Existence of Pareto solutions for vector polynomial optimization problems with constraints. (English) Zbl 1507.90125
This paper deals with a vector polynomial optimization problem over a basic closed semi-algebraic set. The considered problem is a constrained vector polynomial optimization, i.e., \(\min f(x)\), where \(x\) belongs to \(S\), the set of \(n\)-vectors with positive coordinates, subject to polynomial equalities type restrictions. \(S\) is a closed, semi-algebraic set and the assumption is that \(S\) is unbounded. The authors deal with the Pareto value (VPO), Pareto solution, weak Pareto value and weak Pareto solution and denoted by \(\mathrm{sol(VPO)}\) (resp., \(\mathrm{solw(VPO)}\)) as the set of all Pareto solutions (resp., weak Pareto solutions). In the paper, the existence of Pareto solutions to the constrained vector polynomial optimization problem (VPO) under some conditions is proved. The authors do not need any convexity assumptions in the problem (VPO), and they consider the problem (VPO) over a closed (and unbounded) semi-algebraic set \(S\). They define the concepts concerning the Palais-Smale, Cerami condition and establish some relationships between them (Theorem 4.1). All these concepts play an important role in establishing some sufficient conditions for the existence of Pareto solutions to the problem (VPO). In the article, an example is constructed to show that the assumption on the Mangasarian-Fromovitz constraint qualification at infinity of \(S\) cannot be dropped. Besides, several examples are also designed to illustrate some related terminologies. The results are some sufficient conditions for the existence of Pareto solutions to the problem (VPO). The obtained results improve and extend the existing theorems in the polynomial setting.
Reviewer: Doina Carp (Bucureşti)
MSC:
| 90C23 | Polynomial optimization |
| 90C29 | Multi-objective and goal programming |
| 49J30 | Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) |
Keywords:
vector optimization; polynomial optimization; Pareto solutions; Palais-Smale condition; Cerami condition; propernessSoftware:
ParetoImageSDPReferences:
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