| name | optimization-variational-methods |
| description | Formulate, solve, and verify finite-dimensional optimization and calculus-of-variations problems. Use for convex programs, constrained extrema, KKT systems, duality, optimal control, Euler-Lagrange equations, and numerical optimization. |
| license | MIT |
Optimization and Variational Methods
Formulate the problem
Write the objective, variables or admissible functions, domain, equality and inequality constraints,
boundary conditions, regularity class, units, and whether the task seeks a local or global optimum. Remove
arbitrary scale or gauge freedom. Check feasibility and existence before choosing an algorithm.
Select a route
- For smooth unconstrained problems, derive first- and second-order conditions and inspect curvature.
- For convex problems, state the convexity argument and a constraint qualification before using KKT sufficiency.
- For constrained nonconvex problems, use KKT as necessary conditions and compare multiple feasible starts.
- For functionals, derive the first variation, natural boundary terms, and Euler-Lagrange equations.
- Use a dual formulation when it supplies a bound, decomposition, or numerical advantage.
Solve
Scale variables and residuals, choose tolerances tied to the scientific question, and retain iteration history.
For discretized functionals, distinguish discretization error from optimizer termination. Do not infer a global
optimum from a single local solver run unless a theorem or valid bound supports it.
Verify
- Check feasibility, stationarity, complementary slackness, and primal-dual gap where applicable.
- Compare analytic derivatives with finite differences or automatic differentiation at test points.
- Test perturbations around the candidate and verify second-order or convexity conditions.
- Refine discretization and solver tolerances independently; inspect sensitivity to initialization and scaling.
- Substitute the result into the original objective and constraints, not only transformed equations.
Deliver
Report formulation, existence or convexity conditions, solver and initialization, residuals, bounds, sensitivity,
and whether the conclusion is local, global, or heuristic.
Source basis
Original synthesis informed by open calculus-of-variations and numerical-analysis sources recorded in
../../docs/TEXTBOOK_SOURCES.md.
