String Submodular Functions With Curvature Constraints

Consider the problem of choosing a string of actions to optimize an objective function that is string submodular. It was shown in previous papers that the greedy strategy, consisting of a string of actions that only locally maximizes the step-wise gain in the objective function, achieves at least a...

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Veröffentlicht in:IEEE transactions on automatic control Jg. 61; H. 3; S. 601 - 616
Hauptverfasser: Zhenliang Zhang, Chong, Edwin K. P., Pezeshki, Ali, Moran, William
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York IEEE 01.03.2016
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
Approximation
Curvature
Frequency modulation
Gain
Mathematical analysis
Optimization
Strategy
Strings
Tasks
greedy
optimal control
Combinatorial optimization
matroid
ISSN:0018-9286, 1558-2523
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Zusammenfassung:Consider the problem of choosing a string of actions to optimize an objective function that is string submodular. It was shown in previous papers that the greedy strategy, consisting of a string of actions that only locally maximizes the step-wise gain in the objective function, achieves at least a (1 - e -1 )-approximation to the optimal strategy. This paper improves this approximation by introducing additional constraints on curvature, namely, total backward curvature, totalforward curvature, and elemental forward curvature. We show that if the objective function has total backward curvature ϵ, then the greedy strategy achieves at least a (1/σ)(1 - e -σ )-approximation of the optimal strategy. If the objective function has total forward curvature e, then the greedy strategy achieves at least a (1 - ϵ)-approximation of the optimal strategy. Moreover, we consider a generalization of the diminishing-return property by defining the elemental forward curvature. We also introduce the notion of string-matroid and consider the problem of maximizing the objective function subject to a string-matroid constraint. We investigate two applications of string submodular functions with curvature constraints: 1) choosing a string of actions to maximize the expected fraction of accomplished tasks; and 2) designing a string of measurement matrices such that the information gain is maximized.
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ISSN:0018-9286
1558-2523
DOI:10.1109/TAC.2015.2440566