1

The role of convexity in optimization, duality theory, algorithms and duality

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2

Convex sets and functions, epigraphs, closed convex functions, recognizing convex functions

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3

Differentiable convex functions, convex and affine hulls, Caratheodory's theorem, relative interior

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4

Algebra of relative interiors and closures, continuity of convex functions, closures of functions, recession cones and lineality space

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5

Directions of recession of convex functions, local and global minima, existence of optimal solutions

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6

Nonemptiness of closed set intersections, existence of optimal solutions, linear and quadratic programming, preservation of closure under linear transformation

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7

Partial minimization, hyperplane separation, proper separation, nonvertical hyperplanes

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8

Convex conjugate functions, conjugacy theorem, support functions

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9

Min common/max crossing duality, weak duality, constrained optimization and minimax, strong duality

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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

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12

Convex programming duality, optimality conditions, mixtures of linear and convex constraints, existence of optimal primal solutions, Fenchel duality, conic duality

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13

Subgradients, Fenchel inequality, sensitivity in constrained optimization, subdifferential calculus, optimality conditions

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14

Min-max duality, existence of saddle points

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15

Problem structures, conic programming

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16

Conic programming, semidefinite programming, exact penalty functions, descent methods for convex/nondifferentiable optimization, steepest descent method

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17

Subgradient methods, calculations of subgradients, convergence

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18

Approximate subgradient methods, ε-subdifferential, ε-subgradient methods, incremental subgradient methods

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19

Return to descent methods, fixing the convergence problem of steepest descent, ε-descent method, extended monotropic programming

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20

Approximation methods, cutting plane methods, proximal minimization algorithm, proximal cutting plane algorithm, bundle methods

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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

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23

Interior point methods, barrier method, conic programming cases, path following

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24

Review and epilogue

(PDF - 1.8MB)

Homework 1 (PDF)

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Homework 2 (PDF)

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Homework 3 (PDF)

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Homework 4 (PDF)

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Homework 5 (PDF)

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