# Positive Bases in Numerical Optimization

@article{Coope2002PositiveBI, title={Positive Bases in Numerical Optimization}, author={Ian D. Coope and C. J. Price}, journal={Computational Optimization and Applications}, year={2002}, volume={21}, pages={169-175} }

The theory of positive bases introduced by C. Davis in 1954 does not appear in most modern texts on linear algebra but has re-emerged in publications in optimization journals. In this paper some simple properties of this highly useful theory are highlighted and applied to both theoretical and practical aspects of the design and implementation of numerical algorithms for nonlinear optimization.

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## 49 Citations

A short proof on the cardinality of maximal positive bases

- Mathematics, Computer ScienceOptim. Lett.
- 2011

This note provides a simple demonstration that relies on a fundamental property of basic feasible solutions in linear programming theory, which is easily shown that the cardinality of every positive basis is bounded below by n + 1.

On the properties of positive spanning sets and positive bases

- Mathematics
- 2016

The concepts of positive span and positive basis are important in derivative-free optimization. In fact, a well-known result is that if the gradient of a continuously differentiable objective…

Grid-based methods for linearly equality constrained optimization problems

- Mathematics
- 2007

This paper describes a direct search method for a class of linearly constrained optimization problem. Through research we find it can be treated as an unconstrained optimization problem. And with the…

A new quasi-Newton pattern search method based on symmetric rank-one update for unconstrained optimization

- Computer Science, MathematicsComput. Math. Appl.
- 2008

This paper proposes a new robust and quickly convergent pattern search method based on an implementation of OCSSR1 (Optimal Conditioning Based Self-Scaling Symmetric Rank-One) algorithm that is competitive in comparison with some other derivative-free methods.

Two minimal positive bases based direct search conjugate gradient methods for computationally expensive functions

- Mathematics, Computer ScienceNumerical Algorithms
- 2011

Two direct search methods for computational expensive functions are proposed based on the minimal positive bases, the Coope–Price’s frame-based direct search framework and a recently developed descent conjugate gradient method to accelerate convergence.

A deterministic algorithm to compute the cosine measure of a finite positive spanning set

- Mathematics, Computer ScienceOptim. Lett.
- 2020

In this paper, a deterministic algorithm to compute the cosine measure of any positive basis or finite positive spanning set is provided and is proven to return the exact value of thecosine measure in finite time.

Optimization by Direct Search: New Perspectives on Some Classical and Modern Methods

- Mathematics, Computer ScienceSIAM Rev.
- 2003

This review begins by briefly summarizing the history of direct search methods and considering the special properties of problems for which they are well suited, then turns to a broad class of methods for which the underlying principles allow general-ization to handle bound constraints and linear constraints.

Uniform simplex of an arbitrary orientation

- Computer Science, MathematicsOptim. Lett.
- 2020

It is proved that a uniform simplex has the greatest normalized volume of any simplex and it is shown how to create a uniform minimal positive basis from a uniformsimplex.

Nonlinear programming by mesh adaptive direct searches 1

- 2005

This paper is intended not as a survey, but as an introduction to some ideas behind the class of mesh adaptive direct search (MADS) methods. Space limitations dictate a brief description of various…

Introduction: Tools and Challenges in Derivative-Free and Blackbox Optimization

- Computer Science
- 2017

In this introductory chapter, a high-level description of optimization, blackbox optimization, and derivative-free optimization is presented, and some basic optimization notation used throughout this book is introduced.

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