Convexity
Convexity is the property of a set containing the line segment between any two of its points, or of a function whose value between two points is no greater than the linear interpolation of its endpoint values.
Build a Robo-Advisor with Python (From Scratch)
A mathematical property of functions or sets that ensures line-segment combinations remain compatible with global optimization guarantees.
- How it is used
- Frames portfolio optimization around solver efficiency and confidence that an obtained solution is globally optimal.
- How deep it goes
- Defines convex functions and sets mathematically, contrasts local and global minima, and stops short of solver implementation.
- Examples
- Uses function graphs, line segments, convex and nonconvex sets, and unconstrained portfolio optimization.
- What you can do after
- Evaluate solver reliability and apply constraints when nonconvexity or naive portfolio optimization can produce nonoptimal results.
Read how Build a Robo-Advisor with Python (From Scratch) explains convexity →
Math and Architectures of Deep Learning
The property of a function or set that, in this section, is characterized by being above its tangent and by a nonnegative second derivative.
- How it is used
- Frames function shape and curvature for reasoning about local minima and later applications of Jensen’s inequality.
- How deep it goes
- Defines sets and functions geometrically, connects tangents to Taylor expansions, and verifies several functions with second derivatives.
- Examples
- Uses convex and nonconvex regions, x², even powers, eˣ, log x, −log x, positive scaling, and sums.
- What you can do after
- Evaluate/understand convexity by testing line segments, weighted-average inequalities, tangent behavior, or second derivatives.
Read how Math and Architectures of Deep Learning explains convexity →
Related concepts
- linear regressionincludes · Linear regression is a statistical modeling method that estimates a continuous response as a linear combination of predictor variables and fitted coefficients.
- second derivativemeasured by · A second derivative is the derivative of a function’s first derivative with respect to the same variable, describing how its rate of change varies.
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