What is the difference between convex and non-convex optimization problems?

A convex optimization problem is a problem where all of the constraints are convex functions, and the objective is a convex function if minimizing, or a concave function if maximizing. A non-convex optimization problem is any problem where the objective or any of the constraints are non-convex, as pictured below.

Is Submodular function convex?

Theorem 2 (Lovász) A set function f : 2S → R with f(∅)=0 is submodular iff ˆf is convex.

How do you prove a function is Submodular?

A function f : 2N → R is said to be submodular, if it satisfies following property of diminishing marginal returns: for every A ⊆ B ⊆ N and j ∈ B, f(A ∪ {j}) − f(A) ≥ f(B ∪ {j}) − f(B). One way to understand submodularity is to think of f as a utility functions.

What is multilinear extension?

Definition 1. The multilinear extension of f is the function F : [0,1]n → R defined by: F(x) = ∑ S⊆N.

How greedy is submodular optimization?

Submodular function lead to greedy optimization algorithms with solutions equal to at least ( 1 − 1 e) × optimal value. Unify your data with Segment. A single platform helps you create personalized experiences and get the insights you need. Is convex optimization hard? What exactly do you mean by “hard”?

What is a submodular function?

Submodular functions have nice optimization properties, in that it is possible to find near optimal solutions for minimization and maximization problems involving submodularity. Submodular Functions are also closely related to Convexity and Concavity. They have been called the Discrete analog of convex functions.

What is submodularity in Discrete Math?

Submodularity is a discrete version of concavity. Think about it 😉 An example is the set cover problem. You have a set of sets and you have to find the minimum combination of those sets to cover some universe of elements.