Complexity and algorithms for nonlinear optimization problems

Nonlinear optimization algorithms are rarely discussed from a complexity point of view. Even the concept of solving nonlinear problems on digital computers is not well defined. The focus here is on a complexity approach for designing and analyzing algorithms for nonlinear optimization problems provi...

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Vydáno v:Annals of operations research Ročník 153; číslo 1; s. 257 - 296
Hlavní autor: Hochbaum, Dorit S.
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer Nature B.V 01.09.2007
Témata:
Accuracy
Algorithms
Complexity
Digital computers
Hierarchies
Knapsack problem
Linear programming
Lower bounds
Operations management
Operations research
Optimization
Optimization algorithms
Polynomials
Solution space
Studies
ISSN:0254-5330, 1572-9338
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Shrnutí:Nonlinear optimization algorithms are rarely discussed from a complexity point of view. Even the concept of solving nonlinear problems on digital computers is not well defined. The focus here is on a complexity approach for designing and analyzing algorithms for nonlinear optimization problems providing optimal solutions with prespecified accuracy in the solution space. We delineate the complexity status of convex problems over network constraints, dual of flow constraints, dual of multi-commodity, constraints defined by a submodular rank function (a generalized allocation problem), tree networks, diagonal dominant matrices, and nonlinear knapsack problem's constraint. All these problems, except for the latter in integers, have polynomial time algorithms which may be viewed within a unifying framework of a proximity-scaling technique or a threshold technique. The complexity of many of these algorithms is furthermore best possible in that it matches lower bounds on the complexity of the respective problems. In general nonseparable optimization problems are shown to be considerably more difficult than separable problems. We compare the complexity of continuous versus discrete nonlinear problems and list some major open problems in the area of nonlinear optimization. [PUBLICATION ABSTRACT]
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ISSN:0254-5330
1572-9338
DOI:10.1007/s10479-007-0172-6