On using estimates of Lipschitz constants in global optimization
Journal of Optimization Theory and Applications
Mathematical Programming: Series A and B
A deterministic algorithm for global optimization
Mathematical Programming: Series A and B
ACM Transactions on Mathematical Software (TOMS)
Global one-dimensional optimization using smooth auxiliary functions
Mathematical Programming: Series A and B
Journal of Global Optimization
Trilinear Monomials with Mixed Sign Domains: Facets of the Convex and Concave Envelopes
Journal of Global Optimization
Computational Experience with a New Class of Convex Underestimators: Box-constrained NLP Problems
Journal of Global Optimization
Journal of Global Optimization
Deterministic Global Optimization: Theory, Methods and (NONCONVEX OPTIMIZATION AND ITS APPLICATIONS Volume 37) (Nonconvex Optimization and Its Applications)
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In this paper, a new global optimization method is proposed for an optimization problem with twice-differentiable objective and constraint functions of a single variable. The method employs a difference of convex underestimator and a convex cut function, where the former is a continuous piecewise concave quadratic function, and the latter is a convex quadratic function. The main objectives of this research are to determine a quadratic concave underestimator that does not need an iterative local optimizer to determine the lower bounding value of the objective function and to determine a convex cut function that effectively detects infeasible regions for nonconvex constraints. The proposed method is proven to have a finite 驴-convergence to locate the global optimum point. The numerical experiments indicate that the proposed method competes with another covering method, the index branch-and-bound algorithm, which uses the Lipschitz constant.