本项目主要运用凸分析、半定规划和实代数几何中的关键理论和工具研究基本半代数集的凸包的谱多面体表示或近似、基本半代数集上的参数化优化问题、凸集的提升和锥分解以及（半）代数系统求解。 当所考虑的集合紧致和光滑时， 前三个问题的解决主要是基于正零点定理、半正定松弛、KKT条件和凸集的极点表示等。而在非紧致或非光滑情形，正零点定理、KKT条件等可能不成立。本项目将重点研究如何应用提升、投影和分解等技巧以及凸分析和实代数几何中的关键理论和技术将紧致光滑情形下上述问题的相关结论推广到一般情形。当（半）代数系统有无穷多实解时，给出其矩量矩阵半定松弛的终止准则并得到系统相应实根理想的一组基。

The project will focus on studying some problems in convex algebraic geometry using convex analysis, semidefinite programming and real algebraic geometry. We work on problems: spectrahedral representations of convex hulls of basic semalgebraic sets; optimizing a parameteric function over a (semi)-algebraic set; lifting of convex sets and cone factorizations; certificates for semidefinite relaxations in computing positive-dimensional real radical ideals. For the first three problems, when the considered set is smooth and compact, some important results have already been obtained via Positivstellensatz, semidefinite relaxations, KKT conditions and representations of convex sets by extreme points. However, when the considered set is nonsmooth or not compact then the Positivstellensatz and KKT condition might be invalid. This project will apply lifting, projection and decomposition techniques and known results in convex analysis and real algebraic geometry to extend results obtained in smooth and compact case to general case. For a (semi)algebraic system with infinitely many real solutions, we will work on practical termination criteria of its moment relaxations and compute a basis of the real radical ideal corresponding to the input system.