论文标题
平滑凸优化中的好奇心和反例
Curiosities and counterexamples in smooth convex optimization
论文作者
论文摘要
在平滑的凸强制设置中提供了一些旧的优化问题的反例。我们表明，具有精确搜索或Bregman下降方法的区块坐标最陡峭的下降通常不会收敛。建立了各种理想特征的其他故障：凯奇梯度曲线的方向收敛，牛顿流量的收敛，Tikhonov路径的有限长度，中央路径的收敛或光滑的kurdyka-lojasiewicz不等式。所有示例都是平面。这些示例基于一般的平滑凸插值结果。给定平面中正面弯曲的C k凸的紧凑型集的序列减小，我们提供了C k $ \ ge $ 2的c k平滑凸函数的水平集插值。如果将交叉点降低到一点点，我们的插值具有正定的Hessian，否则在溶液集中是正定的。此外，在一系列降低多边形的序列下，我们提供了与顶点一致的插值，并且其梯度与规定的正态相吻合。
Counterexamples to some old-standing optimization problems in the smooth convex coercive setting are provided. We show that block-coordinate, steepest descent with exact search or Bregman descent methods do not generally converge. Other failures of various desirable features are established: directional convergence of Cauchy's gradient curves, convergence of Newton's flow, finite length of Tikhonov path, convergence of central paths, or smooth Kurdyka-Lojasiewicz inequality. All examples are planar. These examples are based on general smooth convex interpolation results. Given a decreasing sequence of positively curved C k convex compact sets in the plane, we provide a level set interpolation of a C k smooth convex function where k $\ge$ 2 is arbitrary. If the intersection is reduced to one point our interpolant has positive definite Hessian, otherwise it is positive definite out of the solution set. Furthermore , given a sequence of decreasing polygons we provide an interpolant agreeing with the vertices and whose gradients coincide with prescribed normals.
