Submodular functions and convexity

Submodular functions and convexity

1983 | L. Lovász
This chapter introduces the concept of submodular functions and their significance in discrete optimization, drawing parallels with convex functions in continuous optimization. Convex functions are central in continuous optimization due to their natural properties, robustness under transformations, and efficient minimization methods. Similarly, submodular set-functions play a crucial role in discrete optimization, despite lacking the geometric simplicity of convex functions. The chapter highlights the four key properties of convexity that submodular functions share: their prevalence in mathematical models, preservation under operations, structural richness, and efficient minimization techniques. It also explores the fundamental connection between submodular and convex functions, enabling the application of convex optimization techniques to submodular functions. The chapter discusses the minimization and maximization problems of submodular functions, the polyhedral properties associated with them, and their role in extending classical graph-theoretical results. The author emphasizes the importance of submodular functions in generalizing and extending existing theorems, leading to new and surprising results in combinatorial studies.This chapter introduces the concept of submodular functions and their significance in discrete optimization, drawing parallels with convex functions in continuous optimization. Convex functions are central in continuous optimization due to their natural properties, robustness under transformations, and efficient minimization methods. Similarly, submodular set-functions play a crucial role in discrete optimization, despite lacking the geometric simplicity of convex functions. The chapter highlights the four key properties of convexity that submodular functions share: their prevalence in mathematical models, preservation under operations, structural richness, and efficient minimization techniques. It also explores the fundamental connection between submodular and convex functions, enabling the application of convex optimization techniques to submodular functions. The chapter discusses the minimization and maximization problems of submodular functions, the polyhedral properties associated with them, and their role in extending classical graph-theoretical results. The author emphasizes the importance of submodular functions in generalizing and extending existing theorems, leading to new and surprising results in combinatorial studies.
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