Special Issue “Nonsmooth Optimization in Honor of the 60th Birthday of Adil M. Bagirov”: Foreword by Guest Editors

Nonsmooth optimization refers to the general problem of minimizing (or maximizing) functions that have discontinuous gradients. This Special Issue contains six research articles that collect together the most recent techniques and applications in the area of nonsmooth optimization. These include novel techniques utilizing some decomposable structures in nonsmooth problems—for instance, the difference-of-convex (DC) structure—and interesting important practical problems, like multiple instance learning, hydrothermal unit-commitment problem, and scheduling the disposal of nuclear waste.

Asymptotics of Decomposable Combinatorial Structures of Alg-Log Type With Positive Log Exponent

International audience We consider the multiset construction of decomposable structures with component generating function $C(z)$ of alg-log type, $\textit{i.e.}$, $C(z) = (1-z)^{-\alpha} (\log \frac{1}{ 1-z})^{\beta}$. We provide asymptotic results for the number of labeled objects of size $n$ in the case when $\alpha$ is positive and $\beta$ is positive and in the case $\alpha = 0$ and $\beta \geq 2$. The case $0<-\alpha <1$ and any $\beta$ and the case $\alpha > 0$ and $\beta = 0$ have been treated in previous papers. Our results extend previous work of Wright.

ON LINEARLY ORDERED STRUCTURES OF FINITE RANK

O-minimal structures have long been thought to occupy the base of a hierarchy of ordered structures, in analogy with the role that strongly minimal structures play with respect to stable theories. This is the first in an anticipated series of papers whose aim is the development of model theory for ordered structures of rank greater than one. A class of ordered structures to which a notion of finite rank can be assigned, the decomposable structures, is introduced here. These include all ordered structures definable (as subsets of n-tuples of the universe) in o-minimal structures. The principal result in this paper, Theorem 5.1, asserts roughly that a decomposable structure [Formula: see text] can be partitioned into finitely many definable subsets such that on each set the restriction of < is a "twisted lexicographic" order. As a consequence (Corollary 5.1), for all n and linear orders ≺ definable on a subset X ⊆ Mn in an o-minimal structure [Formula: see text], there is a definable partition of X such that the restriction of ≺ to each set in the partition is "lexicographic".

The Size of the rth Smallest Component in Decomposable Structures with a Restricted Pattern

International audience In our previous work [paper1], we derived an asymptotic expression for the probability that a random decomposable combinatorial structure of size n in the \exp -\log class has a given restricted pattern. In this paper, under similar conditions, we provide the probability that a random decomposable combinatorial structure has a given restricted pattern and the size of its rth smallest component is bigger than k, for r,k given integers. Our studies apply to labeled and unlabeled structures. We also give several concrete examples.