By Gabriele Eichfelder (auth.), Vincent Barichard, Matthias Ehrgott, Xavier Gandibleux, Vincent T'Kindt (eds.)
This e-book supplies the reader an perception into the state-of-the-art within the box of multiobjective (linear, nonlinear and combinatorial) programming, objective programming and multiobjective metaheuristics. The 26 papers describe all suitable tendencies during this fields of analysis . They disguise quite a lot of subject matters starting from theoretical investigations to algorithms, facing uncertainty, and purposes to genuine global difficulties equivalent to engineering layout, water distribution platforms and portfolio choice. The e-book relies at the papers of the 7th foreign convention on a number of target programming and aim programming (MOPGP06).
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Additional info for Multiobjective Programming and Goal Programming: Theoretical Results and Practical Applications
Example text
Proposition 2 (Polyhedral Cone). For a matrix A ∈ Rl×m , let C(A) ⊆ Rm be the associated polyhedral cone. C(A) is nonempty because 0 ∈ C(A), and always convex. C(A) is pointed if and only if c → Ac is injective. For the pointedness of a polyhedral cone C(A) ⊆ Rm , equivalent conditions are that rank A = m ≥ 2 or, since the mapping c → Ac is linear, that Ac = 0 if and only if c = 0. In particular, for A = I m ∈ Rm×m , the Pareto cone C(I m ) = Rm ≥ is convex and pointed. 38 A. M. Wiecek Proposition 3 (Nonpolyhedral Cone).
Heyde et al. (iv) Consider the function dc : Tc → F , defined by dc (u) := j(Dc (u)) = Cl + Dc (u). Proceeding as in the proof of Proposition 2 (iii), we obtain dc (u) = {y ∈ Rq | Lu, y ≥ u, b } for all u ∈ Tc . One easily verifies the inclusion dc ∑ri=1 λi ui ⊇ ri=1 dc (ui ). Since j is an isotone bijection between (I , ) and (F , ⊇), this is equivalent to the desired assertion. (v) Let u ∈ Tc be given. 5] yields that there are extreme points u1 , . . , uk and extreme directions uk+1 , . . , ul of Tc such that l u = ∑ λi ui , λi ≥ 0 (i = 1, .
J Optim Theory Appl 30:353–381 3. Ecker JG, Kouada IA (1978) Finding all efficient extreme points for multiple objective linear programs. Math Program 14:249–261 4. Ehrgott M, Puerto J, Rodriguez-Chia AM (2006) A primal-dual simplex method for multiobjective linear programming. J Optim Theory Appl 134(3):483–497 5. Evans JP, Steuer RE (1973) A revised simplex method for linear multiple objective programs. Math Program 5:375–377 6. Gal T (1977) A general method for determining the set of all efficient solutions to a linear vectormaximum problem.