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Take $a=1$ and $b=k$, sois the capped simplex. A linear objective over this convex polytope attains its maximum at a zero-one extreme point. Choosing the $k$ coordinates at which $x_i$ is largest givesEquivalently, an exchange of weight from a smaller component to a larger one never decreases the objective. Thus the sum of the largest components is the support function ${\sigma }_C$.

Part c now givesBy the projection onto a box-constrained hyperplane, $v=P_C(y/t)$ hasConsequently the proximal operator is evaluated by solving this one-dimensional equation for $\nu$, then substituting the resulting projection.

SetThe gradient has Lipschitz continuity with constantwhere the norm is the spectral norm. The proximal gradient method is thereforeFor $\lambda >0$, part d makes the second step explicit:where $C$ is the capped simplex; when $\lambda =0$, this proximal step is the identity.

A standard fixed choice is $0<\alpha \le 1/L$; the wider interval $0<\alpha <2/L$ also gives convergence under the usual forward-backward conditions. For a general convex objective, the function-value error is $O(1/r)$. If $A$ has full column rank, the quadratic term is strongly convex and an appropriate fixed step gives a linear convergence rate.

The sum of the $k$ largest components of $x\in {\mathbb{R}}^n$ is the support function of the capped simplex: ${\sum }_{i=1}^k{x}_{[i]}=\max \{x^{T}v:0\le v\le 1,{\sum }_i{v}_i=k\}$.