# American Institute of Mathematical Sciences

April  2012, 8(2): 429-455. doi: 10.3934/jimo.2012.8.429

## An efficient convexification method for solving generalized geometric problems

 1 Department of Information Management, Fu Jen Catholic University, No.510, Zhongzheng Rd., Xinzhuang Dist., New Taipei City 24205, Taiwan

Received  September 2010 Revised  November 2011 Published  April 2012

Convexification transformation is vital for solving Generalized Geometric Problems (GGP) in global optimization. Björk et al. [3] posited that transforming non-convex signomial terms in a GGP into 1-convex functions is currently the most efficient convexification technique. However, to the best of our knowledge, an efficient convexification method based on the concept of 1-convex functions has not been proposed. To address this research gap, we present a Beta method to maximally improve the efficiency of convexification based on the concept of 1-convex functions, and thereby enhance the accuracy of linearization without increasing the number of break points and binary variables in the piecewise linear function. The Beta method yields an excellent solution quality and computational efficiency. We compare its performance, with that of three existing approaches using four numerical examples. The computational results demonstrate that, in terms of solution quality and computation time, the proposed method outperforms the compared approaches.
Citation: Hao-Chun Lu. An efficient convexification method for solving generalized geometric problems. Journal of Industrial & Management Optimization, 2012, 8 (2) : 429-455. doi: 10.3934/jimo.2012.8.429
##### References:

show all references

##### References:
 [1] Liran Rotem. Banach limit in convexity and geometric means for convex bodies. Electronic Research Announcements, 2016, 23: 41-51. doi: 10.3934/era.2016.23.005 [2] Radu Ioan Boţ, Anca Grad, Gert Wanka. Sequential characterization of solutions in convex composite programming and applications to vector optimization. Journal of Industrial & Management Optimization, 2008, 4 (4) : 767-782. doi: 10.3934/jimo.2008.4.767 [3] Hui Zhang, Jian-Feng Cai, Lizhi Cheng, Jubo Zhu. Strongly convex programming for exact matrix completion and robust principal component analysis. Inverse Problems & Imaging, 2012, 6 (2) : 357-372. doi: 10.3934/ipi.2012.6.357 [4] Jian Gu, Xiantao Xiao, Liwei Zhang. A subgradient-based convex approximations method for DC programming and its applications. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1349-1366. doi: 10.3934/jimo.2016.12.1349 [5] Sophie Guillaume. Evolution equations governed by the subdifferential of a convex composite function in finite dimensional spaces. Discrete & Continuous Dynamical Systems - A, 1996, 2 (1) : 23-52. doi: 10.3934/dcds.1996.2.23 [6] Murat Adivar, Shu-Cherng Fang. Convex optimization on mixed domains. Journal of Industrial & Management Optimization, 2012, 8 (1) : 189-227. doi: 10.3934/jimo.2012.8.189 [7] Nelly Point, Silvano Erlicher. Convex analysis and thermodynamics. Kinetic & Related Models, 2013, 6 (4) : 945-954. doi: 10.3934/krm.2013.6.945 [8] Roland Hildebrand. Barriers on projective convex sets. Conference Publications, 2011, 2011 (Special) : 672-683. doi: 10.3934/proc.2011.2011.672 [9] Bingsheng He, Xiaoming Yuan. Linearized alternating direction method of multipliers with Gaussian back substitution for separable convex programming. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 247-260. doi: 10.3934/naco.2013.3.247 [10] Zhi-Bin Deng, Ye Tian, Cheng Lu, Wen-Xun Xing. Globally solving quadratic programs with convex objective and complementarity constraints via completely positive programming. Journal of Industrial & Management Optimization, 2018, 14 (2) : 625-636. doi: 10.3934/jimo.2017064 [11] Jinchuan Zhou, Changyu Wang, Naihua Xiu, Soonyi Wu. First-order optimality conditions for convex semi-infinite min-max programming with noncompact sets. Journal of Industrial & Management Optimization, 2009, 5 (4) : 851-866. doi: 10.3934/jimo.2009.5.851 [12] Silvia Faggian. Boundary control problems with convex cost and dynamic programming in infinite dimension part II: Existence for HJB. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 323-346. doi: 10.3934/dcds.2005.12.323 [13] Su-Hong Jiang, Min Li. A modified strictly contractive peaceman-rachford splitting method for multi-block separable convex programming. Journal of Industrial & Management Optimization, 2018, 14 (1) : 397-412. doi: 10.3934/jimo.2017052 [14] Qingshan You, Qun Wan, Yipeng Liu. A short note on strongly convex programming for exact matrix completion and robust principal component analysis. Inverse Problems & Imaging, 2013, 7 (1) : 305-306. doi: 10.3934/ipi.2013.7.305 [15] Zhongming Wu, Xingju Cai, Deren Han. Linearized block-wise alternating direction method of multipliers for multiple-block convex programming. Journal of Industrial & Management Optimization, 2018, 14 (3) : 833-855. doi: 10.3934/jimo.2017078 [16] Feng Ma, Jiansheng Shu, Yaxiong Li, Jian Wu. The dual step size of the alternating direction method can be larger than 1.618 when one function is strongly convex. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2020016 [17] Mickaël Crampon. Entropies of strictly convex projective manifolds. Journal of Modern Dynamics, 2009, 3 (4) : 511-547. doi: 10.3934/jmd.2009.3.511 [18] Małgorzata Wyrwas, Dorota Mozyrska, Ewa Girejko. Subdifferentials of convex functions on time scales. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 671-691. doi: 10.3934/dcds.2011.29.671 [19] Lorenzo Brasco, Eleonora Cinti. On fractional Hardy inequalities in convex sets. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4019-4040. doi: 10.3934/dcds.2018175 [20] Gang Li, Lipu Zhang, Zhe Liu. The stable duality of DC programs for composite convex functions. Journal of Industrial & Management Optimization, 2017, 13 (1) : 63-79. doi: 10.3934/jimo.2016004

2018 Impact Factor: 1.025

## Metrics

• HTML views (0)
• Cited by (4)

• on AIMS