This paper generalizes the factorization theorem of Gouveia,Parrilo and Thomas to a broader class of convex sets.Given a general convex set,the authors define a slack operator associated to the set and its polar accor...This paper generalizes the factorization theorem of Gouveia,Parrilo and Thomas to a broader class of convex sets.Given a general convex set,the authors define a slack operator associated to the set and its polar according to whether the convex set is full dimensional,whether it is a translated cone and whether it contains lines.The authors strengthen the condition of a cone lift by requiring not only the convex set is the image of an affine slice of a given closed convex cone,but also its recession cone is the image of the linear slice of the closed convex cone.The authors show that the generalized lift of a convex set can also be characterized by the cone factorization of a properly defined slack operator.展开更多
Let f, g_1, ···, g_s be polynomials in R[X_1, ···, X_n]. Based on topological properties of generalized critical values, the authors propose a method to compute the global in?mum f~? of f...Let f, g_1, ···, g_s be polynomials in R[X_1, ···, X_n]. Based on topological properties of generalized critical values, the authors propose a method to compute the global in?mum f~? of f over an arbitrary given real algebraic set V = {x ∈ R^n| g_1(x) = 0, ···, g_s(x) = 0}, where V is not required to be compact or smooth. The authors also generalize this method to solve the problem of optimizing f over a basic closed semi-algebraic set S = {x ∈ R^n| g_1(x) ≥ 0, ···, g_s(x) ≥ 0}.展开更多
基金supported by Equipment Pre-Research Field Fund under Grant Nos.JZX7Y20190258055501,JZX7Y20190243016801the National Natural Science Foundation of China under Grant No.11901544+2 种基金the National Key Research Project of China under Grant No.2018YFA0306702the National Natural Science Foundation of China under Grant No.11571350supported by National Institute for Mathematical Sciences 2014 Thematic Program on Applied Algebraic Geometry in Daejeon,South Korea。
文摘This paper generalizes the factorization theorem of Gouveia,Parrilo and Thomas to a broader class of convex sets.Given a general convex set,the authors define a slack operator associated to the set and its polar according to whether the convex set is full dimensional,whether it is a translated cone and whether it contains lines.The authors strengthen the condition of a cone lift by requiring not only the convex set is the image of an affine slice of a given closed convex cone,but also its recession cone is the image of the linear slice of the closed convex cone.The authors show that the generalized lift of a convex set can also be characterized by the cone factorization of a properly defined slack operator.
基金supported by the National Key Research Project of China under Grant No.2018YFA0306702the National Natural Science Foundation of China under Grant No.11571350
文摘Let f, g_1, ···, g_s be polynomials in R[X_1, ···, X_n]. Based on topological properties of generalized critical values, the authors propose a method to compute the global in?mum f~? of f over an arbitrary given real algebraic set V = {x ∈ R^n| g_1(x) = 0, ···, g_s(x) = 0}, where V is not required to be compact or smooth. The authors also generalize this method to solve the problem of optimizing f over a basic closed semi-algebraic set S = {x ∈ R^n| g_1(x) ≥ 0, ···, g_s(x) ≥ 0}.
