Mathematics / advanced
Nonlinear optimization and local models
Understand practical algorithms through local approximation, constraint qualifications, descent guarantees, and failure modes.
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Module 01 advanced
Nonlinear optimization and local models
Understand practical algorithms through local approximation, constraint qualifications, descent guarantees, and failure modes.
- 01
First- and second-order optimality
Turn derivatives into local certificates, and recognize the cases where a Hessian test leaves the question open.
29 min - 02
Descent directions and Wolfe line search
Use decrease and curvature tests together to reject steps that are either wasteful or dangerously short.
31 min - 03
Trust-region subproblems and model agreement
Constrain a local quadratic model, compare prediction with reality, and handle negative curvature without an unchecked Newton step.
32 min - 04
BFGS updates and curvature conditions
Build a symmetric curvature model from gradient changes and see exactly why the positive-curvature test matters.
32 min - 05
Nonlinear constraints and constraint qualifications
Identify when linearized constraints faithfully support multiplier conditions and diagnose singular representations.
34 min - 06
Sequential quadratic programming
Derive a constrained Newton model and see why satisfying a tangent constraint does not guarantee nonlinear feasibility.
34 min - 07
Augmented Lagrangian methods
Combine multiplier correction with a finite penalty and distinguish inner stationarity from outer feasibility.
33 min - 08
Primal-dual interior methods for nonlinear programs
Follow perturbed complementarity with positive slacks and multipliers while keeping nonconvex guarantees distinct from convex ones.
35 min - 09
Derivative-free models and direct search
Extract local information from function values while accounting for geometry, sample cost, and evaluation noise.
33 min - 10
Nonconvex complexity and stationarity certificates
Derive a first-order iteration bound and state exactly what a small-gradient or approximate-curvature certificate proves.
34 min