CVODE, a stiff/nonstiff ODE solver in C
Computers in Physics
Deterministic Global Optimization in Nonlinear Optimal Control Problems
Journal of Global Optimization
A Rigorous Global Optimization Algorithm for Problems with Ordinary Differential Equations
Journal of Global Optimization
Convex Envelopes of Monomials of Odd Degree
Journal of Global Optimization
Convexification and Global Optimization in Continuous And
Convexification and Global Optimization in Continuous And
Outer approximation algorithms for separable nonconvex mixed-integer nonlinear programs
Mathematical Programming: Series A and B
Methods and Applications of Interval Analysis (SIAM Studies in Applied and Numerical Mathematics) (Siam Studies in Applied Mathematics, 2.)
Journal of Global Optimization
Global Optimization with Nonlinear Ordinary Differential Equations
Journal of Global Optimization
Bounding the Solutions of Parameter Dependent Nonlinear Ordinary Differential Equations
SIAM Journal on Scientific Computing
Validated solutions of initial value problems for parametric ODEs
Applied Numerical Mathematics
McCormick-Based Relaxations of Algorithms
SIAM Journal on Optimization
Discretize-then-relax approach for convex/concave relaxations of the solutions of parametric ODEs
Applied Numerical Mathematics
Improved relaxations for the parametric solutions of ODEs using differential inequalities
Journal of Global Optimization
Journal of Global Optimization
Global optimization of bounded factorable functions with discontinuities
Journal of Global Optimization
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Convex and concave relaxations are used extensively in global optimization algorithms. Among the various techniques available for generating relaxations of a given function, McCormick's relaxations are attractive due to the recursive nature of their definition, which affords wide applicability and easy implementation computationally. Furthermore, these relaxations are typically stronger than those resulting from convexification or linearization procedures. This article leverages the recursive nature of McCormick's relaxations to define a generalized form which both affords a new framework within which to analyze the properties of McCormick's relaxations, and extends the applicability of McCormick's technique to challenging open problems in global optimization. Specifically, relaxations of the parametric solutions of ordinary differential equations are considered in detail, and prospects for relaxations of the parametric solutions of nonlinear algebraic equations are discussed. For the case of ODEs, a complete computational procedure for evaluating convex and concave relaxations of the parametric solutions is described. Through McCormick's composition rule, these relaxations may be used to construct relaxations for very general optimal control problems.