In a capital adequacy framework, risk measures are used to determine the minimal amount of capital that a financial institution has to raise and invest in a portfolio of pre-specified eligible assets in order to pass a given capital adequacy test. From a capital efficiency perspective, it is important to identify the set of portfolios of eligible assets that allow to pass the test by raising the least amount of capital. We study the existence and uniqueness of such optimal portfolios as well as their sensitivity to changes in the underlying capital position. This naturally leads to investigating the continuity properties of the set-valued map associating to each capital position the corresponding set of optimal portfolios. We pay special attention to lower semicontinuity, which is the key continuity property from a financial perspective. This "stability" property is always satisfied if the test is based on a polyhedral risk measure but it generally fails once we depart from polyhedrality even when the reference risk measure is convex. However, lower semicontinuity can be often achieved if one if one is willing to focuses on portfolios that are close to being optimal. Besides capital adequacy, our results have a variety of natural applications to pricing, hedging, and capital allocation problems.The minimal amount of capital that needs to be raised and invested in a portfolio of eligible payoffs to pass the prescribed acceptability test is represented by the risk measure ρ : X → R defined byBy definition, the quantity ρ(X) admits a natural operational interpretation as a capital requirement. The risk measures introduced by Artzner et al. (1999) constitute the prototype of the above capital requirement functionals and correspond to the simplest form of management action, i.e. raising capital and investing it in a single eligible asset. The extension to a multi-asset framework was first dealt with, to the best of our knowledge, in Föllmer and Schied (2002) and later taken up in Frittelli and Scandolo (2006), Artzner et al. (2009) and in the more comprehensive study by Farkas et al. (2015). Optimal eligible payoffsThis paper is concerned with the study of the set-valued map E : X ⇒ M defined by E(X) = {Z ∈ M ; X + Z ∈ A, π(Z) = ρ(X)}.

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In this paper, we derive a randomized version of the Mirror-Prox method for solving some structured matrix saddle-point problems, such as the maximal eigenvalue minimization problem. Deterministic first-order schemes, such as Nesterov's Smoothing Techniques or standard Mirror-Prox methods, require the exact computation of a matrix exponential at every iteration, limiting the size of the problems they can solve. Our method allows us to use stochastic approximations of matrix exponentials. We prove that our randomized scheme decreases significantly the complexity of its deterministic counterpart for large-scale matrix saddle-point problems. Numerical experiments illustrate and confirm our theoretical results.

We extend in two ways the standard Karush-Kuhn-Tucker optimality conditions to problems with a convex objective, convex functional constraints, and the extra requirement that some of the variables must be integral. While the standard Karush-Kuhn-Tucker conditions involve separating hyperplanes, our extension is based on lattice-free polyhedra. Our optimality conditions allow us to define an exact dual of our original mixed-integer convex problem.

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