Convex optimization epigraph

  • How do you find the epigraph of a function?

    The epigraph of a function f : Rn → R is defined as epi f = {(x,t) : x ∈ dom f,f(x) ≤ t}, which is a subset of Rn+1.
    Epi means above, and thus epigraph means above the graph.
    Theorem 1.
    A function is convex if and only of its epigraph is a convex set..

  • What is epigraph in convex optimization?

    Epigraph of a function A function (in black) is convex if and only if the region above its graph (in green) is a convex set.
    This region is the function's epigraph.
    Importantly, although both the graph and epigraph of consists of points in the epigraph consists entirely of points in the subset..

  • The epigraph of a function f:Rn→[−∞,+∞] is the set of points (x,μ)u220.
    1. Rn+1 satisfying f(x)≤μ, and is denoted by epi(f)
Apr 5, 2017So, the problem in the equivalent epigraph representation is still in a standard convex optimization problem form. convex-optimization.General epigraph form of optimization problem whose objective Convert any convex optimization problem to a linear objectiveConvex function, Epigraph, Sublevel setProjection on Epigraph of a convex functionMore results from math.stackexchange.com
Apr 5, 2017To provide the original problem in epigraph standard representation, but preserving the problem to be in convex form, it needs to add an  Convex function, Epigraph, Sublevel setProjection on Epigraph of a convex functionGeneral epigraph form of optimization problem whose objective Epigraphs and convexity - Mathematics Stack ExchangeMore results from math.stackexchange.com

Is equivalent epigraph a convex optimization problem?

So, the problem in the equivalent epigraph representation is still in a standard convex optimization problem form

Furthermore, for straightforward and meaningful analysis of a problem, also designing an efficient algorithm, different equivalent representation of a problem can be used

Note: f f is convex if and only if epi f epi f is convex set

What is the epigraph of a convex function?

Fig 1

Epigraph for functions Formally, given a function f: Rn!R, the epigraph is the set of all points f(y;x) : yf(x); x2Rn; y2Rg 2 So, if the function is convex, then its epigraph is a convex set and vice versa

This gives a geometric interpretation of a function being convex


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