Automatic control systems, electronic circuit design, image registration, SLAM and several other engineering problems all require nonconvex optimization. Many approaches have been developed to carry out such nonconvex optimization, but they suffer drawbacks including large computation time, require tuning of multiple unintuitive parameters and are unable to find multiple local/global minima. In...

In this paper we discuss the global weak sharp minima property for vector optimization problems with polynomial data. Exploiting the imposed polynomial structure together with tools of variational analysis and a quantitative version of Lojasiewicz’s gradient inequality due to D’Acunto and Kurdyka, we establish the Hölder type global weak sharp minima with explicitly calculated exponents.

In the present paper we consider the system ∆u−Wu(u) = 0, where u : R → R, for a class of potentials W : R → R that possess several global minima and are invariant under a general finite reflection group G. We establish existence of nontrivial entire solutions connecting the global minima of W along certain directions at infinity.

In deep learning, depth, as well as nonlinearity, create non-convex loss surfaces. Then, does depth alone create bad local minima? In this paper, we prove that without nonlinearity, depth alone does not create bad local minima, although it induces non-convex loss surface. Using this insight, we greatly simplify a recently proposed proof to show that all of the local minima of feedforward deep l...