There is no variational characterization of the cycles in the method of periodic projections

The method of periodic projections consists in iterating projections onto $m$ closed convex subsets of a Hilbert space according to a periodic sweeping strategy. In the presence of $m\geq 3$ sets, a long-standing question going back to the 1960s is whether the limit cycles obtained by such a process can be characterized as the minimizers of a certain functional. In this paper we answer this question in the negative. Projection algorithms that minimize smooth convex functions over a product of convex sets are also discussed

A Dynamical Approach to Convex Minimization Coupling Approximation with the Steepest Descent Method

AbstractWe study the asymptotic behavior of the solutions to evolution equations of the form 0∈u(t)+∂f(u(t), ε(t));  u(0)=u0, where {f(·, ε):ε>0} is a family of strictly convex functions whose minimum is attained at a unique pointx(ε). Assuming thatx(ε) converges to a pointx* as ε tends to 0, and depending on the behavior of the optimal trajectoryx(ε), we derive sufficient conditions on the parametrization ε(t) which ensure that the solutionu(t) of the evolution equation also converges tox* whent→+∞. The results are illustrated on three different penalty and viscosity-approximation methods for convex minimization

Asymptotic behavior of compositions of under-relaxed nonexpansive operators

In general there exists no relationship between the fixed point sets of the composition and of the average of a family of nonexpansive operators in Hilbert spaces. In this paper, we establish an asymptotic principle connecting the cycles generated by under-relaxed compositions of nonexpansive operators to the fixed points of the average of these operators. In the special case when the operators are projectors onto closed convex sets, we prove a conjecture by De Pierro which has so far been established only for projections onto affine subspaces

Algorithms for flows over time with scheduling costs

Flows over time have received substantial attention from both an optimization and (more recently) a game-theoretic perspective. In this model, each arc has an associated delay for traversing the arc, and a bound on the rate of flow entering the arc; flows are time-varying. We consider a setting which is very standard within the transportation economic literature, but has received little attention from an algorithmic perspective. The flow consists of users who are able to choose their route but also their departure time, and who desire to arrive at their destination at a particular time, incurring a scheduling cost if they arrive earlier or later. The total cost of a user is then a combination of the time they spend commuting, and the scheduling cost they incur. We present a combinatorial algorithm for the natural optimization problem, that of minimizing the average total cost of all users (i.e., maximizing the social welfare). Based on this, we also show how to set tolls so that this optimal flow is induced as an equilibrium of the underlying game

Mitochondrial and nuclear markers in populations of Brazilian Rhipicephalus (Boophilus) microplus.

Comissão organizadora: Renato Andreotti, Fernando Paiva, Wilson Werner Koller, Jacqueline Cavalcante Barros, Luiz Antonio Dias Leal, Jaqueline Matias