The globally convexized filled functions for global optimization
Applied Mathematics and Computation
Testing Unconstrained Optimization Software
ACM Transactions on Mathematical Software (TOMS)
Analysis of Bounds for Multilinear Functions
Journal of Global Optimization
Global Optimization of Nonlinear Bilevel Programming Problems
Journal of Global Optimization
Semidefinite Relaxations of Fractional Programs via Novel Convexification Techniques
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
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
Convex envelopes for edge-concave functions
Mathematical Programming: Series A and B
Journal of Global Optimization
Deterministic Global Optimization: Theory, Methods and (NONCONVEX OPTIMIZATION AND ITS APPLICATIONS Volume 37) (Nonconvex Optimization and Its Applications)
Introduction to Global Optimization (Nonconvex Optimization and Its Applications)
Introduction to Global Optimization (Nonconvex Optimization and Its Applications)
Tight convex underestimators for $${{\mathcal C}^2}$$-continuous problems: I. univariate functions
Journal of Global Optimization
Frontiers in Global Optimization
Frontiers in Global Optimization
Tight convex underestimators for $${{\mathcal C}^2}$$-continuous problems: I. univariate functions
Journal of Global Optimization
A review of recent advances in global optimization
Journal of Global Optimization
Convex relaxation for solving posynomial programs
Journal of Global Optimization
On convex relaxations of quadrilinear terms
Journal of Global Optimization
Convergence rate of McCormick relaxations
Journal of Global Optimization
Convergence analysis of Taylor models and McCormick-Taylor models
Journal of Global Optimization
New methods for calculating $$\alpha $$BB-type underestimators
Journal of Global Optimization
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In Part I (Gounaris, C.E., Floudas, C.A.: Tight convex understimators for $${\mathcal {C}^{2}}$$ -continuous functions: I: Univariate functions. J. Global Optim. (2008). doi: 10.007/s10898-008-9287-9 ), we introduced a novel approach for the underestimation of univariate functions which was based on a piecewise application of the well-known 驴BB underestimator. The resulting underestimators were shown to be very tight and, in fact, can be driven to coincide with the convex envelopes themselves. An approximation by valid linear supports, resulting in piecewise linear underestimators was also presented. In this paper, we demonstrate how one can make use of the high quality results of the approach in the univariate case so as to extend its applicability to functions with a higher number of variables. This is achieved by proper projections of the multivariate 驴BB underestimators into select two-dimensional planes. Furthermore, since our method utilizes projections into lower-dimensional spaces, we explore ways to recover some of the information lost in this process. In particular, we apply our method after having transformed the original problem in an orthonormal fashion. This leads to the construction of even tighter underestimators, through the accumulation of additional valid linear cuts in the relaxation.