| 1 | The role of convexity in optimization, duality theory, algorithms and duality | (PDF) |
| 2 | Convex sets and functions, epigraphs, closed convex functions, recognizing convex functions | (PDF) |
| 3 | Differentiable convex functions, convex and affine hulls, Caratheodory's theorem, relative interior | (PDF) |
| 4 | Algebra of relative interiors and closures, continuity of convex functions, closures of functions, recession cones and lineality space | (PDF) |
| 5 | Directions of recession of convex functions, local and global minima, existence of optimal solutions | (PDF) |
| 6 | Nonemptiness of closed set intersections, existence of optimal solutions, linear and quadratic programming, preservation of closure under linear transformation | (PDF) |
| 7 | Partial minimization, hyperplane separation, proper separation, nonvertical hyperplanes | (PDF) |
| 8 | Convex conjugate functions, conjugacy theorem, support functions | (PDF) |
| 9 | Min common/max crossing duality, weak duality, constrained optimization and minimax, strong duality | (PDF) |
| 10 | Min common/max crossing duality theorems, strong duality conditions, existence of dual optimal solutions, nonlinear Farkas' lemma | (PDF - 1.5MB) |
| 11 | Min common/max crossing theorem III, nonlinear Farkas' lemma/linear constraints, linear programming duality, convex programming duality | (PDF) |
| 12 | Convex programming duality, optimality conditions, mixtures of linear and convex constraints, existence of optimal primal solutions, Fenchel duality, conic duality | (PDF) |
| 13 | Subgradients, Fenchel inequality, sensitivity in constrained optimization, subdifferential calculus, optimality conditions | (PDF) |
| 14 | Min-max duality, existence of saddle points | (PDF) |
| 15 | Problem structures, conic programming | (PDF) |
| 16 | Conic programming, semidefinite programming, exact penalty functions, descent methods for convex/nondifferentiable optimization, steepest descent method | (PDF) |
| 17 | Subgradient methods, calculations of subgradients, convergence | (PDF) |
| 18 | Approximate subgradient methods, ε-subdifferential, ε-subgradient methods, incremental subgradient methods | (PDF) |
| 19 | Return to descent methods, fixing the convergence problem of steepest descent, ε-descent method, extended monotropic programming | (PDF) |
| 20 | Approximation methods, cutting plane methods, proximal minimization algorithm, proximal cutting plane algorithm, bundle methods | (PDF) |
| 21 | Generalized polyhedral approximations in convex optimization | ( PDF - 2.9MB) |
| 22 | Review of Fenchel duality, review of proximal minimization, dual proximal minimization algorithm, augmented Lagrangian methods | (PDF) |
| 23 | Interior point methods, barrier method, conic programming cases, path following | (PDF) |
| 24 | Review and epilogue | (PDF - 1.8MB) |
