Content deleted Content added
added Category:Nonlinear programming using HotCat |
LucasBrown (talk | contribs) Changing short description from "Concept in mathematical opimization" to "Concept in mathematical optimization" |
||
| (115 intermediate revisions by 48 users not shown) | |||
Line 1:
{{Short description|Concept in mathematical optimization}}
In [[mathematical optimization]], '''linear-fractional programming''' ('''LFP
Formally, a linear-fractional program is defined as the problem of maximizing (or minimizing) a ratio of [[affine function]]s over a [[polyhedron]],
:<math>
\begin{align}
\text{maximize} \quad & \frac{\mathbf{c}^T \mathbf{x} + \alpha}{\mathbf{d}^T \mathbf{x} + \beta} \\
\text{subject to} \quad & A\mathbf{x} \leq \mathbf{b},
\end{align}
</math>
where <math>\mathbf{x} \in \mathbb{R}^n</math> represents the vector of variables to be determined, <math>\mathbf{c}, \mathbf{d} \in \mathbb{R}^n</math> and <math>\mathbf{b} \in \mathbb{R}^m</math> are vectors of (known) coefficients, <math>A \in \mathbb{R}^{m \times n}</math> is a (known) matrix of coefficients and <math>\alpha, \beta \in \mathbb{R}</math> are constants. The constraints have to restrict the [[feasible region]] to <math>\{\mathbf{x} | \mathbf{d}^T\mathbf{x} + \beta > 0\}</math>, i.e. the region on which the denominator is positive.<ref name="CC">{{cite journal |last1=Charnes |first1=A. |last2=Cooper |first2=W. W. |author2-link=William W. Cooper |year=1962 |title=Programming with Linear Fractional Functionals |journal=Naval Research Logistics Quarterly |volume=9 |issue=3–4 |pages=181–186 |doi=10.1002/nav.3800090303 |mr=152370}}</ref><ref name="BV">{{cite book |last1=Boyd |first1=Stephen P. |url=https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf |title=Convex Optimization |last2=Vandenberghe |first2=Lieven |publisher=Cambridge University Press |year=2004 |isbn=978-0-521-83378-3 |page=151 |access-date=October 15, 2011}}</ref> Alternatively, the denominator of the objective function has to be strictly negative in the entire feasible region.
==Motivation by comparison to linear programming==
Both linear programming and linear-fractional programming represent optimization problems using linear equations and linear inequalities, which for each problem-instance define a [[feasible set]]. Fractional linear programs have a richer set of objective functions. Informally, linear programming computes a policy delivering the best outcome, such as maximum profit or lowest cost. In contrast, a linear-fractional programming is used to achieve the highest ''ratio'' of outcome to cost, the ratio representing the highest efficiency. For example, in the context of LP we maximize the objective function '''profit = income − cost''' and might obtain maximum profit of $100 (= $1100 of income − $1000 of cost). Thus, in LP we have an efficiency of $100/$1000 = 0.1. Using LFP we might obtain an efficiency of $10/$50 = 0.2 with a profit of only $10, but only requiring $50 of investment.
==Transformation to a linear program==
Fractional-linear programs are [[quasiconvex function|quasiconvex]] [[convex minimization|minimization]] problems with a [[monotonicity|monotone]] property, "[[pseudoconvex function|pseudoconvexity]]" which is a stronger property than [[quasiconvex function|quasiconvexity]]. A fractional-linear objective function has both "pseudoconvexity" and "pseudoconcavity", these properties allowing FLP problems to be solved by a variant of the [[simplex algorithm]] (of [[George B. Dantzig]]).<ref>▼
Any linear-fractional program can be transformed into a linear program, assuming that the feasible region is non-empty and bounded, using the '''Charnes–Cooper transformation'''.<ref name="CC" /> The main idea is to introduce a new non-negative variable <math>t </math> to the program which will be used to rescale the constants involved in the program (<math>\alpha, \beta, \mathbf{b}</math>). This allows us to require that the denominator of the objective function (<math>\mathbf{d}^T \mathbf{x} + \beta</math>) equals 1. (To understand the transformation, it is instructive to consider the simpler special case with <math>\alpha = \beta = 0</math>.)
Chapter five: {{cite book| last=Craven|first=B. D.|title=Fractional programming|series=Sigma Series in Applied Mathematics|volume=4|publisher=Heldermann Verlag|___location=Berlin|year=1988|pages=145|isbn=3-88538-404-3 |id={{MR|949209}}| }}</ref><ref> {{cite article | last1=Kruk | first1=Serge|last2=Wolkowicz|first2=Henry|title=Pseudolinear programming | url=http://www.jstor.org/stable/2653207 |journal=[[SIAM Review]]|volume=41 |year=1999 |number=4 |pages=795-805 |id={{MR|1723002}}.{{jstor|2653207}}.{{doi|DOI:10.1137/S0036144598335259}}| }}▼
</ref><ref> {{cite article | last1=Mathis|first1=Frank H.|last2=Mathis|first2=Lenora Jane|title=A nonlinear programming algorithm for hospital management |url=http://www.jstor.org/stable/2132826|journal=[[SIAM Review]]|volume=37 |year=1995 |number=2 |pages=230-234|id={{MR|1343214}}.{{jstor|2132826}}.{{doi|DOI:10.1137/1037046}}|}}▼
</ref> FLP problems can also be solved by the [[criss-cross algorithm]], which is a "purely combinatorial" [[exchange algorithm|basis-exchange algorithm]].<ref>{{cite journal|title=The finite criss-cross method for hyperbolic programming|journal=European Journal of Operational Research|volume=114|number=1|▼
Formally, the linear program obtained via the Charnes–Cooper transformation uses the transformed variables <math>\mathbf{y} \in \mathbb{R}^n</math> and <math>t \ge 0 </math>:
:<math>
<references/>▼
\begin{align}
\text{maximize} \quad & \mathbf{c}^T \mathbf{y} + \alpha t \\
\text{subject to} \quad & A\mathbf{y} \leq \mathbf{b} t \\
& \mathbf{d}^T \mathbf{y} + \beta t = 1 \\
& t \geq 0.
\end{align}
</math>
A solution <math>\mathbf{x}</math> to the original linear-fractional program can be translated to a solution of the transformed linear program via the equalities
* {{cite book|first=E. B.|last=Bajalinov|title=Linear-Fractional Programming: Theory, Methods, Applications and Software| publisher=Kluwer Academic Publishers|___location=Boston|year=2003}}▼
:<math>\mathbf{y} = \frac{1}{\mathbf{d}^T \mathbf{x} + \beta} \cdot \mathbf{x}\quad \text{and} \quad t = \frac{1}{\mathbf{d}^T \mathbf{x} + \beta}.</math>
Conversely, a solution for <math>\mathbf{y}</math> and <math>t </math> of the transformed linear program can be translated to a solution of the original linear-fractional program via
* {{cite book|last=Barros|first=Ana Isabel|title=Discrete and fractional programming techniques for ___location models|series=Combinatorial Optimization|volume=3|publisher=Kluwer Academic Publishers|___location=Dordrecht|year=1998|pages=xviii+178|isbn=0-7923-5002-2|id={{MR|1626973}}|}}▼
:<math>\mathbf{x}=\frac{1}{t}\mathbf{y}.</math>
==Duality==
Let the [[duality (optimization)|dual variables]] associated with the constraints <math>A\mathbf{y} - \mathbf{b} t \leq \mathbf{0}</math> and <math>\mathbf{d}^T \mathbf{y} + \beta t - 1 = 0</math> be denoted by <math>\mathbf{u}</math> and <math>\lambda</math>, respectively. Then the dual of the LFP above is <ref>{{cite journal|last1=Schaible |first1=Siegfried |title=Parameter-free Convex Equivalent and Dual Programs|journal=Zeitschrift für Operations Research |volume=18 |year=1974 |issue=5 |pages=187–196|doi=10.1007/BF02026600|mr=351464|s2cid=28885670 }}</ref><ref>{{cite journal|title=Fractional programming I: Duality |last1=Schaible |first1=Siegfried | journal=Management Science |volume=22 |issue=8 |pages=858–867 |year=1976|jstor=2630017|mr=421679|doi=10.1287/mnsc.22.8.858}}</ref>
:<math>
\begin{align}
\text{minimize} \quad & \lambda \\
\text{subject to} \quad & A^T\mathbf{u} + \lambda \mathbf{d} = \mathbf{c} \\
& -\mathbf{b}^T \mathbf{u} + \lambda \beta \geq \alpha \\
& \mathbf{u} \in \mathbb{R}_+^m, \lambda \in \mathbb{R},
\end{align}
</math>
which is an LP and which coincides with the dual of the equivalent linear program resulting from the Charnes–Cooper transformation.
==Properties and algorithms==
▲
▲Chapter five: {{cite book| last=Craven|first=B. D.|title=Fractional programming|series=Sigma Series in Applied Mathematics|volume=4|publisher=Heldermann Verlag|___location=Berlin|year=1988|pages=145|isbn=978-3-88538-404-
▲</ref><ref>
▲</ref><ref>{{harvtxt|Murty|1983|loc=Chapter 3.20
pages=198–214|year=1999 <!-- issn=0377-2217 -->|doi=10.1016/S0377-2217(98)00049-6|first1=Tibor|last1=Illés|first2=Ákos|last2=Szirmai|first3=Tamás|last3=Terlaky|zbl=0953.90055|id=[http://www.cas.mcmaster.ca/~terlaky/files/dut-twi-96-103.ps.gz Postscript preprint]|citeseerx=10.1.1.36.7090}}</ref> or [[interior-point method]]s.
==Notes==
* {{cite book|last=Martos|first=Béla|title=Nonlinear programming: Theory and methods|publisher=North-Holland Publishing Co.|___location=Amsterdam-Oxford|year=1975|pages=279|isbn=0-7204-2817-3|id={{MR|496692}}|}}▼
▲<references />
==Sources==
* {{cite book|last=Schaible|first=S.|chapter=Fractional programming|pages=495–608|id={{MR|1377091}}|title=Handbook of global optimization|editor=Reiner Horst and Panos M. Pardalos|▼
series=Nonconvex optimization and its applications|volume=2|publisher=Kluwer Academic Publishers|___location=Dordrecht|year=1995|isbn=0-7923-3120-6}} ▼
*
==Further
▲*
▲*
▲*
▲*
▲series=Nonconvex optimization and its applications|volume=2|publisher=Kluwer Academic Publishers|___location=Dordrecht|year=1995|isbn=978-0-7923-3120-
*{{cite book | last=Stancu-Minasian | first=I. M.| title=Fractional programming: Theory, methods and applications | others=Translated by Victor Giurgiutiu from the 1992 Romanian | series=Mathematics and its applications|volume=409|publisher=Kluwer Academic Publishers Group | ___location=Dordrecht | year=1997 | pages=viii+418 | isbn=978-0-7923-4580-0 | mr=1472981 }}
[[Category:Operations research]]▼
[[Category:Nonlinear programming]]▼
{{DEFAULTSORT:Linear-Fractional Programming}}
[[Category:Optimization algorithms and methods]]
| |||