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On the whole Euclidean space, an $L$-Lipschitz gradient of a convex function is $1/L$-cocoercive. Subtract the supporting affine function at one point, apply the descent lemma to a gradient step at the other point, then interchange the points and add. The conclusion needs convexity but no second derivative.

A differentiable function with an $L$-Lipschitz gradient obeys $f(v)\le f(u)+⟨\nabla f(u),v-u⟩+(L/2)\Vert v-u{\Vert }^2$. Integrate the gradient along the line segment and bound the deviation from its initial value. The lemma does not require convexity.

For $f(x,y)=h(x,y)+\varphi (y)$, where $h$ is convex with an $L_h$-Lipschitz gradient and $\varphi$ is convex, use a proximal $y$-step for the full section and an explicit $x$-step for $h$. Monotonicity of the two selected subgradients of $\varphi$ combines with cocoercivity of $\nabla h$ to make the full update firmly nonexpansive for $0<\tau L_h\le 1$. Existence of a minimizer then gives convergence, even though the full objective is not smooth.

For the Euclidean norm, rotational symmetry and minimizing the radial objective give the radial soft thresholding formulaIndeed, if $r=\Vert x{\Vert }_2$, an optimal $u$ is a nonnegative scalar multiple of $x$; minimizing $s+(r-s{)}^2/(2\tau )$ over $s\ge 0$ gives $s=(r-\tau {)}_{+}$. The gradient of a Moreau envelope is consequentlyAt $x=0$ this is $\{0\}$, and the two expressions agree when $\Vert x{\Vert }_2=\tau$. The Moreau envelope of the Euclidean norm itself isThus a quadratic core replaces the nondifferentiable tip while the outer gradient remains the normalized radial direction. The Euclidean norm has an unbounded effective domain, unlike the earlier bounded-domain hypothesis; the preceding argument explicitly shows that this restriction is unnecessary here.

For convex optimization, this Moreau–Yosida regularisation permits gradient descent and other methods for functions with a Lipschitz gradient, using a gradient Lipschitz constant $1/\tau$. It approximates the norm uniformly: $0\le \Vert x{\Vert }_2-f_{\tau }(x)\le \tau /2$. Smaller $\tau$ improves approximation but makes the permitted gradient steps smaller. Minimizing $f_{\tau }$ alone preserves the minimizers and minimum value of $f$; replacing one term in a larger objective can shift the minimizer, so the smoothing parameter controls that approximation error.

Here $L$-smooth means that $f$ is differentiable on the entire Euclidean space and has an $L$-Lipschitz gradient, with $L>0$. Together with convexity, this implies $1/L$-cocoercivity:First, integration of the Lipschitz gradient along a line segment proves the descent lemma:Indeed, subtract the linear term and integrate $⟨\nabla f(v+t(w-v))-\nabla f(v),w-v⟩$ for $0\le t\le 1$; its absolute value is at most $Lt\Vert w-v{\Vert }_2^2$.

Fix $u$ and define $q(w)=f(w)-⟨\nabla f(u),w⟩$. This is a convex function, has the same Lipschitz gradient constant, and $\nabla q(u)=0$, so $u$ is a global minimizer. Put $d=\nabla f(v)-\nabla f(u)=\nabla q(v)$. The descent lemma at $v$ with $w=v-d/L$ givesThereforeInterchange $u,v$ and add the inequalities. The function values cancel, leaving the asserted cocoercivity. This is a proof of the Baillon–Haddad theorem without assuming a second derivative. Convexity is essential: $f(u)=-\Vert u{\Vert }_2^2/2$ has a $1$-Lipschitz gradient but its gradient is not cocoercive.

Set $r^k=A{x}^k-c$. Completing the square in the $y$-minimization givesThus, with $\sigma =\tau /(1+2\tau )$, the steps in terms of the proximal map of $\varphi$ areThe factor $2$ comes from the squared residual without a $1/2$ prefactor. The proximal map can equivalently be denoted $(I+\sigma \partial \varphi {)}^{-1}$. Since $\varphi$ is a finite convex function, it is continuous, and the quadratic makes the proximal minimizer unique.

A general convex $\varphi$ may be nondifferentiable, so this last example need not satisfy the full $L$-smooth hypothesis of the preceding part. The formulas nevertheless remain well-defined: only the $x$ derivative is used explicitly. If $\varphi$ has an $L_{\varphi }$-Lipschitz gradient, one may take $L=2(\Vert A{\Vert }_2^2+1)+L_{\varphi }$ in the smooth proof, where $\Vert A{\Vert }_2$ is the operator norm.

There is also a useful nonsmooth extension. Put $h(x,y)=\Vert Ax+y-c{\Vert }_2^2$, whose Lipschitz gradient constant is $L_h=2(\Vert A{\Vert }_2^2+1)$. For two updates, retain the mixed-point differences $(p,q)$ above, let $(a,b)=\nabla h(w)-\nabla h(\tilde{w})$, and let $d$ be the difference of the two selected subgradients of $\varphi$ at $y^{+},{\tilde{y}}^{+}$. The input difference is $(p,q+\tau (b+d))$ and the output difference is $(p-\tau a,q)$. Monotonicity of $\partial \varphi$ gives $⟨q,d⟩\ge 0$. The same norm expansion now yieldsFor $0<\tau L_h\le 1$, discard the nonpositive $b$ term and use $2\tau /L_h-{\tau }^2\ge {\tau }^2$. The result is the firmly nonexpansive mapping inequality for the full residual $\tau (a,b+d)$. Fixed points satisfy ${\nabla }_{x}h=0$ and $0\in {\nabla }_{y}h+\partial \varphi$, exactly the subgradient optimality condition for the convex objective. Thus the stated algorithm also converges for nonsmooth $\varphi$ under this sufficient step bound, provided a minimizer exists; this extension uses monotonicity, not an unavailable gradient of $\varphi$.

Assume $\mathrm{argmin}f\ne \varnothing$, as is necessary for convergence to a minimizer. The printed hypotheses alone omit this condition: $f(x,y)=x$ in scalar $x$ is convex with an $L$-Lipschitz gradient for any prescribed $L>0$, but has no minimizer, and the update sends $x^k=x^0-k\tau$ to $-\infty$.

Under existence, define the full update $T(x,y)=(x^{+},y^{+})$. The proximal $y$-step is uniquely defined by strong convexity of its quadratic term and satisfiesThe partial subdifferential in this equation belongs to the scalar function $y\mapsto f(x,y)$; differentiating a subdifferential itself would not define the printed proximal minimization.

We prove directly that the alternating proximal-gradient operator $T$ is averaged. For two inputs $z=(x,y)$ and $\tilde{z}=(\tilde{x},\tilde{y})$, setThe input difference is $(p,q+\tau b)$, the output difference is $(p-\tau a,q)$, and the difference of the two update residuals is $\tau (a,b)$. Exact expansion givesThe first inequality uses cocoercivity at the mixed points $w,\tilde{w}$. Therefore $T$ is $(\tau L/2)$-averaged for the requested $0<\tau L\le 1$. In particular, the endpoint $\tau L=1$ is covered: $T$ is firmly nonexpansive, not merely nonexpansive.

A fixed point has $y^{+}=y$ and $x^{+}=x$, so the two optimality equations give ${\nabla }_{y}f=0$ and ${\nabla }_{x}f=0$. Conversely, at a minimizer both gradients vanish and neither step moves. By convexity,The assumed existence of a minimizer supplies a fixed point. The Browder convergence theorem for averaged operators now givesNo uniqueness of the minimizer is needed. The same estimate even proves averagedness for $0<\tau L<2$ in this smooth setting, but the requested interval already suffices. The existence condition and the use of the gradient at $(x^k,y^{k+1})$ are both essential to the proof as presented.

For a differentiable convex function $F$ with $L$-Lipschitz gradient and a proper lower-semicontinuous convex function $H$, the proximal gradient method isThe proximal operator is ${\mathrm{prox}}_{\tau H}(v)=\arg {\min }_z\{H(z)+\Vert z-v{\Vert }^2/(2\tau )\}$. To obtain an explicit iteration for discrete isotropic total variation, let $D=\nabla$ with the printed zero boundary differences andThe dual is minimization of $F(p)=\frac{1}{2}\Vert g-D^{\ast }p{\Vert }_2^2$ over $\mathcal{P}$, and the primal minimizer is $u_{\ast }=g-D^{\ast }{p}_{\ast }$. Here $D^{\ast }=-\mathrm{div}$ must be the exact adjoint operator of these differences. For example,with the analogous formula in $j$ for $(D_y^{+}{)}^{\ast }$; for $N=1$ both operators are zero. Equivalently use $p_{0j}^1=p_{Nj}^1=p_{i0}^2=p_{iN}^2=0$ in the backward-difference expression, ignoring unused edge components.

The gradient of the dual objective is $D(D^{\ast }p-g)$, with Lipschitz constant $\Vert D{\Vert }^2\le 8$. Taking $H$ to be the indicator functional of a constraint set makes its proximal operator the Euclidean projection onto a convex set. Thus a valid explicit algorithm isThe bound follows by summing $(a-b{)}^2\le 2a^2+2b^2$ over grid differences. Finite-dimensional proximal gradient method convergence gives a dual minimizer and convergence of the recovered $u^k$ to the unique primal minimizer. Projection is pointwise in the dual; applying pointwise soft thresholding directly to $Du$ would not compute the primal TV proximal operator.

Starting from $x_0\in {\mathbb{R}}^n$, gradient descent repeatedly computes the gradient and updatesstopping when the gradient norm, step, or objective decrease is sufficiently small.

The Hessian bounds say that $f$ is $\alpha$-strongly convex and has $\beta$-smooth gradient. With $\eta =1/\beta$,Thus the iteration count isconvergence becomes slower linearly with the condition number $\kappa =\beta /\alpha$.

Forthe Hessian matrix is $\mathrm{diag}(2,200,20000)$, soTakeThenwhose Hessian is $2I$ and whose condition number is $1$.

Define the softmax weightsThey obey $p_i\ge 0$ and ${\sum }_i{p}_i=1$. The gradient is their weighted mean,Differentiating once more gives the covariance-form Hessian matrixFor every unit vector $u$,where $G={\max }_i\Vert a_i{\Vert }_2$. The Hessian is a covariance matrix, so it is positive semidefinite; the displayed upper bound also gives ${\nabla }^2{f}_{\beta }⪯\beta G^{2}I$ in the Loewner order. Consequently $f_{\beta }$ has a Lipschitz gradient with

Chooseso the smooth maximum error is at most $ϵ/2$ and the Lipschitz gradient constant isSuppose a minimizer of $f_{\beta }$ lies within distance $R$ of the starting point. The Nesterov accelerated gradient method can find $x$ such thatiniterations. If $x_{\ast }$ minimizes $f$, then the smoothing inequalities implyThis improves the nonsmooth subgradient method dependence from $O({ϵ}^{-2})$ to $O({ϵ}^{-1})$.

Composing the log-sum-exp function with finitely many affine functions gives a differentiable approximation to their pointwise maximum. Increasing the inverse-temperature parameter $\beta$ reduces the uniform approximation error while increasing the Lipschitz gradient constant.