Hahn-Banach and Sandwich Theorems for Equivariant Vector Lattice-Valued Operators and Applications

We prove Hahn-Banach, sandwich and extension theorems for vector lattice-valued operators, equivariant with respect to a given group G of homomorphisms. As applications and consequences, we present some Fenchel duality and separation theorems, a version of the Moreau-Rockafellar formula and some Farkas and Kuhn-Tucker-type optimization results. Finally, we prove that the obtained results are equivalent to the amenability of G.

Fenchel duality of Cox partial likelihood and its application in survival kernel learning

The Cox proportional hazard model is the most widely used method in modeling time-to-event data in the health sciences. A common form of the loss function in machine learning for survival data is also mainly based on Cox partial likelihood function, due to its simplicity. However, the optimization problem becomes intractable when more complicated regularization is employed with the Cox loss function. In this paper, we show that a convex conjugate function of Cox loss function based on Fenchel Duality exists, and this provides an alternative framework to optimization based on the primal form. Furthermore, the dual form suggests an efficient algorithm for solving the kernel learning problem with censored survival outcomes. We illustrate the application of the derived duality form of Cox partial likelihood loss in the multiple kernel learning setting

Shortfall risk through Fenchel duality

In this paper, we propose a Fenchel duality approach to study the minimization problem of the shortfall risk. We consider a general increasing and strictly convex loss function, which may be more general than the situation of convex risk measures usually assumed in the literature. We first translate the associated stochastic optimization problem to an equivalent static optimization problem, and then obtain the explicit structure of the optimal randomized test for both complete and incomplete markets. For the incomplete market case, to the best of our knowledge, we obtain for the first time the explicit randomized test, while previous literature only established the existence through the supermartingale optional decomposition approach. We also solve the shortfall risk minimization problem for an insider through the enlargement of filtrations approach.

A Forward-Reverse Brascamp-Lieb Inequality: Entropic Duality and Gaussian Optimality

Inspired by the forward and the reverse channels from the image-size characterization problem in network information theory, we introduce a functional inequality which unifies both the Brascamp-Lieb inequality and Barthe's inequality, which is a reverse form of the Brascamp-Lieb inequality. For Polish spaces, we prove its equivalent entropic formulation using the Legendre-Fenchel duality theory. Capitalizing on the entropic formulation, we elaborate on a "doubling trick" used by Lieb and Geng-Nair to prove the Gaussian optimality in this inequality for the case of Gaussian reference measures.