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convex-optimizer

convex and conic optimization

MOSEK, Clarabel, OSQP

SystemTechniques · 7Primitives · 13 convex-optimizer Fill-reducing orderingnumerics · medium: Permute a sparse symmetric matrix before Cholesky/LU so the factor has few new nonzeros; minimum degree, nested dissection and RCM trade quality, speed and parallelism.›Sparse direct solvernumerics · medium: Solve sparse Ax = b exactly by reordering, symbolic analysis, numeric LLᵀ/LU and triangular solves; fill-in, not flops per entry, decides the cost.›Interior-point method (primal-dual,…optimization · medium: Follows the central path with Newton steps on the perturbed KKT system, one sparse Cholesky per iteration; Mehrotra's predictor-corrector makes it fast.›Presolve reductionsoptimization · medium: Cheap, provably safe reductions run before the solver (fix variables, drop rows, tighten bounds and coefficients) so LP and MIP solves start from a smaller, tighter model.›Primal dual hybrid gradientoptimization · medium: First-order LP solving (PDLP): restarted PDHG with preconditioning for very large LPsQuadratic programmingoptimization · medium: Convex QP solvers (active-set, interior-point, ADMM) for constrained optimizationGraph partitioninggraphs · medium: Split a graph into k balanced parts with few edges between them; multilevel coarsen-partition-refine (METIS, KaHIP) for quality, streaming heuristics (LDG, Fennel) for one-pass scale.›Sparse matrices (CSR)numerics · low: Compressed sparse row stores only nonzeros as row pointers, column indices and values; SpMV streams them once, so it is memory-bandwidth bound.›LU and Cholesky factorizationnumerics · low: Factor A = PLU (any square matrix, partial pivoting) or A = LLᵀ (SPD) once in O(n³), then solve each right-hand side in O(n²).›Newton root findingnumerics · low: Linearize at the guess and jump to the tangent root; quadratic convergence near a simple root, made robust by brackets (1D) or line search.›Floating-point errornumerics · low: How IEEE 754 rounding error accumulates and how to contain it: compensated (Kahan/Neumaier) and pairwise summation, error-free transformations with FMA, ULP comparisons.›SIMD programminghardware-os · low: Process 4-16 lanes per instruction with AVX2, AVX-512 or NEON via Rust std::arch, with runtime dispatch and a portable fallback.›CUDA and GPU languageslanguages · low: Programming GPUs: thousands of lockstep threads, bound by memory bandwidth; libraries first, Triton for fused kernels, CUDA C++ for peak speed.›Conjugate gradientnumerics · low: Krylov iteration for symmetric positive definite Ax = b using one matrix-vector product and two dot products per step; converges in about √κ iterations.›Simplex methodoptimization · low: Solves linear programs by walking along the edges of the feasible polytope from vertex to vertex, one basis swap (pivot) at a time; the workhorse behind MIP solvers.›d-ary heapdata-structures · low: Array heap with d children per node (d = 4 typical): half the binary heap's depth, siblings in one cache line, faster priority queues.›Fast hash functionsdata-structures · low: Non-cryptographic hashes built from 64×64→128-bit multiplies and xor-shifts; pick by key size, trust level and whether the hash is persisted.›Interval arithmeticnumerics · low: Rigorous bounds with outward rounding; root isolation and robust geometric testsGPU compute shadersml-training · low: The vendor-neutral GPU launch model of Metal, Vulkan and WebGPU: a grid of workgroups, shared memory, barriers and subgroups, driven by recorded command buffers.›PRNGs (PCG, xoshiro)numerics · low: Small, fast, statistically strong non-cryptographic generators: PCG32 and xoshiro256**, with seeding, parallel streams and unbiased ranges.›

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