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April  2016, 12(2): 739-756. doi: 10.3934/jimo.2016.12.739

## A polynomial-time interior-point method for circular cone programming based on kernel functions

 1 Department of Mathematics, Shanghai University, Shanghai 200444 2 Department of Mathematics, Shanghai University, Shanghai, 200444, China, China

Received  May 2014 Revised  March 2015 Published  June 2015

We present an interior-point method based on kernel functions for circular cone optimization problems, which has been found useful for describing optimal design problems of optimal grasping manipulation for multi-fingered robots. Since the well-known second order cone is a particular circular cone, we first establish an invertible linear mapping between a circular cone and its corresponding second order cone. Then we develop a kernel function based interior-point method to solve circular cone optimization in terms of the corresponding second order cone optimization problem. We derive the complexity bound of the interior-point method and conclude that circular cone optimization is polynomial-time solvable. Finally we illustrate the performance of interior-point method by a real-world quadruped robot example of optimal contact forces taken from the literature [10].
Citation: Yanqin Bai, Pengfei Ma, Jing Zhang. A polynomial-time interior-point method for circular cone programming based on kernel functions. Journal of Industrial and Management Optimization, 2016, 12 (2) : 739-756. doi: 10.3934/jimo.2016.12.739
##### References:
 [1] F. Alizadeh and D. Goldfarb, Second-order cone programming, Mathematical Programming Series B, 95 (2003), 3-51. doi: 10.1007/s10107-002-0339-5. [2] Y. Q. Bai, M. El. Ghami and C. Roos, A comparative study of kernel functions for primal-dual interior-point algorithms in linear optimization, SIAM Journal on Optimization, 15 (2004), 101-128. doi: 10.1137/S1052623403423114. [3] Y. Q. Bai, G. Q. Wang and C. Roos, Primal-dual interior-point algorithms for second-order cone optimization based on kernel functions, Nonlinear Analysis: Theory, Methods & Applications, 70 (2009), 3584-3602. doi: 10.1016/j.na.2008.07.016. [4] Y. Q. Bai, Kernel Function-Based Interior-point Algorithms for Conic Optimization, Science Press, Beijing, 2010. [5] A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms and Engineering Applications, MPS/SIAM Series on Optimization, SIAM Publications, Philadelphia, 2001. doi: 10.1137/1.9780898718829. [6] I. Bomze, Copositive optimization-Recent developments and applications, European Journal of Operational Research, 216 (2012), 509-520. doi: 10.1016/j.ejor.2011.04.026. [7] I. Bomze, M. Dur and C. P. Teo, Copositive optimization, Mathematical Optimization Society Newsletter, Optima 89 (2012), 2-8. doi: 10.1007/978-0-387-74759-0_99. [8] P. H. Borgstrom, M. A. Batalin, G. Sukhatme, et al., Weighted barrier functions for computation of force distributions with friction cone constraints, Robotics and Automation (ICRA), 2010 IEEE International Conference on, IEEE, (2010), 785-792. doi: 10.1109/ROBOT.2010.5509833. [9] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511804441. [10] S. Boyd and B. Wegbreit, Fast computation of optimal contact forces, IEEE Transactions on Robotics, 23 (2007), 1117-1132. [11] S. Burer, On the copositive representation of binary and continuous nonconvex quadratic programs, Mathematical Programming Series A, 120 (2009), 479-495. doi: 10.1007/s10107-008-0223-z. [12] Y. L. Chang, C. Y. Yang and J. S. Chen, Smooth and nonsmooth analyses of vector-valued functions associated with circular cones, Nonlinear Analysis: Theory, Methods and Applications, 85 (2013), 160-173. doi: 10.1016/j.na.2013.01.017. [13] J. S. Chen, X. Chen and P. Tseng, Analysis of nonsmooth vector-valued functions associated with second order cones, Mathematical Programming Series B, 101 (2004), 95-117. doi: 10.1007/s10107-004-0538-3. [14] S. C. Fang and W. X. Xing, Linear Conic Programming: Theory and Applications, Science Press, Beijing, 2013. [15] M. Fukushima, Z. Q. Luo and P. Tseng, Smoothing functions for second-order-cone complementarity problems, SIAM Journal on Optimization, 12 (2001), 436-460. doi: 10.1137/S1052623400380365. [16] O. Güler, Barrier functions in interior point methods, Mathematics of Operations Research, 21 (1996), 860-885. doi: 10.1287/moor.21.4.860. [17] C. H. Ko and J. S. Chen, Optimal grasping manipulation for multifingered robots using semismooth Newton method, em appear in Mathematical Problems in Engineering, 2013 (2013). Available from: http://math.ntnu.edu.tw/~jschen/Publications.html. [18] B. León, A. Morales and J. Sancho-Bru, Robot Grasping Foundations, In From Robot to Human Grasping Simulation, Springer International Publishing, (2014), 15-31. [19] Z. J. Li, S. Z. Sam Ge and S. B. Liu, Contact-force distribution optimization and control for quadruped robots using both gradient and adaptive neural networks, IEEE Transactions on Neural Networks and Learning Systems, 8 (2014), 1460-1473. doi: 10.1109/TNNLS.2013.2293500. [20] Y. Matsukawa and A. Yoshise, A primal barrier function Phase I algorithm for nonsymmetric conic optimization problems, Japan Journal of Industrial and Applied Mathematics, 29 (2012), 499-517. doi: 10.1007/s13160-012-0081-1. [21] A. Nemirovski, Advanced in convex optimization: Conic optimization, In International Congress of Mathematicians, 1 (2006), 413-444. doi: 10.4171/022-1/17. [22] Y. Nesterov, Towards nonsymmetric conic optimization, Optimization Methods and Software, 27 (2012), 893-917. doi: 10.1080/10556788.2011.567270. [23] Y. Nesterov and MJ. Todd, Primal-dual interior-point methods for self-scaled cones, SIAM Journal on Optimization, 8 (1998), 324-364. doi: 10.1137/S1052623495290209. [24] J. Peng, C. Roos and T. Terlaky, New primal-dual algorithms for second-order conic optimization based on self-regular proximities, SIAM Journal on Optimization, 13 (2002), 179-203. doi: 10.1137/S1052623401383236. [25] J. Renegar, A Mathematical View of Interior-Point Methods in Convex Optimization, MPS/SIAM Series on Optimization, SIAM Publications, Philadelphia, 2001. doi: 10.1137/1.9780898718812. [26] A. Skajaa and Y. Ye, Homogeneous interior-point algorithm for nonsymmetric convex conic optimization, To appear mathematical programming, (2014). Available from: http://link.springer.com/journal/10107/onlineFirst/page/1. doi: 10.1007/s10107-014-0773-1. [27] C. J. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai, A new exact penalty function method for continuous inequality constrained optimization problems, Journal of Industrial Management and Optimization, 6 (2010), 895-910. doi: 10.3934/jimo.2010.6.895. [28] C. J. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai, On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem, Journal of Industrial Management and Optimization, 8 (2012), 485-491. doi: 10.3934/jimo.2012.8.485. [29] J. C. Zhou and J. S. Chen, Properties of circular cone and spectral factorization associated with circular cone, Journal of Nonlinear and Convex Analysis, 14 (2014), 807-816.

show all references

##### References:
 [1] F. Alizadeh and D. Goldfarb, Second-order cone programming, Mathematical Programming Series B, 95 (2003), 3-51. doi: 10.1007/s10107-002-0339-5. [2] Y. Q. Bai, M. El. Ghami and C. Roos, A comparative study of kernel functions for primal-dual interior-point algorithms in linear optimization, SIAM Journal on Optimization, 15 (2004), 101-128. doi: 10.1137/S1052623403423114. [3] Y. Q. Bai, G. Q. Wang and C. Roos, Primal-dual interior-point algorithms for second-order cone optimization based on kernel functions, Nonlinear Analysis: Theory, Methods & Applications, 70 (2009), 3584-3602. doi: 10.1016/j.na.2008.07.016. [4] Y. Q. Bai, Kernel Function-Based Interior-point Algorithms for Conic Optimization, Science Press, Beijing, 2010. [5] A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms and Engineering Applications, MPS/SIAM Series on Optimization, SIAM Publications, Philadelphia, 2001. doi: 10.1137/1.9780898718829. [6] I. Bomze, Copositive optimization-Recent developments and applications, European Journal of Operational Research, 216 (2012), 509-520. doi: 10.1016/j.ejor.2011.04.026. [7] I. Bomze, M. Dur and C. P. Teo, Copositive optimization, Mathematical Optimization Society Newsletter, Optima 89 (2012), 2-8. doi: 10.1007/978-0-387-74759-0_99. [8] P. H. Borgstrom, M. A. Batalin, G. Sukhatme, et al., Weighted barrier functions for computation of force distributions with friction cone constraints, Robotics and Automation (ICRA), 2010 IEEE International Conference on, IEEE, (2010), 785-792. doi: 10.1109/ROBOT.2010.5509833. [9] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511804441. [10] S. Boyd and B. Wegbreit, Fast computation of optimal contact forces, IEEE Transactions on Robotics, 23 (2007), 1117-1132. [11] S. Burer, On the copositive representation of binary and continuous nonconvex quadratic programs, Mathematical Programming Series A, 120 (2009), 479-495. doi: 10.1007/s10107-008-0223-z. [12] Y. L. Chang, C. Y. Yang and J. S. Chen, Smooth and nonsmooth analyses of vector-valued functions associated with circular cones, Nonlinear Analysis: Theory, Methods and Applications, 85 (2013), 160-173. doi: 10.1016/j.na.2013.01.017. [13] J. S. Chen, X. Chen and P. Tseng, Analysis of nonsmooth vector-valued functions associated with second order cones, Mathematical Programming Series B, 101 (2004), 95-117. doi: 10.1007/s10107-004-0538-3. [14] S. C. Fang and W. X. Xing, Linear Conic Programming: Theory and Applications, Science Press, Beijing, 2013. [15] M. Fukushima, Z. Q. Luo and P. Tseng, Smoothing functions for second-order-cone complementarity problems, SIAM Journal on Optimization, 12 (2001), 436-460. doi: 10.1137/S1052623400380365. [16] O. Güler, Barrier functions in interior point methods, Mathematics of Operations Research, 21 (1996), 860-885. doi: 10.1287/moor.21.4.860. [17] C. H. Ko and J. S. Chen, Optimal grasping manipulation for multifingered robots using semismooth Newton method, em appear in Mathematical Problems in Engineering, 2013 (2013). Available from: http://math.ntnu.edu.tw/~jschen/Publications.html. [18] B. León, A. Morales and J. Sancho-Bru, Robot Grasping Foundations, In From Robot to Human Grasping Simulation, Springer International Publishing, (2014), 15-31. [19] Z. J. Li, S. Z. Sam Ge and S. B. Liu, Contact-force distribution optimization and control for quadruped robots using both gradient and adaptive neural networks, IEEE Transactions on Neural Networks and Learning Systems, 8 (2014), 1460-1473. doi: 10.1109/TNNLS.2013.2293500. [20] Y. Matsukawa and A. Yoshise, A primal barrier function Phase I algorithm for nonsymmetric conic optimization problems, Japan Journal of Industrial and Applied Mathematics, 29 (2012), 499-517. doi: 10.1007/s13160-012-0081-1. [21] A. Nemirovski, Advanced in convex optimization: Conic optimization, In International Congress of Mathematicians, 1 (2006), 413-444. doi: 10.4171/022-1/17. [22] Y. Nesterov, Towards nonsymmetric conic optimization, Optimization Methods and Software, 27 (2012), 893-917. doi: 10.1080/10556788.2011.567270. [23] Y. Nesterov and MJ. Todd, Primal-dual interior-point methods for self-scaled cones, SIAM Journal on Optimization, 8 (1998), 324-364. doi: 10.1137/S1052623495290209. [24] J. Peng, C. Roos and T. Terlaky, New primal-dual algorithms for second-order conic optimization based on self-regular proximities, SIAM Journal on Optimization, 13 (2002), 179-203. doi: 10.1137/S1052623401383236. [25] J. Renegar, A Mathematical View of Interior-Point Methods in Convex Optimization, MPS/SIAM Series on Optimization, SIAM Publications, Philadelphia, 2001. doi: 10.1137/1.9780898718812. [26] A. Skajaa and Y. Ye, Homogeneous interior-point algorithm for nonsymmetric convex conic optimization, To appear mathematical programming, (2014). Available from: http://link.springer.com/journal/10107/onlineFirst/page/1. doi: 10.1007/s10107-014-0773-1. [27] C. J. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai, A new exact penalty function method for continuous inequality constrained optimization problems, Journal of Industrial Management and Optimization, 6 (2010), 895-910. doi: 10.3934/jimo.2010.6.895. [28] C. J. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai, On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem, Journal of Industrial Management and Optimization, 8 (2012), 485-491. doi: 10.3934/jimo.2012.8.485. [29] J. C. Zhou and J. S. Chen, Properties of circular cone and spectral factorization associated with circular cone, Journal of Nonlinear and Convex Analysis, 14 (2014), 807-816.
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