Applied Large-Scale Convex Optimization: A Complete Course

Understand convex sets, functions, and optimization algorithms with hands-on examples and expert instruction.

What you'll learn
- Understand what convex sets and convex functions are, and why they matter in optimization.
- Explore how convexity helps solve real-world problems like shortest paths, image processing, and machine learning.
- Learn the geometry and structure of high-dimensional spaces using inner-product spaces.
- Discover how to analyze and work with functions that aren’t smooth using subgradients.
- Study key optimization algorithms like the subgradient method and forward-backward splitting.
- Key algorithms - subgradient method, forward-backward splitting, and accelerated techniques for solving large-scale problem
- Learn about accelerated methods that improve the speed of convergence.
- Subdifferentials & Epigraphs: Learn how to analyze non-differentiable functions using subgradients and epigraphs, essential tools in nonsmooth optimization.
- Dive into duality theory to understand the relationship between primal and dual optimization problems.
- Duality Theory: Explore the powerful framework of Fenchel, Lagrange, and strong duality, and how dual problems provide deep insights into primal formulations.
- Applications: Apply convex optimization to real-world problems such as shortest-path computations, support vector machines, and signal/image processing.
- Gain the skills to model, analyze, and solve large-scale convex optimization problems in practical settings.


Requirements
- Mathematical Maturity: Comfort with abstract mathematical reasoning and notation.
- Linear Algebra: Familiarity with vectors, matrices, inner-product spaces, and basic operations.
- Calculus: Understanding of differentiation, continuity, and basic function properties.

Description
Convex optimization is a cornerstone of modern applied mathematics, underpinning a wide range of technologies from machine learning and artificial intelligence to signal processing, control systems, and operations research. This course offers a deep and structured exploration oflarge-scale convex optimization, tailored for learners who seek both theoretical rigor and practical insight.

Through 55 carefully crafted video lectures, this course guides you from the foundational concepts of convex sets and functions to advanced algorithmic techniques for solving high-dimensional optimization problems. It is designed to be accessible to motivated learners while maintaining the depth expected in graduate-level education. This course:

Offers astructured deep dive into large-scale convex optimization, balancing theory and algorithmic practice.

Introducescore concepts: convex sets, convex functions, and their properties—essential for high-dimensional optimization.

Begins withreal-world applications: shortest-path problems, signal/image processing, and support vector machines.

Coversmathematical foundations: inner-product spaces, convex geometry, convexity-preserving operations, differentiability, lower semicontinuity, and closedness.

Focuses onalgorithmic techniques: Subgradient method: convergence, boundedness, and implementation; and Forward-backward splitting and accelerated variants for composite problems.

Exploresduality theory: Primal-dual relationships; Perturbation and infimal value functions; Fenchel and Lagrange duality frameworks.

Equips learners toanalyze, solve, and implementconvex optimization methods in practical and research contexts.

By the end of this course, you will:

Understand the theoretical underpinnings of convex optimization

Be able to formulate and solve large-scale convex problems

Implement key algorithms for optimization in practical settings

Apply convex optimization to real-world problems in engineering and data science

Be prepared for advanced studies or research in optimization and applied mathematics

Why This Course Stands Out

University-Level Instruction: Based on a graduate course taught at a leading European engineering faculty, ensuring academic depth and clarity.

Balanced Approach: Combines intuitive explanations with formal mathematical rigor, making it suitable for both practitioners and researchers.

Algorithmic Focus: Emphasizes practical methods for solving optimization problems, with step-by-step walkthroughs of algorithms and their convergence properties.

Real-World Relevance: Demonstrates how convex optimization is used in cutting-edge fields like machine learning, data science, and engineering.

Who this course is for:
- Graduate students in mathematics, computer science, engineering, or economics
- Researchers and professionals working in optimization, AI, or quantitative fields
- Data scientists and machine learning engineers seeking deeper mathematical foundations
- Anyone interested in understanding and applying convex optimization to complex problems
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