Kurdyka–Łojasiewicz (KL) exponent plays an important role in estimating the convergence rate of many contemporary first-order methods. In particular, a KL exponent of \(\frac{1}{2}\) for a suitable potential function is related to local linear convergence. Nevertheless, KL exponent is in general extremely hard to estimate. In this paper, we show under mild assumptions that KL exponent is preserved via inf-projection. Inf-projection is a fundamental operation that is ubiquitous when reformulating optimization problems via the lift-and-project approach. By studying its operation on KL exponent, we show that the KL exponent is \(\frac{1}{2}\) for several important convex optimization models, including some semidefinite-programming-representable functions and some functions that involve \(C^2\)-cone reducible structures, under conditions such as strict complementarity. Our results are applicable to concrete optimization models such as group-fused Lasso and overlapping group Lasso. In addition, for nonconvex models, we show that the KL exponent of many difference-of-convex functions can be derived from that of their natural majorant functions, and the KL exponent of the Bregman envelope of a function is the same as that of the function itself. Finally, we estimate the KL exponent of the sum of the least squares function and the indicator function of the set of matrices of rank at most k.

Kurdyka–Łojasiewicz (KL) 指数在估计许多当代一阶方法的收敛速度方面起着重要作用。特别是，一个合适的势函数的 KL 指数\(\frac{1}{2}\)与局部线性收敛有关。然而，KL 指数通常极难估计。在本文中，我们在温和的假设下展示了通过 inf 投影保留 KL 指数。Inf-projection 是一种基本操作，在通过提升和投影方法重新制定优化问题时无处不在。通过研究其对 KL 指数的运算，我们表明对于几个重要的凸优化模型，包括一些半定规划可表示函数和一些涉及的函数，KL 指数是\(\frac{1}{2}\)\(C^2\) -锥体可还原结构，在严格互补等条件下。我们的结果适用于具体的优化模型，例如组融合套索和重叠组套索。此外，对于非凸模型，我们证明了许多凸差函数的 KL 指数可以从它们的自然主函数的指数导出，并且函数的 Bregman 包络的 KL 指数与功能本身。最后，我们估计最小二乘函数和秩最大为k的矩阵集合的指示函数之和的 KL 指数。