Globally convergent homotopy method for designing piecewise linear deterministic contractual function
Zhichuan Zhu, Bo Yu, Li Yang · Journal of Industrial and Management Optimization · 2013
In this paper, to design a piecewise linear contractual function, weconsider to solve the single-level nonconvex programming withintegral operator which is equivalent to the principal-agent bilevelprogramming model with continuous distribution. A modifiedconstraint shifting homotopy method for solving theKarush-Kuhn-Tucker system of the discrete nonconvex programming isproposed and the global convergence from any initial point inshifted feasible set is proven under some mild conditions. A simplehomotopy path tracing algorithm is given and is implemented inMatlab. For some typical risk averse utility functions and thetypical distribution functions which simultaneously satisfy monotonelikelihood ratio condition and convexity of the distributionfunction condition, some numerical tests to design the piecewiselinear contract are done by our homotopy method as well as by using fmincon in Matlab, LOQO and MINOS and, as a comparison, thepiecewise constant contracts are also designed by solving thesingle-level nonconvex programming which is equivalent to theprincipal-agent bilevel programming model with correspondingdiscrete distributions. Numerical tests show that: to design apiecewise linear contract, which is much better than a piecewiseconstant contract, it needs only to solve a much lower dimensionaloptimization problem and hence needs much less computing time.Numerical experiences also show that the modified constraintshifting homotopy method is feasible and robust.