Functional Analysis and Mathematical Optimization
A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".
Deadline for manuscript submissions: 31 May 2025 | Viewed by 1536
Special Issue Editors
Interests: functional analysis; data science; optimization; operator theory; operations research
Interests: computational number theory: properties of algebraic integers, rational approximations of irrational numbers, etc.
Special Issue Information
Dear Colleagues,
The theory of mathematical optimization or mathematical programming is at the crossroads of many subjects. The subject grew from a realization that quantitative problems in manifestly different disciplines have important mathematical elements in common. Because of this commonality, many problems can be formulated and solved by using the unified set of ideas and methods that make up the field of optimization. The terms “minimum,” “maximum,” and “optimum” are in line with the mathematical tradition. Historically, linear programs were the focus in the optimization community, and initially, it was thought that the major divide was between linear and nonlinear optimization problems; later, people discovered that some nonlinear problems were much harder than others, and the “right” divide was between convex and nonconvex problems. Optimization is also generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines, from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.
Functional analysis is a branch of mathematical analysis dealing with functionals, or functions of functions. A functional, like a function, is a relationship between objects, but the objects may be numbers, vectors, or functions. Groupings of such objects are called spaces. Functional analysis is a subject that is seen as the study of vector spaces endowed with a topology, and in particular, infinite dimensional spaces. An important part of functional analysis is the extension of the theories of measure, integration, and probability to infinite dimensional spaces.
Optimization and functional analysis are interrelated. Regarding function space methods for optimization problems, much discussion has taken place. For example, the necessary optimality conditions can, in general, be written as nonlinear operator equations for the primal variable and Lagrange multiplier. The Lagrange multiplier theory of a general class of non-smooth and non-convex optimization can be based on functional analysis tools. Many of the constraints for optimization problems may also be governed by partial differential and functional equations, and/or non-smooth and non-convex operator equations.
Dr. Renying Zeng
Prof. Dr. Qiang Wu
Prof. Dr. Chunlei Tang
Guest Editors
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Keywords
- functional analysis
- number theory
- operator theory
- theories of measure, integration, and probability in infinite dimensional spaces
- applications of functional analysis
- discrete optimization
- continuous optimization
- stochastic optimization
- optimization algorithms
- calculus of variations
- optimization techniques and applications
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