15th EUROPT Workshop on Advances in Continuous Optimization
Montréal, Canada, 12 — 14 July 2017
15th EUROPT Workshop on Advances in Continuous Optimization
Montréal, Canada, 12 — 14 July 2017
Variational Analysis and Nonsmooth Optimization III
Jul 12, 2017 03:30 PM – 05:10 PM
Location: Amphithéâtre Banque Nationale
Chaired by Tim Hoheisel
4 Presentations

03:30 PM  03:55 PM
Decentralized Stochastic Optimization with MultiAgent Mirror Descent
We develop a decentralized algorithm for stochastic composite optimization problems, combining ideas from consensusbased multiagent optimization and the celebrated mirror descent algorithm. When the composite regularization term is strongly convex, the proposed method is shown to converge at a rate of $\order{1/k}$ where $k$ is the number of iterations executed. This is known to be the best possible rate of convergence for the class of problems considered. Moreover, theory and experiments show that the speedup of the proposed methodsthe number of iterations required to reach a desired level of accuracyscales linearly with the number of nodes in the network.

03:55 PM  04:20 PM
Epsilon Subdifferential Computation in Computational Convex Analysis
Computational convex analysis focuses on the computation of convex operators that routinely appear in convex analysis e.g. LegendreFenchel conjugate, Moreau envelope, etc. After recalling what computation the (open source) CCA numerical library allows, we will switch the focus from convex transforms like the LegendreFenchel transform to convex operators like the subdifferential. We will present results on efficiently computing the epsilon subdifferential of a convex univariate function.

04:20 PM  04:45 PM
Recent Developments on Augmented Lagrangian Methods in Finite and Infinite Dimensions
The augmented Lagrangian method (ALM) is a classical technique for nonlinear programming which has strong connections to convex analysis. In this talk, we describe some recent developments on ALMs in finite dimensions, and discuss how such methods can be applied to generic (nonconvex) optimization problems in Banach and Hilbert spaces. It turns out that the resulting algorithm is fairly general and can deal with many classes of constraint functions. To illustrate this, we give a variety of examples, including standard problems from variational analysis as well as optimal control problems in infinite dimensions.

04:45 PM  05:10 PM
A FirstOrder Method for Nonconvex Optimization without Lipschitz Gradients
Firstorder algorithms for nonconvex optimization problems travel along gradient directions in order to reach stationary points. The standard assumption imposed upon gradients is that they vary in a Lipschitz manner. This assumption leads to global convergence, but restricts the applicability of convergence guarantees to functions that grow at most quadratically. In this talk, we introduce a method which lifts this quadratic growth restriction, but still maintains global convergence guarantees. We furthermore provide sufficient, easy to check conditions which guarantee global convergence of proposed algorithm, as well as a new approach, based on Ekeland's variational principle, for determining how quickly the proposed algorithm converges.