Wikipedia — Convex optimization

Authoritative encyclopedia treatment of convex-optimization, used to deepen that page beyond the “local = global” headline. Source for the standard-form definition, the named subclass hierarchy, and the optimality conditions.

What it adds

• Standard form. A convex problem minimizes a convex objective subject to convex inequality constraints gᵢ(x) ≤ 0 and affine equality constraints; the feasible set is convex because “the intersection of convex sets is convex.”
• Subclass hierarchy (each more general than the last): LP (linear objective + constraints) ⊂ QP (convex-quadratic objective, linear constraints) ⊂ SOCP ⊂ SDP ⊂ conic optimization.
• Optimality conditions — Lagrange multipliers λ₀…λ_m ≥ 0 with complementary slackness λᵢ·gᵢ(x) = 0; strict convexity gives at most one optimum.
• Tractability provenance. “Many classes of convex optimization problems admit polynomial-time algorithms” via interior-point methods; the article credits Nesterov & Nemirovskii (1994) and their self-concordant barrier functions for making them broadly tractable.

Tier T2 (authoritative tertiary). Cited from convex-optimization.