A diameter criterion tied to a potential function certifies convergence of difference inclusions, enabling discrete proofs for first-order optimization methods with diminishing steps.
On the Finite Number of Directional Stationary Values of Piecewise Programs
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abstract
Extending a fundamental result for (indefinite) quadratic programs, this paper shows that certain non-convex piecewise programs have only a finite number of directional stationary values, and thus, possess only finitely many locally minimum values. We present various special cases of our main results, in particular, an application to a least-squares piecewise affine regression problem for which every directional stationary point is locally minimizing.
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Convergence of difference inclusions via a diameter criterion
A diameter criterion tied to a potential function certifies convergence of difference inclusions, enabling discrete proofs for first-order optimization methods with diminishing steps.