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 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 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 Parametric optimization of steel structures based on gradient projection method

2020 ◽  
pp. 192-220
Author(s):  
Vitalina Yurchenko ◽  
Ivan Peleshko
Keyword(s):  
Parametric Optimization ◽  
Optimization Problems ◽  
Steel Structure ◽  
Steel Structures ◽  
Gradient Projection ◽  
Truss Structures ◽  
Design System ◽  
Geometrical Parameters ◽  
Structural Members ◽  
Load Case

The main research goal is the development of a numerical methodology for solving parametric optimization problems of steel structures with orientation on software implementation in a computer-aided design system. The paper has proposed a new mathematical model for parametric optimization problems of steel structures. The design variable vector includes geometrical parameters of the structure (node coordinates), cross-sectional dimensions of the structural members, as well as initial pre-stressing forces introduced into the specified redundant members of the structure. The system of constraints covers load-carrying capacities constraints formulated for all design sections of structural members of the steel structure subjected to all ultimate load case combinations. The displacements constraints formulated for the specified nodes of the steel structure subjected to all serviceability load case combinations have been also included into the system of constraints. The method of the objective function gradient projection onto the active constraints surface with simultaneous correction of the constraints violations has been used for solving the parametric optimization problem. A numerical algorithm for solving the formulated parametric optimization problems of steel structures has been developed in the paper. The comparison of the optimization results of truss structures presented by the paper confirms the validity of the optimum solutions obtained using the proposed numerical methodology.

Efficient Parametric Optimization for Expensive Single Objective Problems

2021 ◽  
pp. 1-12
Author(s):  
Jonathan Weaver-Rosen ◽  
Richard Malak
Keyword(s):  
Adaptive Sampling ◽  
Parametric Optimization ◽  
Optimization Problems ◽  
Test Problems ◽  
Surface Model ◽  
High Fidelity ◽  
Unknown Parameters ◽  
Element Analysis ◽  
Better Than

Abstract Parametric optimization solves optimization problems as a function of uncontrollable or unknown parameters. Such an approach allows an engineer to gather more information than traditional optimization procedures during design. Existing methods for parametric optimization of computationally or monetarily expensive functions can be too time-consuming or impractical to solve. Therefore, new methods for the parametric optimization of expensive functions need to be explored. This work proposes a novel algorithm that leverages the advantages of two existing optimization algorithms. This new algorithm is called the efficient parametric optimization (EPO) algorithm. EPO enables adaptive sampling of a high-fidelity design space using an inexpensive low-fidelity response surface model. Such an approach largely reduces the required number of expensive high-fidelity computations. The proposed method is benchmarked using analytic test problems and used to evaluate a case study requiring finite element analysis. Results show that EPO performs as well as or better than the existing alternative, P3GA, for these problems given an allowable number of function evaluations.

Parametric Optimization for Structural Design Problems

2019 ◽  
Author(s):  
Krupakaran Ravichandran ◽  
Nafiseh Masoudi ◽  
Georges M. Fadel ◽  
Margaret M. Wiecek
Keyword(s):  
Structural Design ◽  
Optimization Problem ◽  
Parametric Optimization ◽  
Optimization Problems ◽  
Nonlinear Constraint ◽  
Computational Performance ◽  
Design Variables ◽  
Input Parameters ◽  
Parametric Optimization Problem ◽  
Objective Function Values

Abstract Parametric Optimization is used to solve problems that have certain design variables as implicit functions of some independent input parameters. The optimal solutions and optimal objective function values are provided as functions of the input parameters for the entire parameter space of interest. Since exact solutions are available merely for parametric optimization problems that are linear or convex-quadratic, general non-convex non-linear problems require approximations. In the present work, we apply three parametric optimization algorithms to solve a case study of a benchmark structural design problem. The algorithms first approximate the nonlinear constraint(s) and then solve the optimization problem. The accuracy of their results and their computational performance are then compared to identify a suitable algorithm for structural design applications. Using the identified method, sizing optimization of a truss structure for varying load conditions such as a varying load direction is considered and solved as a parametric optimization problem to evaluate the performance of the identified algorithm. The results are also compared with non-parametric optimization to assess the accuracy of the solution and computational performance of the two methods.

OPTIMIZATION PROBLEMS IN DESIGNING AUTOMOBILES

Transport ◽  
2008 ◽  
Vol 23 (4) ◽  
pp. 316-322 ◽  
Author(s):  
Ivan Dyakov ◽  
Olegas Prentkovskis
Keyword(s):  
Mathematical Model ◽  
System Approach ◽  
Parametric Optimization ◽  
Optimality Criterion ◽  
Optimization Problems

A mathematical model of optimization problems in designing automobiles is suggested and requirements raised are defined taking into account the main technical and economic characteristics of the automobile. The optimality criterion of the integral parts of the automobile is presented. The system approach to the theory of parametric optimization based on generalized models is used and the algorithm for solving the considered problem is offered.

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