Optimality conditions for abs-normal NLPs

Structured nonsmoothness is widely present in practical optimization problems. A particularly attractive class of nonsmooth problems, both from a theoretical and from an algorithmic perspective, are nonsmooth NLPs with equality and inequality constraints in abs-normal form, so-called abs-normal NLPs. In this thesis optimality conditions for this particular class are obtained. To this aim, first the theory for the case of unconstrained optimization problems in abs-normal form of Andreas Griewank and Andrea Walther is extended. In particular, similar necessary and sufficient conditions of first and second order are obtained that are directly based on classical Karush-Kuhn-Tucker (KKT) theory for smooth NLPs. Then, it is shown that the class of abs-normal NLPs is equivalent to the class of Mathematical Programs with Equilibrium Constraints (MPECs). Hence, the regularity assumption LIKQ introduced for the abs-normal NLP turns out to be equivalent to MPEC-LICQ. Moreover, stationarity concepts and optimality conditions under these regularity assumptions of linear independece type are equivalent up to technical assumptions. Next, well established constraint qualifications of Mangasarian Fromovitz, Abadie and Guignard type for MPECs are used to define corresponding concepts for abs-normal NLPs. Then, it is shown that kink qualifications and MPEC constraint qualifications of Mangasarian Fromovitz resp. Abadie type are equivalent. As it remains open if this holds for Guignard type kink and constraint qualifications, branch formulations for abs-normal NLPs and MPECs are introduced. Then, equivalence of Abadie’s and Guignard’s constraint qualifications for all branch problems hold. Throughout a reformulation of inequalities with absolute value slacks is considered. It preserves constraint qualifications of linear independence and Abadie type but not of Mangasarian Fromovitz type. For Guignard type it is still an open question but ACQ and GCQ are preserved passing over to branch problems. Further, M-stationarity and B-stationarity concepts for abs-normal NLPs are introduced and corresponding first order optimality con- ditions are proven using the corresponding concepts for MPECs. Moreover, a reformulation to extend the optimality conditions for abs-normal NLPs to those with additional nonsmooth objective functions is given and the preservation of regularity assumptions is considered. Using this, it is shown that the unconstrained abs-normal NLP always satisfies constraint qualifications of Abadie and thus Guignard type. Hence, in this special case every local minimizer satisfies the M-stationarity and B-stationarity concepts for abs-normal NLPs

Relations between Abs-Normal NLPs and MPCCs. Part 2: Weak Constraint Qualifications

This work continues an ongoing effort to compare non-smooth optimization problems in abs-normal form to Mathematical Programs with Complementarity Constraints (MPCCs). We study general Nonlinear Programs with equality and inequality constraints in abs-normal form, so-called Abs-Normal NLPs, and their relation to equivalent MPCC reformulations. We introduce the concepts of Abadie's and Guignard's kink qualification and prove relations to MPCC-ACQ and MPCC-GCQ for the counterpart MPCC formulations. Due to non-uniqueness of a specific slack reformulation suggested in [10], the relations are non-trivial. It turns out that constraint qualifications of Abadie type are preserved. We also prove the weaker result that equivalence of Guginard's (and Abadie's) constraint qualifications for all branch problems hold, while the question of GCQ preservation remains open. Finally, we introduce M-stationarity and B-stationarity concepts for abs-normal NLPs and prove first order optimality conditions corresponding to MPCC counterpart formulations

Relations between Abs-Normal NLPs and MPCCs. Part 1: Strong Constraint Qualifications

This work is part of an ongoing effort of comparing non-smooth optimization problems in abs-normal form to MPCCs. We study the general abs-normal NLP with equality and inequality constraints in relation to an equivalent MPCC reformulation. We show that kink qualifications and MPCC constraint qualifications of linear independence type and Mangasarian-Fromovitz type are equivalent. Then we consider strong stationarity concepts with first and second order optimality conditions, which again turn out to be equivalent for the two problem classes. Throughout we also consider specific slack reformulations suggested in [9], which preserve constraint qualifications of linear independence type but not of Mangasarian-Fromovitz type

On choosing mixture components via non-local priors

Choosing the number of mixture components remains an elusive challenge. Model selection criteria can be either overly liberal or conservative and return poorly-separated components of limited practical use. We formalize non-local priors (NLPs) for mixtures and show how they lead to well-separated components with non-negligible weight, interpretable as distinct subpopulations. We also propose an estimator for posterior model probabilities under local and non-local priors, showing that Bayes factors are ratios of posterior to prior empty-cluster probabilities. The estimator is widely applicable and helps set thresholds to drop unoccupied components in overfitted mixtures. We suggest default prior parameters based on multi-modality for Normal/T mixtures and minimal informativeness for categorical outcomes. We characterise theoretically the NLP-induced sparsity, derive tractable expressions and algorithms. We fully develop Normal, Binomial and product Binomial mixtures but the theory, computation and principles hold more generally. We observed a serious lack of sensitivity of the Bayesian information criterion (BIC), insufficient parsimony of the AIC and a local prior, and a mixed behavior of the singular BIC. We also considered overfitted mixtures, their performance was competitive but depended on tuning parameters. Under our default prior elicitation NLPs offered a good compromise between sparsity and power to detect meaningfully-separated components

Roadmap on optical rogue waves and extreme events

The pioneering paper 'Optical rogue waves' by Solli et al (2007 Nature 450 1054) started the new subfield in optics. This work launched a great deal of activity on this novel subject. As a result, the initial concept has expanded and has been enriched by new ideas. Various approaches have been suggested since then. A fresh look at the older results and new discoveries has been undertaken, stimulated by the concept of 'optical rogue waves'. Presently, there may not by a unique view on how this new scientific term should be used and developed. There is nothing surprising when the opinion of the experts diverge in any new field of research. After all, rogue waves may appear for a multiplicity of reasons and not necessarily only in optical fibers and not only in the process of supercontinuum generation. We know by now that rogue waves may be generated by lasers, appear in wide aperture cavities, in plasmas and in a variety of other optical systems. Theorists, in turn, have suggested many other situations when rogue waves may be observed. The strict definition of a rogue wave is still an open question. For example, it has been suggested that it is defined as 'an optical pulse whose amplitude or intensity is much higher than that of the surrounding pulses'. This definition (as suggested by a peer reviewer) is clear at the intuitive level and can be easily extended to the case of spatial beams although additional clarifications are still needed. An extended definition has been presented earlier by N Akhmediev and E Pelinovsky (2010 Eur. Phys. J. Spec. Top. 185 1-4). Discussions along these lines are always useful and all new approaches stimulate research and encourage discoveries of new phenomena. Despite the potentially existing disagreements, the scientific terms 'optical rogue waves' and 'extreme events' do exist. Therefore coordination of our efforts in either unifying the concept or in introducing alternative definitions must be continued. From this point of view, a number of the scientists who work in this area of research have come together to present their research in a single review article that will greatly benefit all interested parties of this research direction. Whether the authors of this 'roadmap' have similar views or different from the original concept, the potential reader of the review will enrich their knowledge by encountering most of the existing views on the subject. Previously, a special issue on optical rogue waves (2013 J. Opt. 15 060201) was successful in achieving this goal but over two years have passed and more material has been published in this quickly emerging subject. Thus, it is time for a roadmap that may stimulate and encourage further research.Peer ReviewedPostprint (author's final draft

A distributed power-saving framework for LTE HetNets exploiting Almost Blank Subframes

Almost Blank Subframes (ABSs) have been defined in LTE as a means to coordinate transmissions in heterogeneous networks (HetNets), composed of macro and micro eNodeBs: the macro issues ABS periods, and refrains from transmitting during ABSs, thus creating interference-free subframes for the micros. Micros report their capacity demands to the macro via the X2 interface, and the latter provisions the ABS period accordingly. Existing algorithms for ABS provisioning usually share resources proportionally among HetNet nodes in a long-term perspective (e.g., based on traffic forecast). We argue instead that this mechanism can be exploited to save power in the HetNet: in fact, dur-ing ABSs, the macro consumes less power, since it only transmits pilot signals. Dually, the micros may inhibit data transmission themselves in some subframes, and optimally decide when to do this based on knowledge of the ABS period. This allows us to define a power saving framework that works in the short term, mod-ifying the ABS pattern at the fastest possible pace, serving the HetNet traffic at reduced power cost. Our framework is designed using only standard signaling. Simulations show that the algorithm consumes less power than its competitors, especially at low loads, and improves the UE QoS