Mathematics / advanced
Convex analysis and dual structure
Study the geometry that supports optimization through separation, subgradients, conjugacy, and monotone operators.
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Module 01 advanced
Convex analysis and dual structure
Study the geometry that supports optimization through separation, subgradients, conjugacy, and monotone operators.
- 01
Convex hulls and Carathéodory’s theorem
Compress a mixture using affine dependence while keeping its location and nonnegative weights unchanged.
27 min - 02
Separating hyperplanes and support functions
Turn an outside point into a linear certificate and reconstruct a closed convex set from directional bounds.
29 min - 03
Extended-real convex functions
Encode feasibility with infinity while keeping properness, domain geometry, and objective arithmetic precise.
27 min - 04
Epigraphs and lower semicontinuity
Use closed epigraphs and sublevel compactness to explain when a convex infimum is attained.
28 min - 05
Subgradients and supporting hyperplanes
Replace a derivative with global affine lower bounds and handle corners without guessing an ordinary slope.
30 min - 06
Fenchel conjugacy and biconjugation
Encode the best affine lower bounds by slope and recover a closed proper convex objective from them.
31 min - 07
Fenchel–Rockafellar duality
Derive a dual lower bound, verify a relative-interior qualification, and distinguish dual attainment from primal existence.
33 min - 08
Infimal convolution and Moreau envelopes
Split a cost between components and derive smooth regularization from a uniquely attained proximal problem.
32 min - 09
Maximal monotone operators and resolvents
Interpret implicit updates as resolvents and distinguish maximal graph extension from unusually large slopes.
34 min - 10
Convex feasibility and projection geometry
Derive projection certificates, analyze alternating projections, and separate feasibility from nearest-point optimization.
32 min