Second-order subdifferential calculus with applications to tilt stability in optimization

The paper concerns the second-order generalized differentiation theory of variational analysis and new applications of this theory to some problems of constrained optimization in finitedimensional spaces. The main attention is paid to the so-called (full and partial) second-order subdifferentials of extended-real-valued functions, which are dual-type constructions generated by coderivatives of frst-order subdifferential mappings. We develop an extended second-order subdifferential calculus and analyze the basic second-order qualification condition ensuring the fulfillment of the principal secondorder chain rule for strongly and fully amenable compositions. The calculus results obtained in this way and computing the second-order subdifferentials for piecewise linear-quadratic functions and their major specifications are applied then to the study of tilt stability of local minimizers for important classes of problems in constrained optimization that include, in particular, problems of nonlinear programming and certain classes of extended nonlinear programs described in composite terms

On the stability of solution mapping for parametric generalized vector quasiequilibrium problems

AbstractIn this paper, we study the solution stability for a class of parametric generalized vector quasiequilibrium problems. By virtue of the parametric gap function, we obtain a sufficient and necessary condition for the Hausdorff lower semicontinuity of the solution mapping to the parametric generalized vector quasiequilibrium problem. The results presented in this paper generalize and improve some main results of Chen et al. (2010) [34], and Zhong and Huang (2011) [35]

Continuity of the Solution Maps for Generalized Parametric Set-Valued Ky Fan Inequality Problems

Under new assumptions, we provide suffcient conditions for the (upper and lower) semicontinuity and continuity of the solution mappings to a class of generalized parametric set-valued Ky Fan inequality problems in linear metric space. These results extend and improve some known results in the literature (e.g., Gong, 2008; Gong and Yoa, 2008; Chen and Gong, 2010; Li and Fang, 2010). Some examples are given to illustrate our results

KKT reformulation and necessary conditions for optimality in nonsmooth bilevel optimization

For a long time, the bilevel programming problem has essentially been considered as a special case of mathematical programs with equilibrium constraints (MPECs), in particular when the so-called KKT reformulation is in question. Recently though, this widespread believe was shown to be false in general. In this paper, other aspects of the difference between both problems are revealed as we consider the KKT approach for the nonsmooth bilevel program. It turns out that the new inclusion (constraint) which appears as a consequence of the partial subdifferential of the lower-level Lagrangian (PSLLL) places the KKT reformulation of the nonsmooth bilevel program in a new class of mathematical program with both set-valued and complementarity constraints. While highlighting some new features of this problem, we attempt here to establish close links with the standard optimistic bilevel program. Moreover, we discuss possible natural extensions for C-, M-, and S-stationarity concepts. Most of the results rely on a coderivative estimate for the PSLLL that we also provide in this paper

A characterization of nonemptiness and boundedness of the solution set for set-valued vector equilibrium problems via scalarization and stability results

International audienceAttitude is a key concept in social psychology. The paper presents a novel agent-based model to simulate attitude formation by combining a rational and an emotional components based on cognitive, psychological and social theories. Individuals of the artificial population perceive actions taken by actors such as government or brands, they form an attitude toward them and also communicate the events through a social network. The model outputs are first studied through a functional analysis in which some unique macroscopic behaviors have emerged such as the impact of social groups, the resistance of the population toward disinformation campaigns or the social pressure. We then applied our model on a real world scenario depicting the effort of French Forces in their stabilization operations in Kapisa (Afghanistan) between 2010 and 2012. We calibrated the model parameters based on this scenario and the results of opinion polls that were conducted in the area during the same period about the sentiment of the population toward the Forces. Our model was able to reproduce polls results with a global error under 3%. Based on these results, we show the different dynamics tendencies that emerged among the population by applying a non-supervised classification algorithm