Solving ill-posed bilevel programs

This paper deals with ill-posed bilevel programs, i.e., problems admitting multiple lower-level solutions for some upper-level parameters. Many publications have been devoted to the standard optimistic case of this problem, where the difficulty is essentially moved from the objective function to the feasible set. This new problem is simpler but there is no guaranty to obtain local optimal solutions for the original optimistic problem by this process. Considering the intrinsic non-convexity of bilevel programs, computing local optimal solutions is the best one can hope to get in most cases. To achieve this goal, we start by establishing an equivalence between the original optimistic problem an a certain set-valued optimization problem. Next, we develop optimality conditions for the latter problem and show that they generalize all the results currently known in the literature on optimistic bilevel optimization. Our approach is then extended to multiobjective bilevel optimization, and completely new results are derived for problems with vector-valued upper- and lower-level objective functions. Numerical implementations of the results of this paper are provided on some examples, in order to demonstrate how the original optimistic problem can be solved in practice, by means of a special set-valued optimization problem

Habitat properties are key drivers of Borrelia burgdorferi (s.l.) prevalence in Ixodes ricinus populations of deciduous forest fragments

Background: The tick Ixodes ricinus has considerable impact on the health of humans and other terrestrial animals because it transmits several tick-borne pathogens (TBPs) such as B. burgdorferi (sensu lato), which causes Lyme borreliosis (LB). Small forest patches of agricultural landscapes provide many ecosystem services and also the disservice of LB risk. Biotic interactions and environmental filtering shape tick host communities distinctively between specific regions of Europe, which makes evaluating the dilution effect hypothesis and its influence across various scales challenging. Latitude, macroclimate, landscape and habitat properties drive both hosts and ticks and are comparable metrics across Europe. Therefore, we instead assess these environmental drivers as indicators and determine their respective roles for the prevalence of B. burgdorferi in I. ricinus. Methods: We sampled I. ricinus and measured environmental properties of macroclimate, landscape and habitat quality of forest patches in agricultural landscapes along a European macroclimatic gradient. We used linear mixed models to determine significant drivers and their relative importance for nymphal and adult B. burgdorferi prevalence. We suggest a new prevalence index, which is pool-size independent. Results: During summer months, our prevalence index varied between 0 and 0.4 per forest patch, indicating a low to moderate disservice. Habitat properties exerted a fourfold larger influence on B. burgdorferi prevalence than macroclimate and landscape properties combined. Increasingly available ecotone habitat of focal forest patches diluted and edge density at landscape scale amplified B. burgdorferi prevalence. Indicators of habitat attractiveness for tick hosts (food resources and shelter) were the most important predictors within habitat patches. More diverse and abundant macro- and microhabitat had a diluting effect, as it presumably diversifies the niches for tick-hosts and decreases the probability of contact between ticks and their hosts and hence the transmission likelihood.[br/] Conclusions: Diluting effects of more diverse habitat patches would pose another reason to maintain or restore high biodiversity in forest patches of rural landscapes. We suggest classifying habitat patches by their regulating services as dilution and amplification habitat, which predominantly either decrease or increase B. burgdorferi prevalence at local and landscape scale and hence LB risk. Particular emphasis on promoting LB-diluting properties should be put on the management of those habitats that are frequently used by humans. In the light of these findings, climate change may be of little concern for LB risk at local scales, but this should be evaluated further

Convergence of a Scholtes-type Regularization Method for Cardinality-Constrained Optimization Problems with an Application in Sparse Robust Portfolio Optimization

We consider general nonlinear programming problems with cardinality constraints. By relaxing the binary variables which appear in the natural mixed-integer programming formulation, we obtain an almost equivalent nonlinear programming problem, which is thus still difficult to solve. Therefore, we apply a Scholtes-type regularization method to obtain a sequence of easier to solve problems and investigate the convergence of the obtained KKT points. We show that such a sequence converges to an S-stationary point, which corresponds to a local minimizer of the original problem under the assumption of convexity. Additionally, we consider portfolio optimization problems where we minimize a risk measure under a cardinality constraint on the portfolio. Various risk measures are considered, in particular Value-at-Risk and Conditional Value-at-Risk under normal distribution of returns and their robust counterparts under moment conditions. For these investment problems formulated as nonlinear programming problems with cardinality constraints we perform a numerical study on a large number of simulated instances taken from the literature and illuminate the computational performance of the Scholtes-type regularization method in comparison to other considered solution approaches: a mixed-integer solver, a direct continuous reformulation solver and the Kanzow--Schwartz regularization method, which has already been applied to Markowitz portfolio problems

On stability of M-stationary points in MPCCs

We consider parameterized Mathematical Programs with Complementarity Constraints arising, e.g., in modeling of deregulated electricity markets. Using the standard rules of the generalized differential calculus we analyze qualitative stability of solutions to the respective M-stationarity conditions. In particular, we provide characterizations and criteria for the isolated calmness and the Aubin properties of the stationarity map. To this end, we introduce the second-order limiting coderivative of mappings and provide formulas for this notion and for the graphical derivative of the limiting coderivative in the case of the normal cone mapping to ℝn Funding ARC- Australian Research Counci

Possible substellar companions in low-mass eclipsing binaries: GU Bootis and YY Geminorum

We present the next results of our long-term observational project to analyze the variations in the orbital periods of low-mass eclipsing binaries. About 70 new precise mid-eclipse times recorded with a CCD were obtained for two eclipsing binaries with short orbital periods: GU Boo (P = 0.​d49) and YY Gem (0.​d81). Observed-minus-calculated diagrams of the stars were analyzed using all reliable timings, and new parameters of the light-time effect were obtained. We derived for the first time or improved the short orbital periods of possible third bodies of 11 and 54 years for these low-mass binaries, respectively. We calculated that the minimum masses of the third components are close to 50 MJup, which corresponds to the mass of brown dwarfs. The multiplicity of these systems also plays an important role in the precise determination of their physical parameters