Session: Variational Analysis: Theory and Applications I
Chair: Walaa Moursi
Cluster: Nonsmooth Optimization
Talk 1: On Opial’s Lemma
Speaker: Heinz Bauschke
Abstract: Opial's Lemma is a fundamental result in the convergence analysis of sequences generated by optimization algorithms in real Hilbert spaces. We introduce the concept of Opial sequences - sequences for which the limit of the distance to each point in a given set exists. We systematically derive properties of Opial sequences, contrasting them with the well-studied Fejér monotone sequences, and establish conditions for weak and strong convergence. Key results include characterizations of weak convergence via weak cluster points (reaffirming Opial's Lemma), strong convergence via strong cluster points, and the behavior of projections onto Opial sets in terms of asymptotic centers. Special cases and examples are provided to highlight the subtle differences in convergence behaviour and projection properties compared to the Fejér monotone case. Joint work with Aleksandr Arakcheev (UBC).
Talk 2: Recovering Nesterov accelerated dynamics from Heavy Ball dynamics via time rescaling
Speaker: Radu Bot
Abstract: In a real Hilbert space, we consider two classical problems: the global minimization of a smooth and
convex function f (i.e., a convex optimization problem) and finding the zeros of a monotone and continuous
operator V (i.e., a monotone equation). Attached to the optimization problem, first we study the asymptotic
properties of the trajectories generated by a second-order dynamical system which features a constant vis-
cous friction coefficient and a positive, monotonically increasing function b(t) multiplying ∇f. When b(t) is
identically 1, we recover the Heavy Ball with friction dynamics introduced by Polyak in 1964. For a gener-
ated solution trajectory y(t), we show small oconvergence rates dependent on b(t) for f(y(t))−min f, and
the weak convergence of y(t) towards a global minimizer of f. In 2015, Su, Boyd and Cand´ es introduced
a second-order system which could be seen as the continuous-time counterpart of Nesterov’s accelerated
gradient. As the first key point of this talk, we show that for a special choice for b(t), these two seemingly
unrelated dynamical systems are connected: namely, they are time reparametrizations of each other. Every
statement regarding the continuous-time accelerated gradient system may be recovered from its Heavy Ball
counterpart.
As the second key point of this talk, we observe that this connection extends beyond the optimization
setting. Attached to the monotone equation involving the operator V, we again consider a Heavy Ball-
like system which features an additional correction term which is the time derivative of the operator along
the trajectory. We derive small orates for ∥V(y(t))∥, which depend on two rescaling functions µ(t) and
γ(t), and show the weak convergence of y(t) to a zero of V. For special choices for µ(t) and γ(t), we
establish a time reparametrization equivalence with the Fast OGDA dynamics introduced by Bot¸, Csetnek
and Nguyen in 2022, which can be seen as an analog of the continuous accelerated gradient dynamics, but
for monotone operators. Again, every statement regarding the Fast OGDA system may be recovered from a
Heavy Ball-like system.
The talk relies on a joint work with H. Attouch, D.A. Hulett and D.-K. Nguyen.
Talk 3: On the Bauschke-Bendit-Moursi Modulus of Averagedness and Classifications of Firmly Nonexpansive Operators
Speaker: Xianfu Wang
Abstract: Firmly nonexpansive operators are important in Convex Analysis and Optimization and Algorithms. It is a special case of averaged operators.
We propose a classification of averaged operators, firmly nonexpansive operators, and proximal mappings by modulus of averagedness.
Explicit formula for calculating the modulus of averagedness of proximal mappings are given. One amazing result is that a proximal mapping of a convex function
has its modulus of averagedness less than $1/2$ if and only if the function is Lipschitz smooth.
Joint work with Shuang Song
