Important Convex Functions to Know for Convex Geometry

Understanding important convex functions is key in convex geometry. These functions, like linear, quadratic, and exponential, play vital roles in optimization problems, helping to model various scenarios and ensuring solutions stay within feasible regions.

1. Linear functions

   • Defined as functions of the form f(x) = ax + b, where a and b are constants.
   • Convexity is guaranteed since the second derivative is zero, indicating a straight line.
   • Used as the simplest model for optimization problems and in defining hyperplanes in convex sets.

2. Quadratic functions

   • Represented as f(x) = ax² + bx + c, where a > 0 ensures convexity.
   • The second derivative is constant and positive, confirming the function's convex nature.
   • Commonly used in optimization problems, particularly in quadratic programming.

3. Exponential functions

   • Formulated as f(x) = e^(ax), where a is a constant; always convex for any real a.
   • The function grows rapidly, making it useful in modeling growth processes and decay.
   • Plays a significant role in various fields, including economics and biology, due to its properties.

4. Logarithmic functions

   • Expressed as f(x) = log_a(x) for a > 1; these functions are concave, but their negative is convex.
   • Useful in optimization problems involving utility functions and information theory.
   • Often applied in scenarios where diminishing returns are observed.

5. Norm functions

   • Defined as f(x) = ||x||, where ||.|| is a norm (e.g., L1, L2 norms).
   • Convex due to the triangle inequality and positive homogeneity properties.
   • Essential in optimization, particularly in regularization techniques in machine learning.

6. Indicator functions

   • Defined as f(x) = 0 if x is in a set C, and f(x) = ∞ otherwise.
   • Convex since the epigraph of the function is a convex set.
   • Useful in defining feasible regions in optimization problems.

7. Support functions

   • Given by f(x) = sup{⟨x, y⟩ : y ∈ C}, where C is a convex set.
   • Convex as it represents the maximum value of a linear functional over a convex set.
   • Important in duality theory and in characterizing convex sets.

8. Perspective functions

   • Defined as f(x, t) = tf(x/t) for t > 0; these functions maintain convexity.
   • Useful in optimization problems involving scaling and transformations.
   • Helps in extending the concept of convexity to functions of multiple variables.

9. Entropy functions

   • Typically expressed as f(p) = -∑(p_i log(p_i)), where p represents a probability distribution.
   • Convex due to the properties of the logarithm and the nature of probability distributions.
   • Widely used in information theory, statistics, and thermodynamics.

10. Barrier functions

    • Functions that approach infinity as the boundary of a feasible region is approached.
    • Convexity is maintained, making them useful in interior-point methods for optimization.
    • Help in ensuring that solutions remain within a feasible region during optimization processes.