A local convergence result for abstract descent methods is proved. The sequence of iterates is attracted by a local (or global) minimum, stays in its neighborhood and converges. This result allows algorithms to exploit local properties of the objective function: The gradient of the Moreau envelope of a prox-regular functions is locally Lipschitz continuous and expressible in terms of the proxim...

In this talk, a new algorithm to sample from possibly non-smooth log-concave probability measures is introduced. This algorithm uses Moreau-Yosida envelope combined with the Euler-Maruyama discretization of Langevin diffusions. They are applied to a deconvolution problem in image processing, which shows that they can be practically used in a high dimensional setting. Finally, non-asymptotic con...

We give a geometric characterization of compact Riemann surfaces admitting orientation reversing involutions with fixed points. Such surfaces are generally called real surfaces and can be represented by real algebraic curves with non-empty real part. We show that there is a family of disjoint simple closed geodesics that intersect all geodesics of a partition at least twice in uniquely right an...

We introduce two algorithms for nonconvex regularized finite sum minimization, where typical Lipschitz differentiability assumptions are relaxed to the notion of relative smoothness. The first one is a Bregman extension Finito/MISO, studied fully problems when sampling random, or under convexity nonsmooth term it essentially cyclic. second algorithm low-memory variant, in spirit SVRG and SARAH,...

Minimax optimization has become a central tool in machine learning with applications robust optimization, reinforcement learning, GANs, etc. These are often nonconvex–nonconcave, but the existing theory is unable to identify and deal fundamental difficulties this poses. In paper, we study classic proximal point method (PPM) applied nonconvex–nonconcave minimax problems. We find that generalizat...

Abstract We study minimization of a structured objective function, being the sum smooth function and composition weakly convex with linear operator. Applications include image reconstruction problems regularizers that introduce less bias than standard regularizers. develop variable smoothing algorithm, based on Moreau envelope decreasing sequence parameters, prove complexity $${\mathcal {O}}(\e...

Sparse principal component analysis (SPCA) has emerged as a powerful technique for modern data analysis. We discuss a robust and scalable algorithm for computing sparse principal component analysis. Specifically, we model SPCA as a matrix factorization problem with orthogonality constraints, and develop specialized optimization algorithms that partially minimize a subset of the variables (varia...

We develop sufficient optimality conditions for a Moreau-Yosida regularized optimal control problem governed by a semilinear elliptic PDE with pointwise constraints on the state and the control. We make use of the equivalence of a setting of Moreau-Yosida regularization to a special setting of the virtual control concept, for which standard second order sufficient conditions have been shown. Mo...

This paper addresses the minimal subset selection of antennas achieving designated channel capacity. This is one of the most natural approaches to alleviating the power consumption in MIMO systems, while it is a mathematically challenging nonlinearlyconstrained sparse optimization ( -norm minimization) problem. We present an ef cient algorithmic solution, to this highly combinatorial problem, u...

We study certain twisted sums of Orlics spaces with non-trivial type which can be viewed as Fenchel-Orlicz spaces on R2. We then show that a large class of Fenchel-Orlicz spaces on R" can be renormed to have property (M). In particular this gives a new construction of the twisted Hilbert space 22 and shows it has property (M), after an appropriate renorming.