From my point of view, the modern optimization theory contains three most important frameworks——first order methods, cutting plane methods and interior point methods.
First order methods
By first order methods, here I mean gradient descent and variants. As for the deterministic algorithms, Accelerated Gradient Descent (AGD)[Nes83] has been proven to be optimal. As for the randomized algorithm, a technique called variance reduction is introduced, and Katyusha[All17] has the current best running time.
Cutting plane methods
Given a convex function $f$ with its set $\mathcal S$ of minimizers, Cutting plane methods minimizes $f$ by maintaining a convex search set $\mathscr E^{(k)} \supseteq\mathcal S$ in the $k$th iteration, and iteratively shrinking $\mathscr E^{(k)}$ using the subgradients of $f$. The current best algorithm is [JLSW20], which is based on a multi-layered data structure for leverage score maintenance.
Interior point methods
Interior point methods solves the convex program $\min_{\mathbf u∈\mathcal S}\langle c, u\rangle$ by solving a sequence of unconstrained problems $\min_u Ψ_t(\mathbf u)\overset{def}{=} \{t · \langle c, u\rangle + ψ_S (\mathbf u)\}$ parametrized by increasing $t$, where $ψ_S$ is a self-concordant barrier function that enforces feasibility by becoming unbounded as it approaches the boundary of the feasible set $\mathcal S$. Over the past decades, this method has had a significant impact, notably in Linear Programming and the Minimum Cost Flow problem. The best algorithms for these problems are solved with interior point methods, namely [JSWZ21] for LP and [CKLPPS22] for Minimum Cost Flow.
Books
Lecture Notes