A gap between multiobjective optimization and scalar optimization

Minimum Principle-Type Necessary Optimality Conditions in Scalar and Vector Optimization. An Account

“…In the single-objective case this condition holds at an optimal point considering any direction v in the closed convex hull of T (X, x 0 ), i. e. in the cone P(X, x 0 ). This is no longer true in the multiobjective case, as Wang and Yang (1991) have shown. However (see Castellani and Pappalardo (2001)…”

A Note on the Guignard Constraint Qualification and the Guignard Regularity Condition in Vector Optimization

“…If the gradients , are linearly pendent, just set , has been obtained by Lin [18] and by Singh [19]; however, their proofs work only if 0 x is a weak Pareto optimal point for (vop) 1 , and not a local weak Pareto optimal point. The flaw is corr ed in Marusciac [20]; s ect ee also the errata corrige of Singh [19], who, however, does not justify his rectification; see also the paper of Wang [21], more specific on this point.…”

“…has no so Applying t rnative, we get the following (weak) Karush-KuhnTucker-type multiplier rule for (vop) 1 . (6) and (7) hold.…”

“…The condition considered by Maeda [3] and reported in the Introduction of the present paper, is therefore a regularity condition. For other regularity conditions for (vop) 1 and their relationships, see also Jimenez and Novo [4] and Giorgi and Zuccotti [7].…”

“…In discussing a gap between multiobjective optimization and scalar optimization (a gap first pointed out by Wang and Yang [1]), Aghezzaf and Hachimi [2] state that "in multiobjective optimization problems, many authors have derived the first-order and second-order necessary conditions under the Abadie constraint qualification, but never under the Guignard constraint qualification". This deserves some comments.…”