Convex functions play an important role in many areas of mathematics. They are especially important in the study of optimization problems where they are distinguished by a number of convenient …

In this lecture, we shift our focus to the other important player in convex optimization, namely, convex functions. Here are some of the topics that we will touch upon:

Convex functions always lie above their tangent lines (i.e. f(y) ≥ f(x) + ∇f(x)T (y − x)). Smooth convex functions always lie below a parabola which passes through the point (x, f(x)) (defined by the RHS …

Learn about convex functions with simple definitions, key properties, and solved examples. Understand convexity and how to check if a function is convex for easy application in optimization problems.

Jul 3, 2025 · A convex function curves upwards, meaning that a straight line between any two points on the graph will lie above or on the graph. A concave function curves downwards, meaning that the line …

Here we will talk about properties of convex (or concave upward) function. We already noted that if function f (x) f (x) is concave upward then f (x) −f (x) is concave downward.

It is often difficult to analytically tell if a function is strongly convex or smooth. The following theorem, however, offers a simple condition that implies strong convexity or smoothness for twice …

Duality such as min-max relation and separation theorem holds good. This section is to give more specific descriptions of these properties, and to discuss their possible versions for discrete functions. …

Convex functions are real valued functions which visually can be understood as functions which satisfy the fact that the line segment joining any two points on the graph of the function lie above that of the …

May 18, 2025 · Discover the foundations, properties, and examples of convex functions in mathematical analysis, covering definitions, key theorems, and operations that preserve convexity.