scholarly journals On new methods to construct lower bounds in simplicial branch and bound based on interval arithmetic

2021 ◽  
Author(s):  
B. G.-Tóth ◽  
L. G. Casado ◽  
E. M. T. Hendrix ◽  
F. Messine
Keyword(s):  
Global Optimization ◽  
Lower Bounds ◽  
Branch And Bound ◽  
Interval Arithmetic ◽  
Efficient Global Optimization ◽  
Test Problems ◽  
Simplicial Partition ◽  
Monotonicity Test ◽  
Low Dimensional ◽  
Linear Relaxations

AbstractBranch and Bound (B&B) algorithms in Global Optimization are used to perform an exhaustive search over the feasible area. One choice is to use simplicial partition sets. Obtaining sharp and cheap bounds of the objective function over a simplex is very important in the construction of efficient Global Optimization B&B algorithms. Although enclosing a simplex in a box implies an overestimation, boxes are more natural when dealing with individual coordinate bounds, and bounding ranges with Interval Arithmetic (IA) is computationally cheap. This paper introduces several linear relaxations using gradient information and Affine Arithmetic and experimentally studies their efficiency compared to traditional lower bounds obtained by natural and centered IA forms and their adaption to simplices. A Global Optimization B&B algorithm with monotonicity test over a simplex is used to compare their efficiency over a set of low dimensional test problems with instances that either have a box constrained search region or where the feasible set is a simplex. Numerical results show that it is possible to obtain tight lower bounds over simplicial subsets.

AN IMPLEMENTATION OF A PARALLEL GENERALIZED BRANCH AND BOUND TEMPLATE

2007 ◽  
Vol 12 (3) ◽  
pp. 277-289 ◽  
Author(s):  
Milda Baravykaitė ◽  
Raimondas Čiegis
Keyword(s):  
Global Optimization ◽  
Load Balancing ◽  
Branch And Bound ◽  
Optimization Problems ◽  
Test Problems ◽  
Search Spaces ◽  
Parallel Version ◽  
Derivative Free ◽  
Template Library ◽  
Selection Of

Branch and bound (BnB) is a general algorithm to solve optimization problems. We present a template implementation of the BnB paradigm. A BnB template is implemented using C++ object oriented paradigm. MPI is used for underlying communications. A paradigm of domain decomposition (data parallelization) is used to construct a parallel algorithm. To obtain a better load balancing, the BnB template has the load balancing module that allows the redistribution of search spaces among the processors at run time. A parallel version of user's algorithm is obtained automatically. A new derivative-free global optimization algorithm is proposed for solving nonlinear global optimization problems. It is based on the BnB algorithm and its implementation is done by using the developed BnB algorithm template library. The robustness of the new algorithm is demonstrated by solving a selection of test problems.

Efficient Global Surrogate Modeling for Reliability-Based Design Optimization

2012 ◽  
Vol 135 (1) ◽  
Author(s):  
Barron J. Bichon ◽  
Michael S. Eldred ◽  
Sankaran Mahadevan ◽  
John M. McFarland
Keyword(s):  
Global Optimization ◽  
Design Optimization ◽  
Reliability Analysis ◽  
Optimization Problem ◽  
Wide Spectrum ◽  
Surrogate Models ◽  
Efficient Global Optimization ◽  
Test Problems ◽  
Reliability Based Design Optimization ◽  
Reliability Based Design

Determining the optimal (lightest, least expensive, etc.) design for an engineered component or system that meets or exceeds a specified level of reliability is a problem of obvious interest across a wide spectrum of engineering fields. Various formulations and methods for solving this reliability-based design optimization problem have been proposed, but they typically involve accepting a tradeoff between accuracy and efficiency in the reliability analysis. This paper investigates the use of the efficient global optimization and efficient global reliability analysis methods to construct surrogate models at both the design optimization and reliability analysis levels to create methods that are more efficient than existing methods without sacrificing accuracy. Several formulations are proposed and compared through a series of test problems.

Trust Region Based Mode Pursuing Sampling Method for Global Optimization of High Dimensional Design Problems

2015 ◽  
Vol 137 (2) ◽  
Author(s):  
George H. Cheng ◽  
Adel Younis ◽  
Kambiz Haji Hajikolaei ◽  
G. Gary Wang
Keyword(s):  
Global Optimization ◽  
Optimization Problems ◽  
Trust Region ◽  
Standard Test ◽  
Black Box ◽  
High Dimensional ◽  
Efficient Global Optimization ◽  
Test Problems ◽  
Design Problems ◽  
Design Variables

Mode pursuing sampling (MPS) was developed as a global optimization algorithm for design optimization problems involving expensive black box functions. MPS has been found to be effective and efficient for design problems of low dimensionality, i.e., the number of design variables is less than 10. This work integrates the concept of trust regions into the MPS framework to create a new algorithm, trust region based mode pursuing sampling (TRMPS2), with the aim of dramatically improving performance and efficiency for high dimensional problems. TRMPS2 is benchmarked against genetic algorithm (GA), dividing rectangles (DIRECT), efficient global optimization (EGO), and MPS using a suite of standard test problems and an engineering design problem. The results show that TRMPS2 performs better on average than GA, DIRECT, EGO, and MPS for high dimensional, expensive, and black box (HEB) problems.

scholarly journals INFLUENCE OF LIPSCHITZ BOUNDS ON THE SPEED OF GLOBAL OPTIMIZATION

2012 ◽  
Vol 18 (1) ◽  
pp. 54-66 ◽  
Author(s):  
Remigijus Paulavičius ◽  
Julius Žilinskas
Keyword(s):  
Global Optimization ◽  
Branch And Bound ◽  
Lipschitz Function ◽  
Optimization Problems ◽  
Optimization Methods ◽  
Branch And Bound Algorithm ◽  
Test Problems ◽  
Global Optimization Methods

Global optimization methods based on Lipschitz bounds have been analyzed and applied widely to solve various optimization problems. In this paper a bound for Lipschitz function is proposed, which is computed using function values at the vertices of a simplex and the radius of the circumscribed sphere. The efficiency of a branch and bound algorithm with proposed bound and combinations of bounds is evaluated experimentally while solving a number of multidimensional test problems for global optimization. The influence of different bounds on the performance of a branch and bound algorithm has been investigated.

scholarly journals Tabu efficient global optimization with applications in additive manufacturing

2021 ◽  
Author(s):  
Long Wang ◽  
Theodore T. Allen ◽  
Michael A. Groeber
Keyword(s):  
Global Optimization ◽  
Additive Manufacturing ◽  
Optimization Problems ◽  
Direct Methods ◽  
Expected Improvement ◽  
Efficient Global Optimization ◽  
Test Problems ◽  
Gaussian Stochastic Process ◽  
Constrained Problems ◽  
Computational Performance

AbstractMethods based on Gaussian stochastic process (GSP) models and expected improvement (EI) functions have been promising for box-constrained expensive optimization problems. These include robust design problems with environmental variables having set-type constraints. However, the methods that combine GSP and EI sub-optimizations suffer from the following problem, which limits their computational performance. Efficient global optimization (EGO) methods often repeat the same or nearly the same experimental points. We present a novel EGO-type constraint-handling method that maintains a so-called tabu list to avoid past points. Our method includes two types of penalties for the key “infill” optimization, which selects the next test runs. We benchmark our tabu EGO algorithm with five alternative approaches, including DIRECT methods using nine test problems and two engineering examples. The engineering examples are based on additive manufacturing process parameter optimization informed using point-based thermal simulations and robust-type quality constraints. Our test problems span unconstrained, simply constrained, and robust constrained problems. The comparative results imply that tabu EGO offers very promising computational performance for all types of black-box optimization in terms of convergence speed and the quality of the final solution.

An exploitation-enhanced multi-objective efficient global optimization algorithm for expensive aerodynamic shape optimizations

2021 ◽  
pp. 095441002110324
Author(s):  
Feng Deng ◽  
Ning Qin
Keyword(s):  
Global Optimization ◽  
Shape Optimization ◽  
Optimization Algorithm ◽  
Optimization Problems ◽  
Efficient Global Optimization ◽  
Test Problems ◽  
Aerodynamic Shape Optimization ◽  
Aerodynamic Shape ◽  
Multi Objective ◽  
Complex Landscape

The traditional multi-objective efficient global optimization (EGO) algorithms have been hybridized and adapted to solving the expensive aerodynamic shape optimization problems based on high-fidelity numerical simulations. Although the traditional EGO algorithms are highly efficient in solving some of the optimization problems with very complex landscape, it is not preferred to solve most of the aerodynamic shape optimization problems with relatively low-degree multi-modal design spaces. A new infill criterion encouraging more local exploitation has been proposed by hybridizing two traditional multi-objective expected improvements (EIs), namely, statistical multi-objective EI and expected hypervolume improvement, in order to improve their robustness and efficiency in aerodynamic shape optimization. Different analytical test problems and aerodynamic shape optimization problems have been investigated. In comparison with traditional multi-objective EI algorithms and a standard evolutionary multi-objective optimization algorithm, the proposed method is shown to be more robust and efficient in the tests due to its hybrid characteristics, easier handling of sub-optimization problems, and enhanced exploitation capability.

Combination of two underestimators for univariate global optimization

2018 ◽  
Vol 52 (1) ◽  
pp. 177-186
Author(s):  
Mohand Ouanes ◽  
Mohammed Chebbah ◽  
Ahmed Zidna
Keyword(s):  
Global Optimization ◽  
Branch And Bound ◽  
Optimization Problems ◽  
Branch And Bound Algorithm ◽  
Test Problems

In this work, we propose a new underestimator in branch and bound algorithm for solving univariate global optimization problems. The new underestimator is a combination of two underestimators, the classical one used in αBB method (see Androulakis et al. [J. Glob. Optim. 7 (1995) 337–3637]) and the quadratic underestimator developed in Hoai An and Ouanes [RAIRO: OR 40 (2006) 285–302]. We show that the new underestimator is tighter than the two underestimators. A convex/concave test is used to accelerate the convergence of the proposed algorithm. The convergence of our algorithm is shown and a set of test problems given in Casado et al. [J. Glob. Optim. 25 (2003) 345–362] are solved efficiently.

scholarly journals On derivative based bounding for simplicial branch and bound

2021 ◽  
Author(s):  
Eligius M.T. Hendrix ◽  
Boglarka G. -Tóth ◽  
Frederic Messine ◽  
Leocadio G. Casado
Keyword(s):  
Global Optimization ◽  
Objective Function ◽  
Branch And Bound ◽  
Partial Derivative ◽  
Interval Arithmetic ◽  
Objective Function Value ◽  
Search Space ◽  
Second Derivative ◽  
Branch And Bound Methods

Simplicial based Global Optimization branch and bound methods require tight bounds on the objective function value. Recently, a renewed interest appears on bound calculation based on Interval Arithmetic by Karhbet and Kearfott (2017) and on exploiting second derivative bounds by Mohand (2021). The investigated question here is how partial derivative ranges can be used to provide bounds of the objective function value over the simplex. Moreover, we provide theoretical properties of how this information can be used from a monotonicity perspective to reduce the search space in simplicial branch and bound.

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