Majorize-minimize is a general recipe for designing algorithms rather than a single update formula. Its surrogate viewpoint includes EM and connects to proximal-gradient and Newton-like quadratic-majorization schemes; see Hunter and Lange (2004) and Lange (2016). The recipe is simple: replace a difficult objective by an upper bound that touches it at the current iterate, improve that surrogate, and repeat.
To minimize an objective $g(\v\gamma)$, an MM algorithm replaces it at the current iterate $\widetilde{\v\gamma}$ by a simpler surrogate $h(\v\gamma,\widetilde{\v\gamma})$ satisfying
$$ \begin{aligned} g(\widetilde{\v\gamma})&=h(\widetilde{\v\gamma},\widetilde{\v\gamma}), \\ g(\v\gamma)&\le h(\v\gamma,\widetilde{\v\gamma}) \quad\text{for every }\v\gamma. \end{aligned} $$Minimize the surrogate and repeat. If $\v\gamma^{(t+1)}\in\argmin_{\v\gamma}h(\v\gamma,\v\gamma^{(t)})$, then
$$ \begin{aligned} g(\v\gamma^{(t+1)}) &\le h(\v\gamma^{(t+1)},\v\gamma^{(t)})\\ &\le h(\v\gamma^{(t)},\v\gamma^{(t)}) =g(\v\gamma^{(t)}). \end{aligned} $$Consider the smooth one-dimensional objective
$$ g(x)=\cos x,\qquad 0\le x\le2\pi. $$On this interval the unique minimizer is $\widehat x=\pi$, where $g(\widehat x)=-1$. Because $g''(x)=-\cos x\le1$, the quadratic upper-bound principle gives the smooth majorizer
$$ h(x,x^{(t)}) =\cos x^{(t)}-\sin x^{(t)}(x-x^{(t)}) +\frac12(x-x^{(t)})^2 \;\ge\;\cos x. $$The bound touches $g$ and has the same slope at $x^{(t)}$. Its minimum gives the closed-form MM update
$$ x^{(t+1)}=x^{(t)}+\sin x^{(t)}. $$Starting farther away at $x^{(0)}=0.2$ gives a longer visible sequence: $0.2000\to0.3987\to0.7869\to1.4950\to2.4921\to3.0969\to\cdots\to\pi$. The animation settles numerically at the displayed global minimum after seven MM updates.
Orange is the objective and the dashed peach curve is the current surrogate. The white dot marks $x^{(t)}$; the peach vertical guide and dot mark the surrogate minimum selected for the next step. Past iterates remain on the objective so the monotone descent is visible. Dashed vertical guides retain every iterate location, with the current iterate drawn more strongly. The guide at $\widehat x=\pi$ marks the true minimum.
Majorization guarantees descent, but it does not guarantee speed. The choice of majorizer is therefore both a mathematical and computational decision. Common constructions use tangent inequalities for concave terms, quadratic curvature bounds for smooth terms, Jensen's inequality for latent-variable objectives, or matrix inequalities for coupled inverse problems. The mathematics changes from problem to problem; the touch, majorize, improve pattern does not.
Exact MM minimizes the surrogate at every iteration. That is sufficient but not necessary. A generalized MM step only needs to improve the surrogate:
$$ h(\v\gamma^{(t+1)},\v\gamma^{(t)}) \le h(\v\gamma^{(t)},\v\gamma^{(t)}). $$The same two-line argument still gives $g(\v\gamma^{(t+1)})\le g(\v\gamma^{(t)})$. This permits partial inner solves, coordinate updates, or damped steps when the exact surrogate minimum is expensive.
Part 2 left us with the convex auxiliary objective
$$ \begin{aligned} g(\v\gamma) &=\v y^\top \left(\m I+\sum_{j=1}^p\gamma_j\v x_j\v x_j^\top\right)^{-1}\v y\\ &\quad+\alpha^2\v 1^\top\v\gamma. \end{aligned} $$The inverse couples all $p$ scales, so the obvious objective is not separable. The MM design problem is therefore precise: find an upper bound that touches this inverse term at the current iterate, separates across $\gamma_j$, and remains tight enough to make a useful concurrent step.