4d F(4) gauged supergravity and black holes of class ℱ

We perform a consistent reduction of 6d matter-coupled F(4) supergravity on a compact Riemann surface $$ {\Sigma}_{\mathfrak{g}} $$ Σ g of genus $$ \mathfrak{g} $$ g , at the level of the bosonic action. The result is an $$ \mathcal{N} $$ N = 2 gauged supergravity coupled to two vector multiplets and a single hypermultiplet. The four-dimensional model is holographically dual to the 3d superconformal field theories of class ℱ, describing different brane systems in massive type IIA and IIB wrapped on $$ {\Sigma}_{\mathfrak{g}} $$ Σ g . The naive reduction leads to a non-standard 4d mixed duality frame with both electric and magnetic gauge fields, as well as a massive tensor, that can be only described in the embedding tensor formalism. Upon a chain of electromagnetic dualities, we are able to determine the scalar manifolds and electric gaugings that uniquely specify the model in the standard supergravity frame. We then use the result to construct the first examples of static dyonic black holes in AdS6 and perform a microscopic counting of their entropy via the 5d topologically twisted index. Finally, we show the existence of further subtruncations to the massless sector of the 4d theory, such as the Fayet-Iliopoulos gauged T3 model and minimal gauged supergravity. We are in turn able to find new asymptotically AdS4 solutions, providing predictions for the squashed S3 partition functions and the superconformal and refined twisted indices of class ℱ theories.

Duality and Lagrange multipliers for nonsmooth multiobjective programming

Without any constraint qualification, the necessary and sufficient optimality conditions are established in this paper for nonsmooth multiobjective programming involving generalised convex functions. With these optimality conditions, a mixed dual model is constructed which unifies two dual models. Several theorems on mixed duality and Lagrange multipliers are established in this paper.

Approximate convexity in vector optimisation

Approximate convex functions are characterised in terms of Clarke generalised gradient. We apply this characterisation to derive optimality conditions for quasi efficient solutions of nonsmooth vector optimisation problems. Two new classes of generalised approximate convex functions are defined and mixed duality results are obtained.