Discrete convex optimization

  • Can a convex set be discrete?

    The framework of convex analysis is adapted to discrete set- tings and the mathematical results in matroid/submodular function theory are generalized.
    Viewed from the continuous side, it is a theory of convex functions f : Rn → R that have additional combi- natorial properties..

  • Can a discrete function be convex?

    The function f has strong discrete convexity if it satisfies the following conditions: 1. f(x+u)+f(x)⩾f(x∨u)+f(x∧u) where u=(u1,…,un)≠0, ui=0,−1,+1 for each i=1,2,…,n and x∨u=(max{xi,xi+ui}), x∧u=(min{xi,xi+ui}).
    This is a local submodularity condition..

  • What are the main features of discrete convex analysis?

    The main feature of discrete convex analysis is the distinction of two convexity concepts, M-convexity and L-convexity, for functions in integer or binary variables, together with their conjugacy relationship..

  • What is convexity of discrete function?

    Discrete convexity.
    Let S be a subspace of a discrete n-dimensional space.
    A function f:Su219.

    1. R is discretely convex if for all and for α∈(0,1)(1) αf(x)+(1−α)f(y)⩾ min uu220
    2. N(z) f(u), where N(z)={uu220
    3. S:u−z\x26lt;1}, z=αx+(1−α)y, and u=max1⩽i⩽n{ui}

  • What is discrete optimization used for?

    It consists in optimizing a linear objective subject to linear constraints, admits efficient algorithmic solutions, and is often an important building block for other optimization techniques..

  • The main feature of discrete convex analysis is the distinction of two convexity concepts, M-convexity and L-convexity, for functions in integer or binary variables, together with their conjugacy relationship.
  • With a discrete optimization problem approach, during each run of the clustering ensemble, the base learner constructs a “best” partition on the subset of the target data set (subsampling) by optimizing a predefined clustering quality measure.
1 Introduction. Discrete convex analysis [18, 40, 43, 47] aims to establish a general theoretical framework for solv- able discrete optimization problems by 
Convex functions are tractable in optimization (or minimization) problems and this is mainly because of the following properties. 1. Local optimality (or 

How do you solve a convex discrete optimization problem?

First, query the linear discrete optimization oracle presenting S on the trivial linear functional w = 0

If the oracle asserts that there is no optimal solution then S is empty so terminate the algorithm asserting that no optimal solution exists to the convex discrete optimization problem either

So assume the problem is feasible

What is discrete convex analysis?

Bibliographic Explorer ( What is the Explorer?) Litmaps ( What is Litmaps?) scite Smart Citations ( What are Smart Citations?) This paper presents discrete convex analysis as a tool for economics and game theory

Discrete convex analysis is a new framework of discrete mathematics and optimization, developed during the last two decades


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