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In this paper, we are interested in the existence of Pareto solutions to vector polynomial optimization problems over a basic closed semi-algebraic set. By invoking some powerful tools from real semi-algebraic geometry, we first introduce the concept called {it tangency varieties}; then we establish connections of the Palais--Smale condition, Cerami condition, {it M}-tameness, and properness related to the considered problem, in which the condition of regularity at infinity plays an essential role in deriving these connections. According to the obtained connections, we provide some sufficient conditions for existence of Pareto solutions to the problem in consideration, and we also give some examples to illustrate our main findings.
We consider optimization problems with polynomial inequality constraints in non-commuting variables. These non-commuting variables are viewed as bounded operators on a Hilbert space whose dimension is not fixed and the associated polynomial inequalit
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In this article, we view the approximate version of Pareto and weak Pareto solutions of the multiobjective optimization problem through the lens of KKT type conditions. We also focus on an improved version of Geoffrion proper Pareto solutions and cha