Lexicographic quasiconcave multiobjective programming

Linear lexicographic optimization

“…Thus h is a solution for (1). Now, let h be a solution for (1). Then h :# 0 and x* + Xh E S for all x* E S and all X > 0.…”

“…all ~ E S with Cx' ~ C~ for no x' E S. Note that for k = 1 (LLP) reduces to an ordinary linear program. Several mathematical and game-theoretic applications of nonlinear lexicographic optimization are reported in [1 ]. Moreover this contribution establishes a procedure how a lexicographic optimization problem can be reduced to a family of k one-objective programming problems.…”

“…As, by assumption, (1) has no solution,.~ is optimal for (LLP). In order to prove the backward implication of (i), let x" be a solution for (1) and assume, to the contrary, that (LLP) has an optimal solution, say x ~ . As x ~ + x" E S and z(x ~ + x") ~ z(x~ we obtain a contradiction to the assumption.…”

Zur existenz maximaler werte bei quasiordnungswertigen abbildungen

“…1 K o r o l l a r z u S a t z 4.5 (stark halbstetig glelch .5-halhstetlg) Linter den. Yoraussetzungen von Satz 4.5 gilt: f stark halbstetig nnch oben (unten)]+[f A -halbstetig nach oben [ ( u n t e n ) ] o [ f <-halbstetig nnch oben (unten)].Analog zu dem Vorgehen von PIRZL[15] wollen wir nun noch auf die Umkehrung des Maximalelementsatzes in Korollar I und 2 zu Satz 3.2 eingehen.F e s t s t el l u n g 4.4 (Ordnungskompaktheit bei Existenz eines maximalen Elementes) : Hat die Quasiordnung ( Y ,5 ) ein 5 -maximaies Eiement, dann ist Y in der <-Obertopologie und damit in jeder zu dieser groberen Topologie kompakt. PIRZL[15] bewies diese Aussage fur (Y, 5 ) Halbordnung.…”

Sensor Placement Algorithms for Process Efficiency Maximization

“…A lexicographic approach is one such technique that can guarantee Pareto-optimality of multiobjective optimization problems. Several mathematical nonlinear lexicographic optimizations have been reported by Behringer (1977). This type of optimization is studied by arranging objective functions in lexicographic order i.e.…”