Algorithms of bilinear programming

In this article, we discuss an alternative method for deriving conservative approximation models for two-stage robust optimization problems. The method mainly relies on a linearization scheme employed in bilinear programming; therefore, we will say that it gives rise to the linearized robust counterpart models. We identify a close relation between this linearized robust counterpart model and the popular affinely adjustable robust counterpart model. We also describe methods of modifying both types of models to make these approximations less conservative. These methods are heavily inspired by the use of valid linear and conic inequalities in the linearization process for bilinear models. We finally demonstrate how to employ this new scheme in location-transportation and multi-item newsvendor problems to improve the numerical efficiency and performance guarantees of robust optimization.

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Safety Verification of Nonlinear Hybrid Systems Based on Bilinear Programming

IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems ◽

10.1109/tcad.2018.2858383 ◽

2018 ◽

Vol 37 (11) ◽

pp. 2768-2778 ◽

Cited By ~ 2

Author(s):

Yifan Zhang ◽

Zhengfeng Yang ◽

Wang Lin ◽

Huibiao Zhu ◽

Xin Chen ◽

...

Keyword(s):

Hybrid Systems ◽

Bilinear Programming ◽

Safety Verification ◽

Nonlinear Hybrid

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A Global Optimization Algorithm for Sum of Linear Ratios Problem

Journal of Applied Mathematics ◽

10.1155/2013/276245 ◽

2013 ◽

Vol 2013 ◽

pp. 1-7 ◽

Cited By ~ 7

Author(s):

Yuelin Gao ◽

Siqiao Jin

Keyword(s):

Global Optimization ◽

Lower Bound ◽

Programming Problem ◽

Branch And Bound ◽

Numerical Experiments ◽

Original Problem ◽

Bilinear Programming ◽

Global Optimization Algorithm ◽

Product Function ◽

Optimal Value

We equivalently transform the sum of linear ratios programming problem into bilinear programming problem, then by using the linear characteristics of convex envelope and concave envelope of double variables product function, linear relaxation programming of the bilinear programming problem is given, which can determine the lower bound of the optimal value of original problem. Therefore, a branch and bound algorithm for solving sum of linear ratios programming problem is put forward, and the convergence of the algorithm is proved. Numerical experiments are reported to show the effectiveness of the proposed algorithm.

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