Convergent bounds for the range of multivariate polynomials
Proceedings of the International Symposium on interval mathematics on Interval mathematics 1985
Algorithms for polynomials in Bernstein form
Computer Aided Geometric Design
New computer methods for global optimization
New computer methods for global optimization
Arbitrarily high degree elevation of Be´zier representations
Computer Aided Geometric Design
Bezier and B-Spline Techniques
Bezier and B-Spline Techniques
Convexification and Global Optimization in Continuous And
Convexification and Global Optimization in Continuous And
Lower bound functions for polynomials
Journal of Computational and Applied Mathematics
Safe bounds in linear and mixed-integer linear programming
Mathematical Programming: Series A and B
Safe and tight linear estimators for global optimization
Mathematical Programming: Series A and B
Deterministic Global Optimization: Theory, Methods and (NONCONVEX OPTIMIZATION AND ITS APPLICATIONS Volume 37) (Nonconvex Optimization and Its Applications)
Mathematical Modeling And Global Optimization
Mathematical Modeling And Global Optimization
A comparison of methods for the computation of affine lower bound functions for polynomials
COCOS'03 Proceedings of the Second international conference on Global Optimization and Constraint Satisfaction
An efficient algorithm for range computation of polynomials using the Bernstein form
Journal of Global Optimization
Polytope-based computation of polynomial ranges
Computer Aided Geometric Design
Edge Detection by Adaptive Splitting II. The Three-Dimensional Case
Journal of Scientific Computing
The Bernstein polynomial basis: A centennial retrospective
Computer Aided Geometric Design
Formalization of Bernstein Polynomials and Applications to Global Optimization
Journal of Automated Reasoning
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A new method for the representation and computation of Bernstein coefficients of multivariate polynomials is presented. It is known that the coefficients of the Bernstein expansion of a given polynomial over a specified box of interest tightly bound the range of the polynomial over the box. The traditional approach requires that all Bernstein coefficients are computed, and their number is often very large for polynomials with moderately-many variables. The new technique detailed represents the coefficients implicitly and uses lazy evaluation so as to render the approach practical for many types of non-trivial sparse polynomials typically encountered in global optimization problems; the computational complexity becomes nearly linear with respect to the number of terms in the polynomial, instead of exponential with respect to the number of variables. These range-enclosing coefficients can be employed in a branch-and-bound framework for solving constrained global optimization problems involving polynomial functions, either as constant bounds used for box selection, or to construct affine underestimating bound functions. If such functions are used to construct relaxations for a global optimization problem, then sub-problems over boxes can be reduced to linear programming problems, which are easier to solve. Some numerical examples are presented and the software used is briefly introduced.