Exponential transformation
Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49), Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021
Introduction
Exponential transformations are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems.
Geometric Programming
Theory & Methodology
Exponential transformation begins with a posynominal noncovex function of the form [1] :
$ f(x) = \sum_{k=1}^N c_k{x_1}^{a_1k}{x_2}^{a_2k}....{x_n}^{a_nk} $
where $ c_k \geq 0 $ and $ x_n \geq0 $
A transformation of $ x_n = e^u_i $ is applied reference https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf
The transformed function is presented as:
$ f(u) = \sum_{k=1}^N c_k{{e_1}^{u}{a_1k}}{e_2}^{a_2k}....{e_n}^{a_nk} $
Proof
Numerical Example
$ {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4 $ Reformulating to exponents: $ {x_1^3}*{x_2^-4} + {x_1^2} + {x_2^(\fract{2}{3}}} $
Substituting $ x_1 = e^{u_1}, x_2 = e^{u_2} $ $ {{e^u_1}}^3 $
$ $ $ $ $ $
Applications
Exponential transformation can be used for convexification of any Geometric NLP or MINLP that meet the criteria of equation (1). This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation.
Example of Convexification application
Proof of convexity with positive definite test of Hessian
https://www.princeton.edu/~chiangm/gp.pdf
pROOF THAT CHANGING IT DOESNT CHANGE THE BOUNDS OF THE PROBLEM https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf
Example:
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf
Quadratic Geometric Programming
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf
Conclusion
Exponential transformation is a powerful method to convexify Geometric NLP/MINLP to simplify the solution approach.