Abstract
A novel method for the convex underestimation of univariate functions is presented in this paper. The method is based on a piecewise application of the well-known αBB underestimator, which produces an overall underestimator that is piecewise convex. Subsequently, two algorithms are used to identify the linear segments needed for the construction of its \({{\mathcal C}^1}\)-continuous convex envelope, which is itself a valid convex underestimator of the original function. The resulting convex underestimators are very tight, and their tightness benefits from finer partitioning of the initial domain. It is theoretically proven that there is always some finite level of partitioning for which the method yields the convex envelope of the function of interest. The method was applied on a set of univariate test functions previously presented in the literature, and the results indicate that the method produces convex underestimators of high quality in terms of both lower bound and tightness over the whole domain under consideration.
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References
Adjiman C.S. and Floudas C.A. (1996). Rigorous convex underestimators for general twice-differentiable problems. J. Glob. Optim. 9: 23–40
Adjiman C.S., Dallwig S., Floudas C.A. and Neumaier A. (1998a). A global optimization method, αBB, for general twice-differentiable constrained NLPs I. Theoretical advances. Comput. Chem. Eng. 22: 1137–1158
Adjiman C.S., Androulakis I.P. and Floudas C.A. (1998b). A global optimization method, αBB, for general twice-differentiable constrained NLPs II. Implementation and computational results. Comput. Chem. Eng. 22: 1159–1179
Akrotirianakis I.G. and Floudas C.A. (2004a). A new class of improved convex underestimators for twice continuously differentiable constrained NLPs. J. Glob. Optim. 30: 367–390
Akrotirianakis I.G. and Floudas C.A. (2004b). Computational experience with a new class of convex underestimators: box-constrained NLP problems. J. Glob Optim. 29: 249–264
Al-Khayyal F.A. and Falk J.E. (1983). Jointly constrained biconvex programming. Math. Oper. Res. 8: 273–286
Androulakis I.P., Maranas C.D. and Floudas C.A. (1995). αBB: a global optimization method for general constrained nonconvex problems. J. Glob. Optim. 7: 337–363
Caratzoulas S. and Floudas C.A. (2005). Trigonometric convex underestimator for the base functions in Fourier space. J. Optim. Theory Appl. 124: 339–362
Casado L.G., Martinez J.A., Garcia I. and Sergeyev Y.D. (2003). New interval analysis support functions using gradient information in a global minimization algorithm. J. Glob. Optim. 25: 345–362
Esposito W.R. and Floudas C.A. (1998). Global optimization in parameter estimation of nonlinear algebraic models via the error-in-variables approach. Indus. Eng. Chem. Res. 35: 1841–1858
Floudas, C.A.: Deterministic Global Optimization: Theory, Algorithms and Applications. Kluwer Academic Publishers (2000)
Floudas C.A. and Pardalos P.M. (1995). Preface. J. Glob. Optim. 1: 113
Floudas, C.A., Pardalos, P.M.: Frontiers in Global Optimization. Kluwer Academic Publishers (2003)
Floudas C.A. (2005). Research challenges, opportunities and synergism in systems engineering and computational biology. AIChE J. 51: 1872–1884
Floudas C.A., Akrotirianakis I.G., Caratzoulas S., Meyer C.A. and Kallrath J. (2005). Global optimization in the 21st century: advances and challenges. Comput. Chem. Eng. 29: 1185–1202
Floudas, C.A., Kreinovich, V.: Towards optimal techniques for solving global optimization problems: symmetry-based approach. In: Torn A., Zilinskas J. (eds.) Models and Algorithms for Global Optimization, pp. 21–42 Springer (2007a)
Floudas C.A. and Kreinovich V. (2007b). On the functional form of convex understimators for twice continuously differentiable functions. Optim. Lett. 1: 187–192
Gounaris C.E. and Floudas C.A. (2008). Tight convex underestimators for \({{\mathcal C}^2}\)-continuous functions: II. Multivariate functions. J. Glob. Optim. DOI: 10.1007/s10898-008-9288-8
Gümüş Z.H. and Floudas C.A. (2001). Global optimization of nonlinear bilevel programming problems. J. Glob. Optim. 20: 1–31
Harding S.T., Maranas C.D., McDonald C.M. and Floudas C.A. (1997). Locating all homogeneous azeotropes in multicomponent mixtures. Ind. Eng. Chem. Res. 36: 160–178
Hertz D., Adjiman C.S. and Floudas C.A. (1999). Two results on bounding the roots of interval polynomials. Comput. Chem. Eng. 23: 1333–1339
Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization, 2nd edn. Kluwer Academic Publishers (2000)
Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches, 3rd edn. Springer (2003)
Liberti L. and Pantelides C.C. (2003). Convex envelopes of monomials of odd degree. J. Glob. Optim. 25: 157–168
Maranas C.D. and Floudas C.A. (1994). Global minimum potential energy conformations of small molecules. J. Glob. Optim. 4: 135–170
Maranas C.D. and Floudas C.A. (1995). Finding all solutions of nonlinearly constrained systems of equations. J. Glob. Optim. 7: 143–182
Maranas C.D., McDonald C.M., Harding S.T. and Floudas C.A. (1996). Locating all azeotropes in homogeneous azeotropic systems. Comput. Chem. Eng. 20: S413–S418
McCormick G.P. (1976). Computability of global solutions to factorable nonconvex programs: Part I—Convex underestimating problems. Math. Program. 10: 147–175
McDonald C.M. and Floudas C.A. (1994). Decomposition based and branch and bound global optimization approaches for the phase equilibrium problem. J. Glob. Optim. 5: 205–251
McDonald C.M. and Floudas C.A. (1995). Global optimization for the phase and chemical equilibrium problem: application to the NRTL equation. Comput. Chem. Eng. 19: 1111–1141
McDonald C.M. and Floudas C.A. (1997). GLOPEQ: a new computational tool for the phase and chemical equilibrium problem. Comput. Chem. Eng. 21: 1–23
Meyer, C.A., Floudas, C.A.: Trilinear monomials with positive or negative domains: facets of the convex and concave envelopes. In: Floudas C.A., Pardalos P.M. (eds.) Frontiers in Global Optimization. Kluwer Academic Publishers (2003)
Meyer C.A. and Floudas C.A. (2004). Trilinear monomials with mixed sign domains: facets of the convex and concave envelopes. J. Glob. Optim. 29: 125–155
Meyer C.A. and Floudas C.A. (2005). Convex underestimation of twice continuously differentiable functions by piecewise quadratic perturbation: spline αBB underestimators. J. Glob. Optim. 32: 221–258
Meyer C.A. and Floudas C.A. (2006). Global optimization of a combinatorially complex generalized pooling problem.. AIChE J. 52: 1027–1037
O’Rourke J.: Computational Geometry in C 2nd edn. Cambridge University Press (1998)
Ryoo H.S. and Sahinidis N.V. (2001). Analysis of bounds for multilinear functions. J. Glob. Optim. 19: 403–424
Sherali, H.D., Adams, W.P.: Reformulation-Linearization Techniques in Discrete and Continuous Optimization. Kluwer Academic Publishers (1999)
Tawarmalani M. and Sahinidis N.V. (2001). Semidefinite relaxations of fractional programs via novel convexification techniques. J. Glob. Optim. 20: 137–158
Tawarmalani, M., Sahinidis, N.V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming. Kluwer Academic Publishers (2002a)
Tawarmalani M. and Sahinidis N.V. (2002b). Convex extensions and envelopes of lower semi-continuous functions. Math. Program. 93: 247–263
Zabinsky, Z.B.: Stochastic Adaptive Search for Global Optimization. Kluwer Academic Publishers (2003)
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Gounaris, C.E., Floudas, C.A. Tight convex underestimators for \({{\mathcal C}^2}\)-continuous problems: I. univariate functions. J Glob Optim 42, 51–67 (2008). https://doi.org/10.1007/s10898-008-9287-9
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DOI: https://doi.org/10.1007/s10898-008-9287-9