Elementary Convexity with Optimization
- 2023
- Buch
- Verfasst von
- Vivek S. Borkar
- K. S. Mallikarjuna Rao
- Buchreihe
- Texts and Readings in Mathematics
- Verlag
- Springer Nature Singapore
Über dieses Buch
Über dieses Buch
This book develops the concepts of fundamental convex analysis and optimization by using advanced calculus and real analysis. Brief accounts of advanced calculus and real analysis are included within the book. The emphasis is on building a geometric intuition for the subject, which is aided further by supporting figures. Two distinguishing features of this book are the use of elementary alternative proofs of many results and an eclectic collection of useful concepts from optimization and convexity often needed by researchers in optimization, game theory, control theory, and mathematical economics. A full chapter on optimization algorithms gives an overview of the field, touching upon many current themes. The book is useful to advanced undergraduate and graduate students as well as researchers in the fields mentioned above and in various engineering disciplines.
Inhaltsverzeichnis
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Frontmatter
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Chapter 1. Continuity and Existence of Optima
Vivek S. Borkar, K. S. Mallikarjuna RaoAbstractOptimization theory in finite dimensional spaces may be viewed as ‘applied real analysis’, since it depends on the analytic tools of the latter discipline for most of its foundations. -
Chapter 2. Differentiability and Local Optimality
Vivek S. Borkar, K. S. Mallikarjuna RaoAbstractIn this chapter we give an overview of a variety of facts concerning optimality of a point relative to a neighborhood of it, in terms of ‘local’ objects such as derivatives. We first introduce the different notions of derivatives and beginning with some familiar conditions for optimality from calculus, build up various generalizations thereof. -
Chapter 3. Convex Sets
Vivek S. Borkar, K. S. Mallikarjuna RaoAbstractRecall that a convex set \(C \subset \mathcal {R}^d\) is a set such that any line segment joining two distinct points in C lies entirely in C, i.e., \(x, y \in C, \ 0 \le \alpha \le 1\) implies \(\alpha x + (1 - \alpha )y \in C\). -
Chapter 4. Convex Functions
Vivek S. Borkar, K. S. Mallikarjuna RaoAbstractThis chapter is devoted to convex functions, the rock star of optimization theory. In this section, we recall their key properties that matter for convex optimization. -
Chapter 5. Convex Optimization
Vivek S. Borkar, K. S. Mallikarjuna RaoAbstractConvex optimization or convex programming refers to the problem of minimizing convex functions over convex sets. Observe that we have been careful to say only minimization. Maximization of convex functions is a different kettle of fish; altogether, these problems can be extremely hard. -
Chapter 6. Optimization Algorithms: An Overview
Vivek S. Borkar, K. S. Mallikarjuna RaoAbstractOptimization algorithms is a vast research area in its own right, with multiple strands. In this chapter we do not attempt anything close to a comprehensive overview, but limit ourselves to giving just a taste of the subject in broad strokes. -
Chapter 7. Epilogue
Vivek S. Borkar, K. S. Mallikarjuna RaoAbstractHere we try to sketch briefly many challenging directions one can take from here. These are, however, only ‘teasers’, without any significant detail. -
Backmatter
- Titel
- Elementary Convexity with Optimization
- Verfasst von
-
Vivek S. Borkar
K. S. Mallikarjuna Rao
- Copyright-Jahr
- 2023
- Verlag
- Springer Nature Singapore
- Electronic ISBN
- 978-981-9916-52-8
- Print ISBN
- 978-981-9916-51-1
- DOI
- https://doi.org/10.1007/978-981-99-1652-8
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