968) 618–630.
M.A. Hanson, “On sufficiency of the Kuhn—Tucker conditions,”Journal of Mathematical Analysis and Applications 80 (1981) 644–550
R. Hartley, “Vector and parametric programming,”Journal of Operations Research Society 36 (1985) 423–432.
A.D. Ioffe, “Necessary and sufficient conditions for a local minimum 2: conditions of Levitin—Milutin—Osmoloviskii type,”SIAM Journal on Control and Optimization 17 (1979) 251–265.
J. Jahn, “Scalarization in multi-objective optimization,”Mathematical Programming 29 (1984) 203–219.
J. Jahn and W. Krabs, “Applications of multicriteria optimization in approximation theory,” in: W. Stadler, ed.,Multicriteria Optimization in Engineering and in the Sciences (Plenum Press, New York, 1988) pp. 49–75.
J. Jahn and E. Sachs, “Generalized quasiconvex mappings and vector optimization,”SIAM Journal on Control and Optimization 24 (1986) 306–322.
V. Jeyakumar, “Compsosite nonsmooth programming with Gâteaux differentiability,”SIAM Journal on Optimization 1(1991) 30–41.
V. Jeyakumar, “On optimality conditions in nonsmooth inequality constrained minimization,”Numerical Functional Analysis and Optimization 9 (1987) 535–546.
V. Jeyakumar, “Convexlike alternative theorems and mathematical programming,”Optimization 16 (1985) 643–652.
V. Jeyakumar, “Infinite dimensional convex programming with applications to constrained approximation,”Journal of Optimization Theory and Applications 75 (1992).
V. Jeyakumar and H. Wolkowicz, “Zero duality gaps in infinite dimensional programming,”Journal of Optimization Theory and Applications 67 (1990) 87–108.
D.T. Luc,Theory of Vector Optimization (Springer, Berlin, 1988).
O.L. Mangasarian, “A simple characterization of solution sets of convex programs,”Operations Research Letters 7 (1988) 21–26.
R.R. Merkovsky and D.E. Ward, “Constraint qualifications in nondifferentiable programming,”Mathematical Programming 47 (1990) 389–405.
P. Michel and J.P. Penot, “Calcul sous-differentiel pour des fonctions lipschitziennes et non lipschitziennes,”Comptes Rendus de l'Académie des Sciences Paris 298 (1984) 269–272.
R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1969).
R.T. Rockafellar, “First and second order epi-differentiability in nonlinear programming,”Transactions of the American Mathematical Society 307 (1988) 75–108.
Y. Sawaragi, H. Nakayama and T. Tanino,Theory of Multiobjective Optimization (Academic Press, New York, 1985).
A. Schy and D.P. Giesy, “Multicriteria optimization techniques for design of aircraft control systems,” in: W. Stadler, ed.,Multicriteria Optimization in Engineering and in the Sciences (Plenum Press, New York, 1988) pp. 225–262.
W. Stadler,Multicriteria Optimization in Engineering and in the Sciences (Plenum Press, New York, 1988).
W. Stadler, “Multicriteria optimization in mechanics: A survey,”Applied Mechanics Reviews 37 (1984) 277–286.
T. Weir and B. Mond, “Generalized convexity and duality in multiple objective programming,”Bulletin of the Australian Mathematical Society 39 (1989) 287–299.
D.J. White,Optimality and Efficiency (Wiley, New York, 1982).
D.J. White, “Multi-objective programming and penalty functions,”Journal of Optimization Theory and Applications 43 (1984) 583–599.
W.I. Zangwill,Nonlinear Programming (Prentice-Hall, Englewood Cliffs, NJ, 1969).
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This research was partially supported by the Australian Research Council grant A68930162.
This author wishes to acknowledge the financial support of the Australian Research Council.
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Jeyakumar, V., Yang, X.Q. Convex composite multi-objective nonsmooth programming. Mathematical Programming 59, 325–343 (1993). https://doi.org/10.1007/BF01581251
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DOI: https://doi.org/10.1007/BF01581251