Abstract
This article is an introduction to recent development of optimization theory on set functions, the nonsubmodular optimization, which contains two interesting results, DS (difference of submodular) functions decomposition and sandwich theorem, together with iterated sandwich method and data-dependent approximation. Some potential research problems will be mentioned.
Highlights
In recent development of computer technology, such as wireless networks [1,2], cloud computing [3,4], sentiment analysis [5,6,7,8], and machine learning [9], many nonlinear optimization problems come out with discrete structure. They form a large group of new problems, which belong to a research area, nonlinear combinatorial optimization
The nonlinear combinatorial optimization has been studied for a long time, but recently becomes very active
[26] Every set function f : 2X → R can be expressed as the difference of two monotone nondecreasing submodular functions g and h, i.e., f = g − h, where X is a finite set
Summary
In recent development of computer technology, such as wireless networks [1,2], cloud computing [3,4], sentiment analysis [5,6,7,8], and machine learning [9], many nonlinear optimization problems come out with discrete structure. The research works came mainly from researchers in operations research Those works are mainly on submodular function optimization, often with monotone nondecreasing property. Theorem 2.1 [26] Every set function f : 2X → R can be expressed as the difference of two monotone nondecreasing submodular functions g and h, i.e., f = g − h, where X is a finite set. To prove this theorem, we first show two lemmas. Proof of Theorem 2.1 By Lemma 2.1, f can be expressed as f = g − h where g and h are submodular functions. When the seed cost is a submodular function with respect to seed set S, the profit becomes a DS function
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