scholarly journals An approximation algorithm for a general class of parametric optimization problems

2020 ◽  
Author(s):  
Cristina Bazgan ◽  
Arne Herzel ◽  
Stefan Ruzika ◽  
Clemens Thielen ◽  
Daniel Vanderpooten
Keyword(s):  
Approximation Algorithm ◽  
Polynomial Time ◽  
General Class ◽  
Parametric Optimization ◽  
Optimization Problems ◽  
Parametric Problem ◽  
Running Time ◽  
Objective Value ◽  
Approximation Guarantee ◽  
Non Parametric

Abstract In a (linear) parametric optimization problem, the objective value of each feasible solution is an affine function of a real-valued parameter and one is interested in computing a solution for each possible value of the parameter. For many important parametric optimization problems including the parametric versions of the shortest path problem, the assignment problem, and the minimum cost flow problem, however, the piecewise linear function mapping the parameter to the optimal objective value of the corresponding non-parametric instance (the optimal value function) can have super-polynomially many breakpoints (points of slope change). This implies that any optimal algorithm for such a problem must output a super-polynomial number of solutions. We provide a method for lifting approximation algorithms for non-parametric optimization problems to their parametric counterparts that is applicable to a general class of parametric optimization problems. The approximation guarantee achieved by this method for a parametric problem is arbitrarily close to the approximation guarantee of the algorithm for the corresponding non-parametric problem. It outputs polynomially many solutions and has polynomial running time if the non-parametric algorithm has polynomial running time. In the case that the non-parametric problem can be solved exactly in polynomial time or that an FPTAS is available, the method yields an FPTAS. In particular, under mild assumptions, we obtain the first parametric FPTAS for each of the specific problems mentioned above and a $$(3/2 + \varepsilon )$$ ( 3 / 2 + ε ) -approximation algorithm for the parametric metric traveling salesman problem. Moreover, we describe a post-processing procedure that, if the non-parametric problem can be solved exactly in polynomial time, further decreases the number of returned solutions such that the method outputs at most twice as many solutions as needed at minimum for achieving the desired approximation guarantee.

scholarly journals One-exact approximate Pareto sets

2020 ◽  
Author(s):  
Arne Herzel ◽  
Cristina Bazgan ◽  
Stefan Ruzika ◽  
Clemens Thielen ◽  
Daniel Vanderpooten
Keyword(s):  
Multiobjective Optimization ◽  
Polynomial Time ◽  
Optimization Problems ◽  
Auxiliary Problem ◽  
Constant Factor ◽  
Pareto Set ◽  
Multiobjective Optimization Problems ◽  
Annual Ieee Symposium ◽  
Biobjective Optimization ◽  
Approximation Guarantee

AbstractPapadimitriou and Yannakakis (Proceedings of the 41st annual IEEE symposium on the Foundations of Computer Science (FOCS), pp 86–92, 2000) show that the polynomial-time solvability of a certain auxiliary problem determines the class of multiobjective optimization problems that admit a polynomial-time computable $$(1+\varepsilon , \dots , 1+\varepsilon )$$ ( 1 + ε , ⋯ , 1 + ε ) -approximate Pareto set (also called an $$\varepsilon $$ ε -Pareto set). Similarly, in this article, we characterize the class of multiobjective optimization problems having a polynomial-time computable approximate $$\varepsilon $$ ε -Pareto set that is exact in one objective by the efficient solvability of an appropriate auxiliary problem. This class includes important problems such as multiobjective shortest path and spanning tree, and the approximation guarantee we provide is, in general, best possible. Furthermore, for biobjective optimization problems from this class, we provide an algorithm that computes a one-exact $$\varepsilon $$ ε -Pareto set of cardinality at most twice the cardinality of a smallest such set and show that this factor of 2 is best possible. For three or more objective functions, however, we prove that no constant-factor approximation on the cardinality of the set can be obtained efficiently.

APPROXIMATING THE DIAMETER, WIDTH, SMALLEST ENCLOSING CYLINDER, AND MINIMUM-WIDTH ANNULUS

2002 ◽  
Vol 12 (01n02) ◽  
pp. 67-85 ◽  
Author(s):  
TIMOTHY M. CHAN
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Optimization Problems ◽  
Linear Time ◽  
Minimum Width ◽  
Time Approximation ◽  
Running Time ◽  
Fixed Dimension ◽  
Smallest Enclosing Cylinder ◽  
Point Set

We study (1+ε)-factor approximation algorithms for several well-known optimization problems on a given n-point set: (a) diameter, (b) width, (c) smallest enclosing cylinder, and (d) minimum-width annulus. Among our results are new simple algorithms for (a) and (c) with an improved dependence of the running time on ε, as well as the first linear-time approximation algorithm for (d) in any fixed dimension. All four problems can be solved within a time bound of the form O(n+ε-c) or O(n log (1/ε)+ε-c).

An FPTAS for a General Class of Parametric Optimization Problems

2019 ◽  
pp. 25-37 ◽  
Author(s):  
Cristina Bazgan ◽  
Arne Herzel ◽  
Stefan Ruzika ◽  
Clemens Thielen ◽  
Daniel Vanderpooten
Keyword(s):  
General Class ◽  
Parametric Optimization ◽  
Optimization Problems

scholarly journals R-Regularity of Set-Valued Mappings Under the Relaxed Constant Positive Linear Dependence Constraint Qualification with Applications to Parametric and Bilevel Optimization

2021 ◽  
Author(s):  
Patrick Mehlitz ◽  
Leonid I. Minchenko
Keyword(s):  
Linear Dependence ◽  
Constraint Qualification ◽  
Parametric Optimization ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Bilevel Optimization ◽  
Regularity Property ◽  
Constant Rank Constraint Qualification ◽  
Existence Criterion ◽  
Set Valued Mappings

AbstractThe presence of Lipschitzian properties for solution mappings associated with nonlinear parametric optimization problems is desirable in the context of, e.g., stability analysis or bilevel optimization. An example of such a Lipschitzian property for set-valued mappings, whose graph is the solution set of a system of nonlinear inequalities and equations, is R-regularity. Based on the so-called relaxed constant positive linear dependence constraint qualification, we provide a criterion ensuring the presence of the R-regularity property. In this regard, our analysis generalizes earlier results of that type which exploited the stronger Mangasarian–Fromovitz or constant rank constraint qualification. Afterwards, we apply our findings in order to derive new sufficient conditions which guarantee the presence of R-regularity for solution mappings in parametric optimization. Finally, our results are used to derive an existence criterion for solutions in pessimistic bilevel optimization and a sufficient condition for the presence of the so-called partial calmness property in optimistic bilevel optimization.

scholarly journals Inapproximability and Polynomial-Time Approximation Algorithm for UET Tasks on Structured Processor Networks

2011 ◽  
Vol 2011 ◽  
pp. 1-20 ◽  
Author(s):  
M. Bouznif ◽  
R. Giroudeau
Keyword(s):  
Approximation Algorithm ◽  
Large Class ◽  
Bipartite Graph ◽  
Polynomial Time ◽  
Performance Guarantee ◽  
Makespan Minimization ◽  
Time Approximation ◽  
Polynomial Time Approximation Algorithm ◽  
Precedence Graph ◽  
Polynomial Time Approximation

We investigate complexity and approximation results on a processor networks where the communication delay depends on the distance between the processors performing tasks. We then prove that there is no heuristic with a performance guarantee smaller than 4/3 for makespan minimization for precedence graph on a large class of processor networks like hypercube, grid, torus, and so forth, with a fixed diameter . We extend complexity results when the precedence graph is a bipartite graph. We also design an efficient polynomial-time -approximation algorithm for the makespan minimization on processor networks with diameter .

A Numerical Scheme for a Class of Parametric Problem of Fractional Variational Calculus

2012 ◽  
Vol 7 (2) ◽  
Author(s):  
Om P. Agrawal ◽  
M. Mehedi Hasan ◽  
X. W. Tangpong
Keyword(s):  
Analytical Solutions ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Optimization Problems ◽  
Practical Interest ◽  
Variational Calculus ◽  
Functional Minimization ◽  
Orthogonal Functions ◽  
Order Problem ◽  
Parametric Problem

Fractional derivatives (FDs) or derivatives of arbitrary order have been used in many applications, and it is envisioned that in the future they will appear in many functional minimization problems of practical interest. Since fractional derivatives have such properties as being non-local, it can be extremely challenging to find analytical solutions for fractional parametric optimization problems, and in many cases, analytical solutions may not exist. Therefore, it is of great importance to develop numerical methods for such problems. This paper presents a numerical scheme for a linear functional minimization problem that involves FD terms. The FD is defined in terms of the Riemann-Liouville definition; however, the scheme will also apply to Caputo derivatives, as well as other definitions of fractional derivatives. In this scheme, the spatial domain is discretized into several subdomains and 2-node one-dimensional linear elements are adopted to approximate the solution and its fractional derivative at point within the domain. The fractional optimization problem is converted to an eigenvalue problem, the solution of which leads to fractional orthogonal functions. Convergence study of the number of elements and error analysis of the results ensure that the algorithm yields stable results. Various fractional orders of derivative are considered, and as the order approaches the integer value of 1, the solution recovers the analytical result for the corresponding integer order problem.

scholarly journals ЭФФЕКТИВНОСТЬ МЕТОДОВ ОПРЕДЕЛЕНИЯ ГРУППОВЫХ СИСТЕМ ПРЕДПОЧТЕНИЙ ДИСПЕТЧЕРОВ НА ОПАСНОСТИ ХАРАКТЕРНЫХ ОШИБОК, СОВЕРШАЕМЫХ В ПРОЦЕССЕ УПРАВЛЕНИЯ ВОЗДУШНЫМ ДВИЖЕНИЕМ

2018 ◽  
pp. 93-103
Author(s):  
Алексей Николаевич Рева ◽  
Шахин Шахвели-оглы Насиров ◽  
Бала Мушгюль-оглы Мирзоев
Keyword(s):  
Decision Making ◽  
Parametric Optimization ◽  
Professional Activity ◽  
Air Traffic Controllers ◽  
Main Emphasis ◽  
Decision Making Processes ◽  
Complex Decision Making ◽  
Coefficient Of Concordance ◽  
Complex Decision ◽  
Non Parametric

The human factor problem should be solved by identifying, qualifying and preventing the erroneous actions of the air traffic controllers. It is presented two schemes explaining the structure of human qualimetry factor and the interaction of the components of the ICAO safety concept, where the main emphasis is on an aviation personnel’ attitude to dangerous actions or conditions, which is revealed by the qualimetry of the decision-making processes’ characteristics: the attitude towards risk (the main dominants and fuzzy assessments), levels of claims, dangerous qualities and preferences systems. The preferences systems are considered as ordered characteristics and indicators of professional activity, which are subjectively compared with the positions of influence on flight safety. The spectrum of n = 21 characteristic errors was formed considering the recommendations of ICAO, EUROCONTROL and accident statistics. It is determined that procedures of collecting the information of errors danger contribute their recognition, memorization, and avoidance: controllers who passed the test according to the proposed method before training made by one third fewer errors in its process. Two criteria for assessing group preferences are realized: the level of consensus (known as Kendall’s coefficient of concordance) and the severity of the ranking, determined by the presence of "related" ranks, for which a special indicator is introduced. It is defined that this indicator should be determined both for the sample of respondents and for the preferences group systems of developed with the chosen method of individual opinions’ aggregation. It was performed the comparative analysis of complex decision-making strategies of effectiveness in the construction of a preferences group systems m = 65 controllers: sum and averaging of ranks, classical criteria (Wald's, Savage's and Laplace's criterion), optimal prediction, applying the non-parametric optimization of the preferences group systems. The non-parametric optimization of the group system of pre-readings was carried out by Kemeny median and it was proved that it was the closest to all the results obtained by other methods and strategies

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