🔎Convex Geometry Unit 11 – Convex Functions and Their Properties

What's the Big Idea?

• Convex functions play a central role in convex geometry and optimization
• Characterized by the property that a line segment between any two points on the graph of the function lies above or on the graph
• Convexity ensures a unique global minimum for optimization problems
• Convex sets are closely related to convex functions
  • A set is convex if the line segment between any two points in the set is also contained within the set
• Convex functions have important properties such as continuity, differentiability, and subdifferentiability
• The study of convex functions and their properties forms the foundation for solving many optimization problems in various fields (machine learning, economics, engineering)

Key Concepts to Grasp

• Definition of a convex function
  • For all $x, y$ in the domain and $t \in [0, 1]$, $f(tx + (1-t)y) \leq tf(x) + (1-t)f(y)$
• Epigraph of a convex function is a convex set
  • Epigraph: $\{(x, y) : x \in \text{dom}(f), y \geq f(x)\}$
• First-order condition for convexity
  • A differentiable function $f$ is convex if and only if $f(y) \geq f(x) + \nabla f(x)^T(y - x)$ for all $x, y$ in the domain
• Second-order condition for convexity
  • A twice differentiable function $f$ is convex if and only if its Hessian matrix $\nabla^2 f(x)$ is positive semidefinite for all $x$ in the domain
• Jensen's inequality for convex functions
  • For a convex function $f$ and points $x_1, \ldots, x_n$ in its domain, $f(\sum_{i=1}^n \lambda_i x_i) \leq \sum_{i=1}^n \lambda_i f(x_i)$ where $\lambda_i \geq 0$ and $\sum_{i=1}^n \lambda_i = 1$
• Properties of convex functions (continuity, differentiability, subdifferentiability)

The Math Behind It

• Convex optimization problems involve minimizing a convex function over a convex set
  • Minimize $f(x)$ subject to $x \in C$, where $f$ is convex and $C$ is a convex set
• Convex functions can be characterized using the first-order and second-order conditions
  • First-order condition: $f(y) \geq f(x) + \nabla f(x)^T(y - x)$
  • Second-order condition: $\nabla^2 f(x)$ is positive semidefinite
• Subdifferential of a convex function at a point $x$ is the set of all subgradients at $x$
  • Subgradient: A vector $g$ such that $f(y) \geq f(x) + g^T(y - x)$ for all $y$ in the domain
• Convex conjugate (Fenchel conjugate) of a function $f$ is defined as $f^*(y) = \sup_{x \in \text{dom}(f)} (y^Tx - f(x))$
  • Convex conjugate is always convex, even if the original function is not
• Legendre-Fenchel transform relates a convex function to its conjugate
  • $f^{**} = f$ for convex, lower semicontinuous functions

Real-World Applications

• Convex optimization is widely used in machine learning for training models (support vector machines, logistic regression)
• Portfolio optimization in finance relies on convex optimization techniques to minimize risk while maximizing returns
• Signal processing applications (image denoising, compressed sensing) often involve solving convex optimization problems
• Control systems use convex optimization for model predictive control and trajectory planning
• Structural optimization in engineering design employs convex optimization to find optimal shapes and topologies
• Resource allocation problems in economics and operations research can be formulated as convex optimization problems
• Convex relaxation techniques are used to approximate non-convex problems by convex ones (semidefinite relaxation, linear matrix inequalities)

Common Pitfalls and How to Avoid Them

• Mistaking a non-convex function for a convex one
  • Always check the definition or use the first-order or second-order conditions to verify convexity
• Applying convex optimization techniques to non-convex problems
  • Be aware of the limitations of convex optimization and consider using global optimization methods for non-convex problems
• Numerical instability in solving convex optimization problems
  • Choose appropriate solvers and adjust tolerance settings if needed
  • Preconditioning or scaling the problem can improve numerical stability
• Overlooking constraints or feasibility issues
  • Ensure that the constraints form a convex set and that the problem is feasible
  • Use constraint qualifications to check regularity conditions
• Misinterpreting results due to local minima
  • Convex functions have no local minima that are not global, but non-convex functions may have multiple local minima
  • Verify that the obtained solution is indeed the global minimum

Practice Problems and Tips

• Prove that the sum of two convex functions is convex
  • Hint: Use the definition of convexity and properties of inequalities
• Show that the negative logarithm $-\log(x)$ is a convex function
  • Hint: Compute the second derivative and show that it is non-negative
• Determine if the function $f(x, y) = x^2 - y^2$ is convex
  • Hint: Check the eigenvalues of the Hessian matrix
• Find the convex conjugate of the function $f(x) = e^x$
  • Hint: Use the definition of the convex conjugate and solve the optimization problem
• Practice solving convex optimization problems using various methods (gradient descent, interior-point methods)
  • Use software tools like CVXPY, CVXOPT, or MOSEK to solve convex optimization problems
• Analyze the convergence properties of different optimization algorithms on convex functions

Going Beyond the Basics

• Study advanced topics in convex analysis (monotone operators, proximal operators, Moreau envelopes)
• Explore generalized convexity concepts (quasiconvexity, pseudoconvexity, invexity)
  • These relaxations of convexity can be useful in certain optimization problems
• Learn about duality in convex optimization (Lagrange duality, Fenchel duality, conic duality)
  • Duality provides a powerful framework for analyzing and solving convex optimization problems
• Investigate the connections between convex functions and monotone operators
  • Monotone operators generalize the concept of monotonicity to vector-valued functions
• Study the properties of convex risk measures in financial mathematics
  • Convex risk measures quantify the risk associated with financial positions or portfolios
• Explore the role of convexity in game theory and mechanism design
  • Convex analysis plays a crucial role in the study of Nash equilibria and incentive-compatible mechanisms

Connecting the Dots

• Understand the relationships between convex sets, convex functions, and convex optimization
  • Convex sets form the feasible regions for convex optimization problems
  • Convex functions are the objective functions and constraints in convex optimization
• See how convex analysis provides a unified framework for studying optimization problems across various domains
  • Convexity is a key property that enables efficient solving of optimization problems
• Recognize the interplay between convex functions and their conjugates
  • The convex conjugate provides a dual representation of a convex function
  • Many properties of convex functions can be studied through their conjugates
• Appreciate the role of convex relaxations in approximating non-convex problems
  • Convex relaxations provide tractable approximations to difficult non-convex problems
  • Techniques like semidefinite programming and sum-of-squares optimization rely on convex relaxations
• Understand how convex optimization algorithms exploit the properties of convex functions
  • Gradient-based methods use the first-order condition for convexity
  • Interior-point methods leverage the second-order condition and self-concordance of convex functions