We consider the mathematical programming problem: find $\inf \{ f(x)\mid x \in C\} $ where f is an arbitrary extended-real-valued function, and C a subset of a finite dimensional space. We give necessary and sufficient optimality conditions for this problem, generalizing previous results of A. Auslendex (this Journal, 22 (1984), pp. 239–254).

The problem of identifying weak sharp minima of order m, an important class of (possibly) nonisolated minima, is investigated in this paper. Some sufficient conditions for weak sharp minimality in nonsmooth mathematical programming are presented, and some characterizations of weak sharp minimality are obtained, with special attention given to orders one and two. It is also demonstrated that two of these sufficient conditions guarantee exactness of an l ₁ penalty function. A key role in this paper is played by two geometric concepts: the limiting proximal normal cone and a generalization of the contingent cone.

Background: Osteoporosis is a socio-economic problem of modern aging societies. Bone fractures and the related treatments generate the highest costs. The occurrence of osteoporotic fractures is a cause of chronic disability, many complications, reduced quality of life, and often premature death. Aim of the study: The aim of the study was to determine which of the patient’s potential risk factors pertaining to various diseases and lifestyle have an essential impact on the occurrence of low-energy fractures and the hierarchy of these factors. Methods: The study was retrospective. The documentation of 222 patients (206 women and 16 men) from an osteoporosis treatment clinic in Łódź, Poland was analyzed. Each patient was described by a vector consisting of 27 features, where each feature was a different risk factor. Using artificial neural networks, an attempt was made to create a model that, based on the available data, would be able to predict whether the patient would be exposed to low-energy fractures. We developed a neural network model that achieved the best result for the testing data. In addition, we used other methods to solve the classification problem, i.e., correctly dividing patients into two groups: those with fractures and those without fractures. These methods were logistic regression, k-nearest neighbors and SVM. Results: The obtained results gave us the opportunity to assess the effectiveness of various methods and the importance of the features describing patients. Using logistic regression and the recursive elimination of features, a ranking of risk factors was obtained in which the most important were age, chronic kidney disease, neck T-score, and serum phosphate level. Then, we repeated the learning procedure of the neural network considering only these four most important features. The average mean squared error on the test set was about $27\%$ for the best variant of the model. Conclusions: The comparison of the rankings with different numbers of patients shows that the applied method is very sensitive to changes in the considered data (adding new patients significantly changes the result). Further cohort studies with more patients and more advanced methods of machine learning may be needed to identify other significant risk factors and to develop a reliable fracture risk system. The obtained results may contribute to the improved identification patients at risk of low-energy fractures and early implementation of comprehensive treatment.