# 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 →

# Quadratic constraints to second-order conic ones

In the previous post we discussed how we can project onto the epigraph of the squared norm. However, when in an optimisation problem we encounter constraints of the form

\begin{aligned} (x,t)\in \mathrm{epi}_{\|{}\cdot{}\|^2} \end{aligned}

That is, quadratic constraints (of the form $\|x\|^2 \leq t$), these can be converted to second-order conic constraints. Continue reading →

# Projection on the epigraph of the squared Euclidean norm

As a follow-up on the previous post titled Projection on an epigraph, we here discuss how we can project on the epigraph of the squared norm function. Continue reading →

# Projection on an epigraph

Here we study the problem of projecting onto the epigraph of a convex continuous function. Unlike the computation of the proximal operator of a function or the projection on its sublevel sets, the projection onto epigraphs is more complex and there exist only a few functions for which semi-explicit formulas are available.

Projection of a point onto the epigraph of a convex function.

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