Vector maximisation and lagrange multipliers

The Lagrangean function for scalar constrained optimisation problems is extended in a directly analogous manner to constrained vector optimisation problems. Some simple saddle point results are presented for vector maxima sets. Conditions are given for the characterisation of the vector maximum set of the original vector problem in terms of the vector maximum sets with respect to the vector Lagrangeans.Finally some attention is given to Lagrangean relaxation for vector optimisation problems as an extension of a result of Everett.

[1]  D. J. White,et al.  A branch and bound method for multi-objective boolean problems , 1984 .

[2]  A. M. Geoffrion.  Lagrangean Relaxation and Its Uses in Integer Programming , 1972 .

[3]  Y. Sawaragi,et al.  Duality theory in multiobjective programming , 1979 .

[4]  Samuel Karlin,et al.  Mathematical Methods and Theory in Games, Programming, and Economics , 1961 .

[5]  D. White.  Optimality and efficiency , 1982 .

[6]  A. Charnes,et al.  Management Models and Industrial Applications of Linear Programming , 1961 .

[7]  E. Johnsen.  A. Charnes and W. W. Cooper: Management Models and Industrial Applications of linear Programming, Vol. I. and Vol. II, ialt 859 sider. John Wiley and Sons, Inc. London 1961. lait 183 sh/-. , 1962 .

[8]  G. R. Bitran,et al.  Duality for nonlinear multiple-criteria optimization problems , 1981 .

[9]  Hidefumi Kawasaki,et al.  A Duality Theorem in Multiobjective Nonlinear Programming , 1982, Math. Oper. Res..

[10]  Shelby Brumelle,et al.  Duality for Multiple Objective Convex Programs , 1981, Math. Oper. Res..

[11]  Charles Gide,et al.  Cours d'économie politique , 1911 .

[12]  Y. Sawaragi,et al.  Conjugate maps and duality in multiobjective optimization , 1980 .

[13]  W. Rödder.  A generalized saddlepoint theory , 1977 .

[14]  W. Stadler.  A survey of multicriteria optimization or the vector maximum problem, part I: 1776–1960 , 1979 .

[15]  Harvey J. Everett.  Generalized Lagrange Multiplier Method for Solving Problems of Optimum Allocation of Resources , 1963 .

[16]  D. J. White,et al.  Multiobjective programming and penalty functions , 1984 .

[17]  W. Rodder.  Book reviewPrognoserechnung: P. MERTENS, Physica, Würzburg-Wien, 1976, Second ed., 304 pages, DM 56.00 , 1977 .

[18]  George L. Nemhauser,et al.  Letter to the Editor - A Note on the Generalized Lagrange Multiplier Solution to an Integer Programming Problem , 1968, Oper. Res..

[19]  Samuel Karlin,et al.  Matrix games, programming, and mathematical economics , 1959 .