On the Primal-Dual Geometry of Level Sets in Linear and Conic Optimization
研究了锥优化问题中原始目标函数水平集与对偶目标函数水平集之间的几何关系，发现原始水平集的最大范数与对偶水平集的最大内切半径近似成反比。
For a conic optimization problem $$ \begin{array}{lclr} P: & {\rm minimize}_x & c^{T}x \\ & \mbox{s.t. } & Ax=b,\\ & & x \in C \\ \end{array} $$ \noindent and its dual $$ \begin{array}{lclr} D: & {\rm supremum}_{y,s} & b^{T}y\\ & \mbox{ s.t. } & A^Ty+s=c,\\ & & s \in C^* ,\\ \end{array} $$ we present a geometric relationship between the primal objective function level sets and the dual objective function level sets, which shows that the maximum norms of the primal objective function level sets are nearly inversely proportional to the maximum inscribed radii of the dual objective function level sets.