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Sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers. Here, nonsmooth approximate gradient projection and complementary approximate Karush-Kuhn-Tucker conditions are presented. These sequential optimality conditions are satisfied by local minimizers of optimization problems independently of the fulfillment of constraint qualifications. It is proved that nonsmooth complementary approximate Karush-Kuhn-Tucker conditions are stronger than nonsmooth approximate gradient projection conditions. Sufficiency for differentiable generalized convex programming is established.

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The goal of this study is to use the grounded theory technique to uncover the characteristics that influence personal branding in the food business. The current study is quantitative in terms of approach and applied in terms of goal. The systematic technique of Strauss and Corbin was applied in this study, which involves three basic procedures: open coding, axial coding, and selective coding. The snowball sampling approach was utilised in this study for sampling. Snowball sampling is generally the first step, and it continues until saturation is reached. Based on this, 19 specialists and managers in Iran's food business who were familiar with the term personal branding were interviewed in-depth and semi-structured interviews. MAXQDA 11.1.4 software was utilised to analyse the data in this investigation. The results showed the components identified in personal branding in casual conditions of this study including personal characteristics, business characteristics, strategies including identity tools defined in behavior, appropriate tools for illustration, social networking, social responsibility, use different and distinctive methods, focus on goal, market research, contextual conditions including cultural values, political, economic and social issues, ideas and beliefs, dynamic world and intervening conditions including criticisms and suggestions, difference between cyberspace and real world, audience expectation level, social norms and outcomes include advancing the company's goals, saving time and money, gaining internal satisfaction, attracting and retaining audiences, gaining reputation, and improving the quality of communication.

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The basic purpose of this study is to investigate and display causal relationships among collaboration components in supply chain 4.0 using a fuzzy framework. The power of collaboration increases with the effect of Industry 4.0 technologies for the improvement of supply chain performance, so supply chain 4.0 is the context of this study. To achieve the research purpose, after reviewing articles and extracting indicators, a collaboration model with trust, initiators, barriers, dimensions, and outcomes was designed. Then using the fuzzy DEMATEL method, the effect of each variable and its position were determined. To collect data targeted sampling and snowball methods were used. 20 questionnaires were distributed to supply chain and digital technologies experts. Findings show that Trust and Information and Communication Technology infrastructure are closely related and are considered the most fundamental factors of the collaboration conceptual model, and can lead to more serious and effective changes in SC 4.0 such as improved Economic and Social performance. SC 4.0 managers can facilitate the development of collaborative trust across the SC By investing in communication and technology infrastructure.