Trimester 07 · Course detail

Optimization Theory & Functional Analysis

Convex analysis and non-linear optimization, advanced functional analysis and spectral theory, stochastic calculus and continuous-time dynamics, and a production optimization lab with capstone seminar

Credits: 15
Sequence: 7 of 8
Structure: 4 subjects
Status: Full syllabus

Curriculum focus

Evidence of mastery

Optimization engine + methodology brief

Subject 01

Convex Analysis & Non-Linear Optimization

Official syllabus code: MIMUW 1000-2M09BNO

The modern theory of convex optimization: convex geometry and subdifferential calculus, Fenchel duality, KKT optimality, and the monotone-operator machinery behind proximal and splitting algorithms.

Convex Geometry & Duality

Convex Sets, Cones, Epigraphs & Subdifferential Calculus

Develop the geometry of convex sets, cones and epigraphs, and build subdifferential calculus as the bridge between convex analysis and optimality conditions.

Convex Conjugates (Fenchel–Legendre) & The Fenchel–Moreau Theorem

Define the Fenchel–Legendre conjugate, prove the Fenchel–Moreau biconjugation theorem, and use conjugate pairs to transform optimization problems.

Lagrangian Duality, Slater's Condition & Saddle-Point Geometries

Construct the Lagrangian dual, prove strong duality under Slater's condition, and interpret optimal primal-dual pairs as saddle points of the Lagrangian.

Karush–Kuhn–Tucker (KKT) Optimality & Constraint Qualifications

Derive the KKT conditions as necessary and sufficient optimality criteria, and analyze the constraint qualifications under which they hold.

Monotone Operators & Algorithmic Methods

Monotone Operators, Resolvents, Proximal Algorithms & ADMM

Study monotone operator theory and resolvents, then build proximal algorithms and ADMM as the workhorses of large-scale structured convex optimization.

Subject 02

Advanced Functional Analysis & Spectral Theory

Official syllabus code: MIMUW 1000-135AF

Functional analysis at full depth: Banach and Fréchet space theorems, weak topologies, operator spectra, and the spectral theorem with its functional calculus for self-adjoint operators.

Banach Space Theory & Duality

Banach & Fréchet Spaces: Baire Category, Open Mapping & Closed Graph

Apply the Baire category theorem to prove the open mapping and closed graph theorems, the pillars of stability for linear operators on Banach and Fréchet spaces.

Duality Pairs, Weak & Weak-* Topologies, Banach–Alaoglu & Eberlein–Šmulian

Master weak and weak-* topologies, prove the Banach–Alaoglu compactness theorem, and characterize weak compactness through Eberlein–Šmulian.

Operator Theory & The Spectral Theorem

Bounded & Compact Operators, Fredholm Theory & Riesz–Schauder Spectrum

Analyze bounded and compact operators, develop Fredholm theory for equations of the second kind, and characterize the Riesz–Schauder spectrum of compact operators.

Unbounded Operators on Hilbert Spaces: Closed, Symmetric & Self-Adjoint

Handle unbounded operators through domain analysis, distinguishing closed, symmetric, and self-adjoint operators and the role of deficiency indices.

The Spectral Theorem for Self-Adjoint Operators & Functional Calculus

Prove the spectral theorem in its projection-valued measure form and build the measurable functional calculus that powers quantum mechanics and PDE theory.

Subject 03

Stochastic Calculus & Continuous-Time Dynamics

Official syllabus code: MIMUW 1000-214bRP2

Continuous-time probability: martingales and stopping times, Brownian motion, the Itô integral and calculus, and the theory of stochastic differential equations with their Fokker–Planck and Feynman–Kac connections.

Martingales & Brownian Motion

Continuous-Time Martingales, Stopping Times & Doob–Meyer Decomposition

Develop continuous-time martingale theory, analyze stopping times, and prove the Doob–Meyer decomposition of submartingales into martingale and predictable parts.

Brownian Motion / Wiener Process: Path Properties & Non-Differentiability

Construct the Wiener process and establish its almost-sure path properties — continuity, unbounded variation, and nowhere differentiability.

Itô Calculus & Stochastic Differential Equations

The Itô Stochastic Integral: Quadratic Variation & Itô Isometry

Construct the Itô integral against Brownian motion, use quadratic variation to control it, and prove the Itô isometry underlying stochastic integration.

Itô's Lemma for Multidimensional Diffusions & Stratonovich Calculus

Prove Itô's lemma for multidimensional diffusions, compare the Itô and Stratonovich calculi, and convert between the two conventions.

Stochastic Differential Equations (SDEs), Fokker–Planck & Feynman–Kac

State and solve stochastic differential equations, connect them to the Fokker–Planck equation for transition densities, and prove the Feynman–Kac representation.

Subject 04

Production Lab & Capstone Seminar

Official syllabus code: PLEY-401 / Optimization Suite

The trimester's production laboratory: implement state-of-the-art optimization solvers, stress-test PLEYSMART production systems, and deliver the high-dimensional constrained convex optimizer as the panel's capstone.

Solver Implementation & Capstone

Implementation of First-Order & Primal-Dual Interior-Point Solvers

Implement first-order methods and primal-dual interior-point solvers from scratch, with convergence analysis and benchmarking on problem suites.

Proximal Gradient Methods, Douglas–Rachford Splitting & Mirror Descent

Build proximal gradient methods, Douglas–Rachford splitting, and mirror descent for structured and non-Euclidean problem geometries.

Reverse-Engineering & Formal Stress-Testing of PLEYSMART Production Systems

Reverse-engineer and formally stress-test the PLEYSMART production optimization systems, documenting failure modes and robustness envelopes.

Panel 7 Capstone Deliverable: High-Dimensional Constrained Convex Optimizer

Assemble the capstone: a high-dimensional constrained convex optimizer combining the trimester's duality theory, operator methods, and solver implementations.