Inexact Conditions Research Articles

Bregman Proximal Point Algorithm Revisited: A New Inexact Version and Its Inertial Variant

We study a general convex optimization problem, which covers various classic problems in different areas and particularly includes many optimal transport related problems arising in recent years. To solve this problem, we revisit the classic Bregman proximal point algorithm (BPPA) and introduce a new inexact stopping condition for solving the subproblems, which can circumvent the underlying feasibility difficulty often appearing in existing inexact conditions when the problem has a complex feasible set. Our inexact condition also covers several existing inexact conditions as special cases and hence makes our inexact BPPA (iBPPA) more flexible to fit different scenarios in practice. As an application to the standard optimal transport (OT) problem, our iBPPA with the entropic proximal term can bypass some numerical instability issues that usually plague the popular Sinkhorn's algorithm in the OT community, since our iBPPA does not require the proximal parameter to be very small for obtaining an accurate approximate solution. The iteration complexity of $O(1/k)$ and the convergence of the sequence are also established for our iBPPA under some mild conditions. Moreover, inspired by Nesterov's acceleration technique, we develop an inertial variant of our iBPPA, denoted by V-iBPPA, and establish the iteration complexity of $O(1/k^{\lambda})$, where $\lambda\geq1$ is a quadrangle scaling exponent of the kernel function. In particular, when the proximal parameter is a constant and the kernel function is strongly convex with Lipschitz continuous gradient (hence $\lambda=2$), our V-iBPPA achieves a faster rate of $O(1/k^2)$ just as existing accelerated inexact proximal point algorithms. Some preliminary numerical experiments for solving the standard OT problem are conducted to show the convergence behaviors of our iBPPA and V-iBPPA under different inexactness settings. The experiments also empirically verify the potential of our V-iBPPA for improving the convergence speed.

A note on inexact gradient and Hessian conditions for cubic regularized Newton’s method

The inexact cubic-regularized Newton’s method (CR) proposed by Cartis, Gould and Toint achieves the same convergence rate as exact CR proposed by Nesterov and Polyak, but the inexact condition is not implementable due to its dependence on a future variable. This note establishes the same convergence rate under a similar but implementable inexact condition, which depends on only current variables. Our proof bounds the function-value decrease over total iterations rather than each iteration in the previous studies.

Using Bond Valence sums as restraints in XAFS analysis

The Bond Valence model gives a simple, accurate relationship between the bond distance, co-ordination number, and formal valence of ions in many systems. This empirical model thus gives complementary information to EXAFS and XANES, and has been used as an independent check of XAFS results by many workers. More generally, the Bond Valence represents prior knowledge of the system that can be used to constrain the XAFS analysis itself. Since Bond Valence sums are not perfectly satisfied, their use as exact constraints in XAFS analysis may not be warranted and they are probably best suited as restraints – inexact conditions placed on the analysis that act as additional data or prior knowledge of the system. The practical use of Bond Valence sums as constraints and restraints in EXAFS analysis is demonstrated for iron oxides, and the use of Bond Valence sums as prior knowledge that can be applied to a wide-range of XAFS analyses is discussed.