A representation theorem for the Lorentz cone automorphisms

仅使用洛伦兹锥约束的二阶锥规划的一个精确对偶

An Exact Dual for Second-Order Cone Programming Using Only Lorentz-Cone Constraints

针对无约束规范时二阶锥规划的对偶间隙问题，本文仅用洛伦兹锥约束构造精确对偶，并证明其可行性与对偶值可达性，回答了Pólik-Terlaky问题。

Extended Lorentz cones and variational inequalities on cylinders

Solutions of a variational inequality are found by giving conditions for the monotone convergence with respect to a cone of the Picard iteration corresponding to its natural map. One of these conditions is the isotonicity of the projection onto the closed convex set in the definition of the variational inequality. If the closed convex set is a cylinder and the cone is an extented Lorentz cone, then this condition can be dropped because it is automatically satisfied. The obtained result is further particularized for unbounded box constrained variational inequalities. For this case a numerical e…

秩一生成双曲率锥及其导数松弛的自同构

Automorphisms of rank-one generated hyperbolicity cones and their derivative relaxations

本文研究了秩一生成（ROG）双曲率锥及其导数松弛的自同构，证明导数松弛的自同构即为原锥固定某子空间的自同构，并揭示了两者自同构群之间的包含关系。

The automorphism group and the non-self-duality of $p$-cones

In this paper, we determine the automorphism group of the $p$-cones ($p\neq 2$) in dimension greater than two. In particular, we show that the automorphism group of those $p$-cones are the positive scalar multiples of the generalized permutation matrices that fix the main axis of the cone. Next, we take a look at a problem related to the duality theory of the $p$-cones. Under the Euclidean inner product it is well-known that a $p$-cone is self-dual only when $p=2$. However, it was not known whether it is possible to construct an inner product depending on $p$ which makes the $p$-cone self-dual…

Linear optimization over homogeneous matrix cones

A convex cone is homogeneous if its automorphism group acts transitively on the interior of the cone, i.e., for every pair of points in the interior of the cone, there exists a cone automorphism that maps one point to the other. Cones that are homogeneous and self-dual are called symmetric. The symmetric cones include the positive semidefinite matrix cone and the second order cone as important practical examples. In this paper, we consider the less well-studied conic optimization problems over cones that are homogeneous but not necessarily self-dual. We start with cones of positive semidefinit…

齐次锥的面结构及其在齐次弦性中的应用

Faces of homogeneous cones and applications to homogeneous chordality

本文阐明齐次锥的面结构，证明每个面可由自同构映射到主面，并利用广义Cholesky分解算法构造，进而证明齐次锥投影暴露，分析齐次弦性PSD矩阵锥的面结构及其对补全问题的影响。

The monotone extended second order cone and mixed complementarity problems

In this paper, we study a new generalization of the Lorentz cone, called the Monotone Extended Second Order Cone (MESOC). We investigate basic properties of MESOC including computation of its Lyapunov rank and proving its reducibility. Moreover, we show that in an ambient space, a cylinder is an isotonic projection set with respect to MESOC. We also examine a Nonlinear Complementarity Problem on a cylinder, which is equivalent to a suitable Mixed Complementarity Problem and provide a computational example illustrating applicability of MESOC.

对称锥中Lyapunov秩的唯一性

The uniqueness of Lyapunov rank among symmetric cones

本文研究对称锥中Lyapunov秩的唯一识别问题，完整刻画了不可约锥中可由Lyapunov秩唯一确定的情形，并在一般情形下取得部分进展。