# Research

**Current research paper: **

Title: Solving parametric linear programming by partition of the parametric space — only one of b and c being parametric.

Abstract: We propose an algorithm for solving parametric linear programming (PLP). The solution of PLP is expressed on a partition of the parametric space. The partition consists of a finite number of cells. Each cell is a polytope in the parametric space. We find the closed form of these polytopes, namely these cells, explicitly in terms of PLP coefficients. This closed form is one of the results acquired through a long study starting from the paper

Gongyun Zhao, “Representing the space of linear programs as the Grassmann manifold”, {Math Programming}, 121 (2010) 2, 353-386.

By virtue of this closed form, all we need to do for constructing the partition is to compute extreme points and rays of a number of polytopes in the parametric space.

A package of codes can be downloaded from

https://blog.nus.edu.sg/zhaogongyun/files/2020/03/PLP1-web.zip

**Research Interests**

- Linear and nonlinear programming;
- Stochastic programming;
- Airline revenue management and pricing;
- Game theory;
- Parametric linear programming.

**Publications**

*Mathematical Programming*, 121 (2010)2, 353-386.- Xinwei Liu, Kim-Chuan Toh and Gongyun Zhao, “On implementation of a log-barrier progressive hedging method for multistage stochastic programs”,
*Journal of Computational and Applied Mathematics,*234(2010) 579-592. - Wanmei Soon, Jieping Zhang and Gongyun Zhao, “Complementarity demand functions and pricing models for multi-product markets”, European Journal of Applied Mathematics, 20(2009)5, 399-430.
- Fanwen Meng, Gongyun Zhao, Mark Goh and Robert De Souza, “Lagrangian-dual Functions and Moreau-Yosida Regularization”,
*SIAM J Optimization,*19(2008)1, 39-61. - Chee-Khian Sim and Gongyun Zhao, “Asymptotic Behavior of HKM Paths in Interior Point Method for Monotone Semidefinite Linear Complementarity Problem: General Theory”,
*Journal of Optimization Theory and Applications.*137 (2008) 1, 11-25. - John Birge and Gongyun Zhao, “Successive Linear Approximation Solution of Infinite Horizon Dynamic Stochastic Programs”,
*SIAM**J Optimization.*18 (2007) 4, 1165-1186. - Chee-Khian Sim and Gongyun Zhao, “Underlying Paths in Interior Point Method for Monotone Semidefinite Linear Complementarity Problem”,
*Math Programming,*110 (2007) 3, 475-499. - Zheng-Hai Huang, Defeng Sun and Gongyun Zhao, “A Smoothing Newton-Type Algorithm of Stronger Convergence for the Quadratically Constrained Convex Quadratic Programming”
*Computational Optimization and Applications,*35 (2006) 2, 199-237. - Chee-Khian Sim and Gongyun Zhao, “A Note on Treating Second Order Cone Problem as a Special Case of Semidefinite Problem”,
*Math Programming,*102 (2005) 3, 609-613. - Fanwen Meng, Defeng Sun and Gongyun Zhao, “Semismoothness of Solutions to Generalized Equations and the Moreau-Yosida Regularization”
*Math Programming,*104 (2005) 561–581. - Gongyun Zhao, “A Lagrangian dual method with self-concordant barrier for multi-stage stochastic convex nonlinear programming,”
*Math. Programming*102 (2005) 1, 1-24. - S.K. Chua, K.C. Toh and G. Zhao, “An analytic center cutting plane method with deep cuts for semidefinite feasibility problems”,
*Journal of Optimization Theory and Applications*, 123 (2004), pp. 291–318. - F. Meng, R. Tan and G. Zhao, “A superlinearly convergent algorithm for large scale multi-stage stochastic nonlinear programming”,
*International Journal of Computational Engineering Science,*5 (2004) 327-344. - Guanglu Zhou, Kim-Chuan Toh and Gongyun Zhao, “Convergence analysis of an infeasible interior point algorithm based on a regularized central path for linear complementarity problems”,
*Computational Optimization and Applications*, 27 (2004) 3, 269-283. - Fanwen Meng and Gongyun Zhao, “On second-order properties of the Moreau-Yosida regularization for constrained nonsmooth convex programs”,
*Numerical Functional Analysis and Optimization.*25 (2004) pp.515-530. - Xinwei Liu and Gongyun Zhao, “A decomposition method based on SQP for a class of multi-stage stochastic nonlinear programs,”
*SIAM J. Optimization,*14 (2003) 1, 200-222. - Kali Rath and Gongyun Zhao, “Nonminimal product differentiation as a bargaining outcome,”
*Games and Economic Behavior*, 42 (2003) 267-280. - Kim-Chuan Toh, Gongyun Zhao and Jie Sun, “A multiple-cut analytic center cutting plane method for semidefinite feasibility problems,”
*SIAM J Optimization*, 12 (2002) 4, pp. 1126-1146. - Jie Sun, Kim-Chuan Toh and Gongyun Zhao, “An analytic center cutting plane method for semidefinite feasibility problems,”
*Mathematics of Operations Research*. 27 (2002) 2, pp332-346. - Kali P. Rath and Gongyun Zhao, “Two stage equilibrium and product choice with elastic demand,”
*International Journal of Industrial Organization*, 19 (2001) 1441-1455. - Gongyun Zhao, “A Log-barrier method with Benders decomposition for solving two-stage stochastic linear programs,”
*Mathematical Programming*, 90 (2001) 507-536. - J. Sun and G. Zhao, “A quadratically convergent polynomial long-step algorithm for a class of nonlinear monotone complementarity problems”,
*Optimization*, 48 (2000) 453-475. - Gongyun Zhao and Jie Sun, “On the rate of local convergence of high-order infeasible-path-following algorithms for $P_*$-LCP with or without strictly complementary solutions,”
*Computational Optimization and Applications*, 14 (1999) 293-307. - Gongyun Zhao, “Interior point methods with decomposition for solving large scale linear programs,”
*Journal of Optimization Theory and Applications*, 102 (1999) 169-192. - J. Sun and G. Zhao, “The quadratic convergence of a long-step interior point method for nonlinear monotone variational inequality problems based on invariance conditions of range spaces,”
*Journal of Optimization Theory and Applications*, 97 (1998) 471-491. - J. Sun and G. Zhao, “Global linear and local quadratic convergence of a long-step adaptive-mode interior point method for some monotone variational inequality problems”,
*SIAM J. Optimization*, 8 (1998) 123-139. - G. Sonnevend, J. Stoer and G. Zhao, “Subspace methods for solving linear programming problems”,
*Pure Mathematics and Applications*, 9 (1998) 193-212. - Gongyun Zhao, “Interior point algorithms for linear complementarity problems based on large neighborhoods of the central path”,
*SIAM Journal on Optimization*, 8 (1998) 397-413. - J. Sun and J. Zhu and G. Zhao, “A predictor-corrector algorithm for a class of nonlinear saddle point problems”,
*SIAM Journal on Control and Optimization*, 35 (1997) 532-551. - Gongyun Zhao, and Jishan Zhu, “The curvature integral and the complexity of linear complementarity problems,”
*Mathematical Programming*70 (1996) 107-122. - Gongyun Zhao, “On the relationship between the curvature integral and the complexity of path-following methods in linear programming”,
*SIAM Journal on Optimization*, 6 (1996) 57-73. - G. Zhao, J. Sun and J. Zhu, “A primal-dual affine scaling algorithm with necessary centering as a safeguard”,
*Optimization*, 35 (1995) 333-343. - Gongyun Zhao, “On the choice of parameters for power-series interior point algorithms in linear programming”,
*Mathematical Programming*68 (1995) 49-71. - G. Zhao and J. Zhu, “Analytical properties of the central trajectory in interior point methods”, in:
*Advances in Optimization and Approximation*, D. Du and J. Sun (Eds), Kluwer Academic Publishers, The Netherlands, (1994) 362-375. - G. Zhao and J. Stoer, “Estimating the complexity of path-following methods for solving linear programs by curvature integrals”,
*Applied Math. and Optimizations*27 (1993) 86-103. - G. Sonnevend, J. Stoer and G. Zhao, “On the complexity of following the central path of linear programs by linear extrapolation II”,
*Mathematical Programming*52 (1991) 527-553. - G. Sonnevend, J. Stoer and G. Zhao, “On the complexity of following the central path by linear extrapolation in linear programs”, in:
*Methods of Operations Research*, Proceedings of the 14th Symposium on Operations Research, U. Rieder and P. Kleinschmidt (Eds). Germany, (1989) 19-31. - Gongyun Zhao, “Diagonalization of singularly perturbed linear ordinary differential equations with multi-parameters”,
*Journal of Xiamen University (national science)*, 26 (1987) 5, 517-524.