Optimality conditions for set-valued optimization problems

Abstract

The problem of scheduling identical jobs with chain precedence constraints on two uniform machines is considered. It is shown that the corresponding makespan problem can be solved in linear time.

A multiplier rule in set-valued optimisation

“…Since (x, y) is a proper minimiser of (P n ), it has to be a minimiser of (P n ) with respect to some K £ K. Therefore, it follows from Lemma 3.2 and the imposed conditions that for such K G /C, we have It is clear that in Theorem 3.1 we have not imposed any differentiability assumption on the map H. Thus it would be of interest to obtain a variant of the well-known theorem of Lyusternik ( [10]), so that the cone M* contain information about some derivative of H. In fact, this is completely true if H is single-valued and sufficiently smooth ( [17]). Moreover, we can also define a variant of the generalised contingent epiderivative (see [3,8,11]) by taking the minimal points of Di(F + C)(x,y) with respect to the cone C. We mention that our results will remain valid for such an epiderivative.…”

“…In recent years a great deal of attention has been given to the characterisation of the weak-minimality for (Po) and (Pj) by employing various notions of derivatives for set-valued maps, see [2,3,4,7,8,11,12,13] and the references therein. A common strategy adopted in these works is to use direct arguments, based on the derivatives chosen for the set-valued maps involved, to verify a claim that the images of the derivatives do not intersect with certain open cones.…”

“…In general, this important property is not shared by the contingent cones. For some S C Z, [3] A multiplier rule 9 5…”

Pointwise well-posedness and scalarization in set optimization

“…There are two approaches to formulate optimality notions for these problems: the vector approach and the set approach. In the vector approach, optimal solutions are defined as the efficient points of the union of all images of the set-valued objective map (see, e.g., Corley 1987;Chen and Jahn 1998;; Khoshkhabar-amiranloo and Soleimani-damaneh 2012 and the references therein). In this approach the set-valued optimization problem is called set-valued vector optimization problem.…”

Tightly Proper Efficiency in Vector Optimization with Nearly Cone-Subconvexlike Set-Valued Maps

“…Recently, several authors have turned their interests to vector optimization of setvalued maps, for instance, see [13][14][15][16][17][18] . Gong 19 discussed set-valued constrained vector optimization problems under the constraint ordering cone with empty interior.…”