31 research outputs found
Solutions of max-plus linear equations and large deviations
We generalise the Gartner-Ellis theorem of large deviations theory. Our
results allow us to derive large deviation type results in stochastic optimal
control from the convergence of generalised logarithmic moment generating
functions. They rely on the characterisation of the uniqueness of the solutions
of max-plus linear equations. We give an illustration for a simple investment
model, in which logarithmic moment generating functions represent
risk-sensitive values.Comment: 6 page
Constant Along Primal Rays Conjugacies and the l0 Pseudonorm
The so-called l0 pseudonorm on Rd counts the number of nonzero components of a vector. It is used in sparse optimization, either as criterion or in the constraints, to obtain solutions with few nonzero entries. For such problems, the Fenchel conjugacy fails to provide relevant analysis: indeed, the Fenchel conjugate of the characteristic function of the level sets of the l0 pseudonorm is minus infinity, and the Fenchel biconjugate of the l0 pseudonorm is zero. In this paper, we display a class of conjugacies that are suitable for the l0 pseudonorm. For this purpose, we suppose given a (source) norm on Rd. With this norm, we define, on the one hand, a sequence of so-called coordinate-k norms and, on the other hand, a coupling between Rd and Rd , called Capra (constant along primal rays). Then, we provide formulas for the Capra-conjugate and biconjugate, and for the Capra subdifferentials, of functions of the l0 pseudonorm (hence, in particular, of the l0 pseudonorm itself and of the characteristic functions of its level sets), in terms of the coordinate-k norms. As an application, we provide a new family of lower bounds for the l0 pseudonorm, as a fraction between two norms, the denominator being any norm
Set optimization - a rather short introduction
Recent developments in set optimization are surveyed and extended including
various set relations as well as fundamental constructions of a convex analysis
for set- and vector-valued functions, and duality for set optimization
problems. Extensive sections with bibliographical comments summarize the state
of the art. Applications to vector optimization and financial risk measures are
discussed along with algorithmic approaches to set optimization problems
Conditions for global minimum through abstract convexity
The theory of abstract convexity generalizes ideas of convex analysis by using the notion of global supports and the global definition of subdifferential. In order to apply this theory to optimization, we need to extend subdifferential calculus and separation properties into the area of abstract convexity.Doctor of Philosoph
Weak Minimizers, Minimizers and Variational Inequalities for set valued Functions. A blooming wreath?
In the literature, necessary and sufficient conditions in terms of
variational inequalities are introduced to characterize minimizers of convex
set valued functions with values in a conlinear space. Similar results are
proved for a weaker concept of minimizers and weaker variational inequalities.
The implications are proved using scalarization techniques that eventually
provide original problems, not fully equivalent to the set-valued counterparts.
Therefore, we try, in the course of this note, to close the network among the
various notions proposed. More specifically, we prove that a minimizer is
always a weak minimizer, and a solution to the stronger variational inequality
always also a solution to the weak variational inequality of the same type. As
a special case we obtain a complete characterization of efficiency and weak
efficiency in vector optimization by set-valued variational inequalities and
their scalarizations. Indeed this might eventually prove the usefulness of the
set-optimization approach to renew the study of vector optimization