Continuity of argmin

Where here we ask what happens to the infima and sets of minimisers of sequences of functions $\{f_n\}_n$? under what conditions do these converge? what is an appropriate notion of convergence for functions which transfers the convergence to the corresponding sequence of its minima and minimizers? This poses a question of continuity for the infimum (as an operator) as well as the set of minimisers (as a multi-valued operator). We aim at characterising the continuity of these operators. Continue reading →

Metric subregularity for monotone inclusions

Metric sub-regularity is a local property of set-valued operators which turns out to be a key enabler for linear convergence in several operator-based algorithms. However, conditions are often stated for the fixed-point residual operator and become rather difficult to verify in practice. In this post we state sufficient metric sub-regularity conditions for a monotone inclusion which are easier to verify and we focus on the preconditioned proximal point method (P3M). Continue reading →

Do no generic termination criteria exist for steepest descent?

Where here we wonder why there are no generic termination criteria for the gradient method which guarantee a desired sub-optimality when $f$ is strictly (not strongly) convex and has $L$-Lipschitz gradient. Does it make sense to terminate when $\|\nabla f(x_k)\|$ is small? And when should we terminate? Continue reading →

Convergence of the iterates of the gradient method with constant stepsize

The gradient method with constant step length is the simplest method for solving unconstrained optimisation problems involving a continuously differentiable function with Lipschitz-continuous gradient. The motivation for this post came after reading this Wikipedia article where it is stated that under certain assumptions the sequence $\{x_k\}$ converges to a local optimum, but it is no further discussion is provided. Continue reading →

Weak closure points not attainable as limits of sequences

Where in this post we discover an uncanny property of the weak topology: the points of the weak closure of a set cannot always be attained as limits of elements of the set. Naturally, the w-closure of a set is weakly closed. The so-called weak sequential closure of a set, on the other hand, is the set of cluster points of sequences made with elements from that set. The new set is not, however, weakly sequentially closed, which means that there may arise new cluster points; we may, in fact, have to take the weak sequential closure trans-finitely many time to obtain a set which is weakly sequentially closed – still, this may not be weakly closed. Continue reading →

Graphical convergence

Where in this post we wonder whether pointwise convergence makes sense for sequences of set-valued mappings. We present some essential limitations of the pointwise limit and we motivate the notion of graphical convergence. What is more, we use animated GIFs to better illustrate and understand these new notions. Continue reading →

On Set Convergence I

We give the definitions of inner and outer limits for sequences of sets in topological and normed spaces and we provide some important facts on set convergence on topological and normed spaces. We juxtapose the notions of the limit superior and limit inferior for sequences of sets and we outline some facts regarding the Painlevé-Kuratowski convergence of set sequences. Continue reading →

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