Convex Optimization & Iterative Methods

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Abstractions about solving structured optimization and linear-system problems, including iterative linear-system solvers (Gauss-Seidel method, Jacobi method), convexity properties enabling efficient optimization (conic optimization, self-concordant function), and combinatorial structures related to polyhedra and graphs (Balinski's theorem, integer points in convex polyhedra, maximum matching).

8 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Balinski's theorem — In polyhedral combinatorics, a branch of mathematics, Balinski's theorem is a statement about the graph-theoretic structure of three-dimensional convex polyhedra and higher-dimensional convex polytopes.
• Conic Optimization — Conic optimization is a subfield of convex optimization that studies problems consisting of minimizing a convex function over the intersection of an affine subspace and a convex cone.
• Convex bipartite graph — In the mathematical field of graph theory, a convex bipartite graph is a bipartite graph with specific properties.
• Gauss–Seidel Method — A stationary linear-system iteration that sweeps coordinates in order and immediately reuses each newly computed component within the same sweep.
• Integer points in convex polyhedra — The study of integer points in convex polyhedra is motivated by questions such as "how many nonnegative integer-valued solutions does a system of linear equations with nonnegative coefficients have" or "how many solutions does an integer linear program have".
• Jacobi Method — Solve a linear system by isolating its diagonal and synchronously recomputing every component from the same previous iterate, with the resulting iteration matrix governing convergence.
• Maximum matching — In graph theory, a maximum-cardinality matching is a special kind of subgraph useful in many computational contexts.
• Self-concordant function — A self-concordant function is a function satisfying a certain differential inequality, which makes it particularly easy for optimization using Newton's method A self-concordant barrier is a particular self-concordant function, that is also a barrier function for a particular convex set.