Convex Algebraic Geometry and Polynomial Optimization

Workshop Overview

This intensive workshop bridges classical real algebraic geometry with modern computational semidefinite programming (SDP). Designed for graduate students and researchers in Control Engineering, Optimization, and Machine Learning, it covers the theoretical guarantees of positivity certificates and demonstrates how to exploit algebraic structures to bypass traditional computational scaling barriers[cite: 1].

Duration: 20 Hours Lecture + 5 Laboratory Sessions[cite: 1]

Dates & Venue

Resource Persons & Speakers

Main Speaker

• Professor Harish Pillai – Department of Electrical Engineering, IIT Bombay[cite: 1] (Covering core workshop topics[cite: 1])

Organizers & Lecturers

• Dr. Swanand R. Khare[cite: 1] (Co-Organizer & Lecturer)
• Dr. Mousumi Mandal[cite: 1] (Co-Organizer & Lecturer)

Plenary Speakers

Four external researchers working in this area are requested (confirmation pending) to deliver plenary talks in online mode during this workshop[cite: 1].

Schedule & Syllabus

Part 1: Real-World Applications & Positivity Foundations (7 Hours)[cite: 1]

• Hour 1: Gallery of Polynomial Matrix Optimization (PMO) Problems – Case studies in engineering, control systems, and machine learning[cite: 1].
• Hour 2: Geometry of Polynomial Optimization & Semialgebraic Sets – Formulating global optimization, understanding NP-hardness, and coordinate rings[cite: 1].
• Hour 3: Primal Certificates: Sum of Squares (SOS) Matrices – SOS polynomials, Hilbert’s 1888 theorem, and Gram Matrix SDP mapping[cite: 1].
• Hour 4: Hand-Solving 1D and 2D SOS Problems – Coefficient matching, Sylvester’s criterion, and exact global lower bounds[cite: 1].
• Hour 5: Intelligent Basis Reduction: The Newton Polytope – Pruning basis elements and inspecting polynomial support[cite: 1].
• Hour 6: Systematic SOS Factorization: The Cholesky Factorization – Explicit polynomial square representations from Gram matrices[cite: 1].
• Hour 7: Matrix-Valued SOS Constraints – Transitioning to scalar/matrix polynomials and non-linear LMIs[cite: 1].

Part 2: The Dual Universe & The Moment Problem (6 Hours)[cite: 1]

• Hour 8: The Dual Problem: Riesz Functionals & Measures – Expected values, moment sequences, and infinite-dimensional linear programs[cite: 1].
• Hour 9: Building and Analyzing the Moment Matrix – Hankel-structured moment matrices and cone duality mechanics[cite: 1].
• Hour 10: Putinar’s Positivstellensatz & Localizing Matrices – Archimedean property and localizing matrix constraints[cite: 1].
• Hour 11: The Lasserre Hierarchy Sandwich – Primal-dual relaxation framework and optimality gaps[cite: 1].
• Hour 12: Dual Point Extraction via Flat Extension – Curto-Fialkow flat extension theorem and Henrion-Lasserre algorithm[cite: 1].

Part 3: Advanced Coordinate Rings & Rekha Thomas’s Framework (7 Hours)[cite: 1]

• Hour 13: Structural Stalls in Scalar Relaxations – Odd-degree obstructions and non-convex matrix manifolds[cite: 1].
• Hour 14: Rekha Thomas’s Framework: Spectrahedra & Coordinate Rings – Structured coordinate rings and Theta Bodies ($TH_k(I)$)[cite: 1].
• Hour 15: Mathematical Construction of the First Theta Body $TH_1(I)$ – Projecting moment structures and convex hull collapse[cite: 1].
• Hour 16: Facial Topology of Spectrahedra – Boundaries, extreme rays, and algebraic rank stratification[cite: 1].
• Hour 17: Semidefinite Lifting of Determinantal Varieties – Replacing scalar constraints with symbolic null-space matrix systems[cite: 1].
• Hour 18: Structural Sparsity & Chordal Graphs – Exploiting structural sparsity and decomposing massive Gram matrices[cite: 1].
• Hour 19: Symmetry Reduction in SOS – Group theory applications and collapsing redundant symmetric rows/columns[cite: 1].
• Hour 20: Future Horizons in Polynomial Matrix Optimization – Non-commutative SOS, quantum information theory, and DRO[cite: 1].

Extended Laboratory Syllabus (5 Sessions)[cite: 1]

• Lab Session 1: Toolchain Setup and Primal-Dual Verification (2 Hours)[cite: 1]
• Lab Session 2: SOS Programming for Non-linear Control (2 Hours)[cite: 1]
• Lab Session 3: Bypassing the Determinantal Variety Stall (2 Hours)[cite: 1]
• Lab Session 4: Structured Matrix Approximations & Neural Net Verification (2 Hours)[cite: 1]
• Lab Session 5: Large-Scale Scalability: Sparsity and Symmetries (2 Hours)[cite: 1]

Registration

Registration for the advanced graduate workshop is open to graduate students, researchers, and professionals in Control Engineering, Optimization, and Machine Learning.

Click below to access the registration portal: