Optimization can involve differential equations , for instance in the formulation of constraints. Interval analysis provides methods for computing interval-valued functions, for example polynomials with interval coefficients, guaranteed to contain solutions to differential equations. Methods have been developed for initial and boundary value problems for both ordinary (ODE) and partial (PDE) differential equations [1]–[32].
For the initial value problem in ODEs, the Cauchy—Peano approach of classical analysis can be made into a constructive method using interval analysis. With interval arithmetic and interval extensions to standard functions, we can computationally verify sufficient conditions for existence of solutions, as well as construct upper and lower bounds on solutions. The techniques of automatic differentiation provide for efficient use of and (using interval arithmetic) bounding of remainder terms in Taylor series expansions, making interval Taylor seriesan effective...
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Moore, R.E. (2001). Automatic differentiation: Point and interval; Automatic differentiation: Point and interval Taylor operators; Bounding derivative ranges; Global optimization: Application to phase equilibrium problems; Interval analysis: Application to chemical engineering design problems; Interval analysis: Eigenvalue bounds of interval matrices; Interval analysis: Intermediate terms; Interval analysis: Nondifferentiable problems; Interval analysis: Parallel methods for global optimization; Interval analysis: Subdivision directions in interval branch and bound methods; Interval analysis: Systems of nonlinear equations; Interval analysis: Unconstrained and constrained optimization; Interval analysis: Verifying feasibility; Interval constraints; Interval fixed point theory; Interval global optimization; Interval linear systems; Interval Newton methods INTERVAL ANALYSIS: DIFFERENTIAL EQUATIONS . In: Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/0-306-48332-7_221
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DOI: https://doi.org/10.1007/0-306-48332-7_221
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