Second-Order Composed Radial Derivatives of the Benson Proper Perturbation Map for Parametric Multi-Objective Optimization Problems

In this paper, we introduce second-order composed radial derivatives of set-valued maps and establish some of its properties. By applying this second-order derivative, we obtain second-order sensitivity results for parametric multi-objective optimization problems under the Benson proper efficiency without assumptions of cone-convexity and Lipschitz continuity. Some of our results improve and derive the recent corresponding ones in the literature.

Higher-order variational set of the benson proper perturbation map in set-valued optimization

In the paper, we study sensitivity analysis in set-valued optimization, a research direction has been attracting much attention of many mathematicians in the world recently. The main derivative used in the paper is higher-order variational set (introduced by Khanh and Tuan in 2008) which is considered as a generalization of the contingent derivative (known as the first and the most popular derivative in set-valued optimization). Firstly, we establish relationships between higher-order variational sets of a given set-valued map and those of its profile (extended by a ordering cone). Then, we give results on higher-order variational set of the Benson proper perturbation map for a kind of set-valued optimization problem, the perturbation map is defined in the objective space. Finally, we apply the obtained results to sensitivity analysis for optimal-value map of a parametrized constrained set-valued optimization problem whose the objective map and constrained maps depends on some parameter. More precisely, some results on sensitivity analysis for parametrized constrained set-valued optimization problem are obtained. The content of the paper gives us more applications of higher-order variational set in set-valued optimization.

Second-Order Composed Contingent Derivative of the Perturbation Map in Multiobjective Optimization

In view of the structural advantage of second-order composed derivatives, the purpose of this paper is to analyze quantitatively the behavior of perturbation maps for the first time by using this concept. First, new concepts of the second-order composed adjacent derivative and the second-order composed lower Dini derivative are introduced. Some relationships among the second-order composed contingent derivative, the second-order composed adjacent derivative and the second-order composed lower Dini derivative are discussed. Second, the relationships between second-order composed lower Dini derivable and Aubin property are provided. Third, by virtue of second-order composed contingent derivatives and the above relationships, some results concerning second-order sensitivity analysis are established without the assumption of the locally Lipschitz property or the locally Hölder continuity. Finally, we give some complete characterizations of second-order composed contingent derivatives of the perturbation maps.

Second-order sensitivity analysis for parametric equilibrium problems in set-valued optimization

In the paper, we first establish relationships between second-order contingent derivatives of a given set-valued map and that of the weak perturbation map. Then, these results are applied to sensitivity analysis for parametric equilibrium problems in set-valued optimization.

Probing Protein Allostery as a Residue-specific Concept via Residue Perturbation Maps

Allosteric regulation is a well-established phenomenon classically defined as conformational or dynamical change of a small number of allosteric residues of the protein upon allosteric effector binding at a distance. Here, we developed a novel approach to delineate allosteric effects in proteins. In this approach, we applied robust machine learning methods, including Deep Neural Network and Random Forest, on extensive molecular dynamics (MD) simulations to distinguish otherwise similar allosteric states of proteins. Using PDZ3 domain of PDS-95 as a model protein, we demonstrated that the allosteric effects could be represented as residue-specific properties through two-dimensional property-residue maps, which we refer as “residue perturbation maps”. These maps were constructed through two machine learning methods and could accurately describe how different properties of various residues are affected upon allosteric perturbation on protein. Based on the “residue perturbation maps”, we propose allostery as a residue-specific concept, suggesting all residues could be considered as allosteric residues because each residue “senses” the allosteric events through perturbation of its one or multiple attributes in a quantitatively unique way. The “residue perturbation maps” could be used to fingerprint a protein based on the unique patterns of residue perturbations upon binding events, providing a novel way to systematically describe the protein allosteric effects of each residue upon perturbation.Author SummaryAllostery is protein regulation at distance. A perturbation at one site of the protein could distantly affect another site. The residues involved in these sites are considered as allosteric residues. The allostery concept has been widely used to understand protein mechanisms and to design allosteric drugs. It is long believed only a small number of residues are allosteric residues. Here, we argue that all residues in a protein are allosteric residues. Upon the perturbation of the allosteric events, the different properties of each residue are affected at the distinct extend. We used hybrid models including molecular dynamics simulations and machine learning components to reveal that not only many properties of residues are affected upon ligand binding, but also each residue is affected through perturbation of its various properties, which makes the residue distinguishable from other residues. According to our findings in a model protein, we defined a “residue perturbation map” as a two-dimensional map that fingerprint a protein based on the extent of perturbation in different properties of all its residues in a quantitative fashion. This “residue perturbation map” provides a novel way to systematically describe the protein allosteric effects of each residue upon perturbation.