Classical optimization (the tractable core)

Parent: optimization

The math you already have, plus the boundary of what's solvable.

Unconstrained

• First-order condition: ∇f(x)=0 (stationary point);
• Second-order: Hessian ∇²f ⪰ 0 (local min); PSD everywhere ⟺ convex.
• Convexity is the key property — a convex function has no bad local minima (every stationary point is global).

Constrained — what you know ("取 μ")

Minimize f(x) s.t. equality hᵢ(x)=0 and inequality gⱼ(x)≤0. Price the constraints with multipliers into the lagrangian L; optimality is the kkt conditions; the dual is always concave (strong duality under Slater for convex problems). Full treatment → lagrangian, kkt.

The named convex classes (nested)

LP ⊂ QP ⊂ SOCP ⊂ SDP — linear, quadratic, second-order-cone, semidefinite programs.

• LP: simplex (Dantzig) / interior-point; LP duality.
• Convex optimization (Boyd–Vandenberghe): the modern synthesis — if you can cast it as a convex cone program, it's poly-time solvable by interior-point methods.

The one idea that matters

Convexity is the dividing line. Convex → globally solvable, duality tight, well-understood. Non-convex → hard (NP-hard in general), local minima, no free convergence — which is where everything after this note goes.