On global optimization using an estimate of Lipschitz constant and simplicial partition

2016 ◽  
Author(s):  
Albertas Gimbutas ◽  
Antanas Žilinskas
Keyword(s):  
Global Optimization ◽  
Lipschitz Constant ◽  
Simplicial Partition

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.

scholarly journals On the experimental investigation of Pareto–Lipschitzian optimization

2011 ◽  
Vol 52 ◽  
Author(s):  
Jonas Mockus ◽  
Justas Stašionis
Keyword(s):  
Experimental Investigation ◽  
Global Optimization ◽  
Pareto Optimality ◽  
Lipschitz Constant ◽  
Real Life ◽  
Multiple Criteria ◽  
Life Problems ◽  
Lipschitzian Optimization ◽  
Lipschitz Constants ◽  
Novel Method

A well-known example of global optimization that provides solutions within fixed error limits is optimization of functions with a known Lipschitz constant. In many real-life problems this constant is unknown. To address that, we propose a novel method called Pareto Lipschitzian Optimization (PLO) that provides solutions within fixed error limits for functions with unknown Lipschitzconstants.In the proposed approach, a set of all unknown Lipschitz constants is regarded as multiple criteria using the concept of Pareto Optimality (PO).

scholarly journals ANALYSIS OF DIFFERENT NORMS AND CORRESPONDING LIPSCHITZ CONSTANTS FOR GLOBAL OPTIMIZATION

2006 ◽  
Vol 12 (4) ◽  
pp. 301-306 ◽  
Author(s):  
Remigijus Paulavičius ◽  
Julius Žilinskas
Keyword(s):  
Global Optimization ◽  
Branch And Bound ◽  
Lipschitz Constant ◽  
Branch And Bound Algorithm ◽  
Test Functions ◽  
Lipschitz Optimization ◽  
Lipschitz Constants

The paper discusses how the used norm and corresponding Lipschitz constant influence the speed of algorithms for global optimization. For this reason Lipschitz constants corresponding to different norms were estimated. Different test functions for global optimization were solved using branch‐and‐bound algorithm for Lipschitz optimization with different norms. Experiments have shown that the best results are achieved when combination of extreme (infinite and first) and sometimes Euclidean norms is used.

Global Optimization

1993 ◽  
Author(s):  
Reiner Horst ◽  
Hoang Tuy
Keyword(s):  
Global Optimization

Probability Models in Global Optimization

Informatica ◽  
2016 ◽  
Vol 27 (2) ◽  
pp. 323-334 ◽  
Author(s):  
James M. Calvin
Keyword(s):  
Global Optimization ◽  
Probability Models
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