Reference 0011

Convexity Cheat Sheet

Convex problem: any local optimum is global, KKT certifies it, duals are trustworthy. One non-convex term poisons all three guarantees.

Definitions

convex set:       for any x, y in S and 0 <= t <= 1,
                  t*x + (1-t)*y is also in S
                  (the segment between members stays inside)

convex function:  f(t*x + (1-t)*y) <= t*f(x) + (1-t)*f(y)
                  (the chord never dips below the curve)

convex problem:   minimize convex f over a convex feasible set

Tests

chord test:      draw any chord; it must stay on or above the curve
2nd-order (1D):  f''(x) >= 0 everywhere
2nd-order (nD):  Hessian PSD everywhere
portfolio risk:  f(w) = w'Σw, Hessian 2Σ, PSD -> convex, always
note:            convex does NOT require smooth (|x| is convex)

Convexity Calculus

convex + convex                      -> convex
(nonnegative scalar) * convex        -> convex
max(convex, convex)                  -> convex
convex composed with affine (A*w+b)  -> convex
intersection of convex sets          -> convex set

WARNING: convex - convex, convex * convex,
min(convex, convex) preserve NOTHING.

Portfolio Terms: Which Side of the Line

Convex

variance w'Σw · expected return -μ'w · proportional cost c'|Δw| · quadratic impact · box caps · budget 1'w = 1 · leverage ||w||_1 <= L · risk cone ||Σ^½w|| <= s

Non-convex

fixed ticket fee per traded name · cardinality (“at most K names”) · minimum position size (w=0 or w >= 2%) · round lots · maximize variance · w'Σw >= s (wrong-side quadratic)

Solver-Class Map (for reading papers)

all terms convex:
  linear only               -> LP
  + quadratic objective     -> QP
  + norm / cone constraints -> SOCP
  guarantee: global optimum, meaningful duals

any non-convex term:
  integer modeling          -> MIQP / MILP (exact but slow)
  otherwise                 -> relaxation or heuristic
  guarantee: none; duals not trustworthy shadow prices

Reading Rule

1. Find objective + constraints.
2. Walk term by term: convex?
3. All convex  -> solution global, duals = sensitivities.
4. Any failure -> expect "mixed-integer", "relaxation",
   or "heuristic" in the paper; distrust reported duals.

Use this page with Lesson 0011, Reference 0002 (why Σ is PSD), and Reference 0004 (KKT). Code-side twin: CVXPY DCP rules.