Comments on “A Note on the Paper “Optimality Conditions for Optimistic Bilevel Programming Problem Using Convexifactors””

Journal of Optimization Theory and Applications - Necessary optimality conditions for a bilevel optimization problem are given in the paper by Kohli (J Optim Theory Appl 152: 632–651, 2012)....     

Essential Oils Derived from Cistus Species Activate Mitochondria by Inducing SIRT1 Expression in Human Keratinocytes,Leading to Senescence Inhibition

Cistus L. is a genus of dicotyledonous perennial herbaceous plants. Cistus species have been commonly used in folk medicine in the Mediterranean region. In the present study, the biological activities of essential oils derived from Cistus species (Cistus laurifolius, C. monspeliensis, C. creticus, and C. salviifolius) were evaluated. Essential oils derived from C. laurifolius and C. monspeliensis were found to augment the expression of SIRT1, an anti-aging gene, in the normal culture of HaCaT cells. Furthermore, these essential oils increased the number and size of mitochondria and augmented their activity. These effects were thought to be caused by the up- and downregulated expression of MITOL and Drp1 in HaCaT cells, respectively, in response to the essential oil treatment. In addition, these essential oils were found to attenuate ultraviolet-B-induced mitochondrial damage and cellular senescence in HaCaT cells. These findings indicate that essential oils derived from C. laurifolius and C. monspeliensis may inhibit skin aging through mitochondrial regulation via SIRT1 activation.     

On the Regularity Condition for Vector Programming Problems

In this work, we use a notion of approximation derived from Jourani and Thibault [13] to ascertain optimality conditions analogous to those that established but applicable to larger class of vector valued objective mappings and constraint set-valued mappings. To this end, we introduce an appropriate regularity condition to help us discern the Karush-Kuhn-Tucker multipliers.     

Necessary optimality conditions for bilevel set optimization problems

Bilevel programming problems are hierarchical optimization problems where in the upper level problem a function is minimized subject to the graph of the solution set mapping of the lower level problem. In this paper necessary optimality conditions for such problems are derived using the notion of a convexificator by Luc and Jeyakumar. Convexificators are subsets of many other generalized derivatives. Hence, our optimality conditions are stronger than those using e.g., the generalized derivative due to Clarke or Michel-Penot. Using a certain regularity condition Karush-Kuhn-Tucker conditions are obtained.      

Optimality Conditions for D.C. Vector Optimization Problems Under Reverse Convex Constraints

In this paper, we establish global necessary and sufficient optimality conditions for D.C. vector optimization problems under reverse convex constraints. An application to vector fractional mathematical programming is also given. Mathematics Subject Classifications (1991). Primary 90C29, Secondary 49K30.     

Second order optimality conditions for bilevel set optimization problems

In this paper, we introduce a new notion of augmenting function known as indicator augmenting function to establish a minmax type duality relation, existence of a path of solution converging to optimal value and a zero duality gap relation for a nonconvex primal problem and the corresponding Lagrangian dual problem. We also obtain necessary and sufficient conditions for an exact penalty representation in the framework of indicator augmented Lagrangian.     

Set-valued mapping monotonicity as characterization of D.C functions

In this paper, using a result of Kung-Ching Chang [6], we give a characterization of locally Lipschitz functions which are differences of convex functions defined on a Banach space (not necessarily Asplund) in terms on maximal cyclically monotone set-valued mappings. A subdifferential integration of locally D.C functions. is also given.     

Optimality conditions for pessimistic bilevel problems using convexificator

Positivity - The purpose of this paper is to study the pessimistic version of bilevel programming problems in finite-dimensional spaces. Problems of this type are intrinsically nonsmooth (even for...     

Sufficient Optimality Condition for Vector Optimization Problems under D.C. Data

In this paper, we establish sufficient optimality conditions for D.C. vector optimization problems. We also give an application to vector fractional mathematical programming in a ordred separable Hilbert space.     

An equivalent one level optimization problem to a semivectorial bilevel problem

In this paper, we are concerned with a bilevel optimization problem \(P_{0}\), where the lower level problem is a vector optimization problem. First, we give an equivalent one level optimization problem for which the nonsmooth Mangasarian–Fromowitz constraint qualification can hold at feasible solution. Using a special scalarization function, one deduces necessary optimality condition for the initial problem.