Publication

Optimization Notes

2019
Reports

Abstract

While optimization is well studied for real-valued functions $f : \mathbb{R}^{N} \to \mathbb{R}$ , many physical problems are (partially) specified in terms of complex-valued functions $f_{c} : \mathbb{C}^{N} \to \mathbb{C}^{M}$ . Current optimization packages have limited support for such functions. In particular it is unclear how to define algorithmic differentiation w.r.t. complex-valued functions and arguments. This document is a collection of working notes on the topic.

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2019
Reports

Abstract

Official source

Related concepts (32)

Multivalued function

In mathematics, a multivalued function, also called multifunction and many-valued function, is a set-valued function with continuity properties that allow considering it locally as an ordinary function. Multivalued functions arise commonly in applications of the implicit function theorem, since this theorem can be viewed as asserting the existence of a multivalued function. In particular, the inverse function of a differentiable function is a multivalued function, and is single-valued only when the original function is monotonic.

Holomorphic function

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space Cn. The existence of a complex derivative in a neighbourhood is a very strong condition: it implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (analytic). Holomorphic functions are the central objects of study in complex analysis.

Vector-valued function

A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could be a scalar or a vector (that is, the dimension of the domain could be 1 or greater than 1); the dimension of the function's domain has no relation to the dimension of its range. A common example of a vector-valued function is one that depends on a single real parameter t, often representing time, producing a vector v(t) as the result.

Related publications (6)

Optimization Notes

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