This paper aims to find efficient solutions to a multi-objective optimization problem (MP) with convex polynomial data. To this end, a hybrid method, which allows us to transform problem (MP) into a scalar convex polynomial optimization problem \(({\mathrm{P}}_{z})\) and does not destroy the properties of convexity, is considered. First, we show an existence result for efficient solutions to problem (MP) under some mild assumption. Then, for problem \((P_{z})\), we establish two kinds of representations of non-negativity of convex polynomials over convex semi-algebraic sets, and propose two kinds of finite convergence results of the Lasserre-type hierarchy of semidefinite programming relaxations for problem \(({\mathrm{P}}_{z})\) under suitable assumptions. Finally, we show that finding efficient solutions to problem (MP) can be achieved successfully by solving hierarchies of semidefinite programming relaxations and checking a flat truncation condition.

本文旨在寻找具有凸多项式数据的多目标优化问题（MP）的有效解决方案。为此，采用了一种混合方法，该方法允许我们将问题（MP）转换为标量凸多项式优化问题 \（（{{mathrm {P}} _ {z}）\），而不会破坏凸性，被认为。首先，我们给出了一个温和假设下有效解决问题（MP）问题的存在结果。然后，针对问题 \（（P_ {z}）\），我们建立了凸半代数集上凸多项式的非负性的两种表示，并提出了Lasserre型层次结构的两种有限收敛结果。问题\（（{{mathrm {P}} _ {z}）\）的半定编程松弛 在适当的假设下。最后，我们表明，通过解决半定程序松弛的层次结构并检查平坦的截断条件，可以成功地找到有效的问题解决方案（MP）。