Isolated minimizers, proper efficiency and stability for C0,1 constrained vector optimization problems

In this paper we consider the vector optimization problem minC f(x), g(x) 2 -K, where f : Rn ! Rm and g : Rn Rp are C0,1 functions and C Rm and K Rp are closed convex cones. We give several notions of solutions (efficiency concepts), among them the notion of a properly efficient point (p-minimizer) of order k and the notion of an isolated minimizer of order k. We show that each isolated minimizer of order k > = 1 is a p-minimizer of order k. The possible reversal of this statement in the case k = 1 is the main subject of the investigation. For this study we apply some first order necessary and sufficient conditions in terms of Dini derivatives. We show that the given optimality conditions are important to solve the posed problem, and a satisfactory solution leads to two approaches toward efficiency concepts, called respectively sense I and sense II concepts. Relations between sense I and sense II isolated minimizers and p-minimizers are obtained. In particular, we are concerned in the stability properties of the p-minimizers and the isolated minimizers. By stability, we mean that they still remain the same type of solutions under small perturbations of the problem data. We show that the p-minimizers are stable under perturbations of the cones, while the isolated minimizers are stable under perturbations both of the cones and the functions in the data set. Further, we show that the sense I concepts are stable under perturbations of the objective data, while the sense II concepts are stable under perturbations both of the objective and the constraints.Vector optimization, Locally Lipschitz data, Properly efficient points, Isolated minimizers, Optimality conditions, Stability.

First-Order Conditions for C0,1 Constrained vector optimization

For a Fritz John type vector optimization problem with C0,1 data we define different type of solutions, give their scalar characterizations applying the so called oriented distance, and give necessary and sufficient first order optimality conditions in terms of the Dini derivative. While establishing the sufficiency, we introduce new type of efficient points referred to as isolated minimizers of first order, and show their relation to properly efficient points. More precisely, the obtained necessary conditions are necessary for weakly efficiency, and the sufficient conditions are both sufficient and necessary for a point to be an isolated minimizer of first order.vector optimization, nonsmooth optimization, C0,1 functions, Dini derivatives, first-order optimality conditions, lagrange multipliers

Letter from the editors

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Editoriale

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A Multicenter Retrospective Survey regarding Diabetic Ketoacidosis Management in Italian Children with Type 1 Diabetes

We conducted a retrospective survey in pediatric centers belonging to the Italian Society for Pediatric Diabetology and Endocrinology. The following data were collected for all new-onset diabetes patients aged 0-18 years: DKA (pH < 7.30), severe DKA (pH < 7.1), DKA in preschool children, DKA treatment according to ISPAD protocol, type of rehydrating solution used, bicarbonates use, and amount of insulin infused. Records (n = 2453) of children with newly diagnosed diabetes were collected from 68/77 centers (87%), 39 of which are tertiary referral centers, the majority of whom (n = 1536, 89.4%) were diagnosed in the tertiary referral centers. DKA was observed in 38.5% and severe DKA in 10.3%. Considering preschool children, DKA was observed in 72%, and severe DKA in 16.7%. Cerebral edema following DKA treatment was observed in 5 (0.5%). DKA treatment according to ISPAD guidelines was adopted in 68% of the centers. In the first 2 hours, rehydration was started with normal saline in all centers, but with different amount. Bicarbonate was quite never been used. Insulin was infused starting from third hour at the rate of 0.05-0.1 U/kg/h in 72% of centers. Despite prevention campaign, DKA is still observed in Italian children at onset, with significant variability in DKA treatment, underlying the need to share guidelines among centers

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Diabetic ketoacidosis at the onset of disease during a national awareness campaign: a 2-year observational study in children aged 0-18 years

After a previous survey on the incidence of diabetic ketoacidosis (DKA) at onset of type 1 diabetes in children in 2013-2014 in Italy, we aimed to verify a possible decline in the incidence of DKA at onset during a national prevention campaign