标题： 凸二次映射的尺度变换与无约束最小化 显示英文标题

标题： Rescaling and unconstrained minimisation of convex quadratic maps

摘要： 我们研究了一类分段分数映射的性质，这类映射来源于在凸二次映射中引入尺度不变性。 随后得到的映射在特定的凸锥上是拟凸和伪凸的；可以通过沿着容许方向的精确线搜索来优化这些映射，并且迭代点继承了二维最优性属性。 我们通过基于梯度的规则研究了此类松弛映射的坐标下降最小化方法，特别强调了在与潜在正半定矩阵相关的再生核希尔伯特空间中验证最大对齐属性的坐标方向。 在这种情况下，我们证明，在某些情况下，考虑迭代点的最佳尺度调整可以显著加速凸二次映射的无约束最小化。 显示英文摘要

摘要： We investigate the properties of a class of piecewise-fractional maps arising from the introduction of an invariance under rescaling into convex quadratic maps. The subsequent maps are quasiconvex, and pseudoconvex on specific convex cones; they can be optimised via exact line search along admissible directions, and the iterates then inherit a bidimensional optimality property. We study the minimisation of such relaxed maps via coordinate descents with gradient-based rules, placing a special emphasis on coordinate directions verifying a maximum-alignment property in the reproducing kernel Hilbert spaces related to the underlying positive-semidefinite matrices. In this setting, we illustrate that accounting for the optimal rescaling of the iterates can in certain situations substantially accelerate the unconstrained minimisation of convex quadratic maps.