# A modified homotopy method for solving the principal-agent bilevel programming problem

## Abstract

In this paper, the principal-agent bilevel programming problem with integral operator is considered, in which the upper-level object is the agent that maximizes its expected utility with respect to an agreed compensation contract. The constraints are the principal’s participation and the agent’s incentive compatibility. The latter is a lower-level optimization problem with respect to its private action. To solve an equivalent single-level nonconvex programming problem with integral operator, a modified homotopy method for solving the Karush–Kuhn–Tucker system is proposed. This method requires only an interior point and, not necessarily, a feasible initial approximation for the constraint shifting set. Global convergence is proven under some mild conditions. Numerical experiments were performed by our homotopy method as well as by fmincon in Matlab, LOQO and MINOS. The experiments showed that: designing a piecewise linear contract is much better than designing a piecewise constant contract and only needs to solve a much lower-dimensional optimization problem and hence needs much less computation time; the optimal value of the principal-agent model with designing piecewise linear contract tends to a limitation, while the discrete segments gradually increase; and finally, the proposed modified homotopy method is feasible and effective.

- Authors:

- Jilin University of Finance and Economics, Faculty of Statistics (China)
- Dalian University of Technology, School of Mathematical Sciences (China)

- Publication Date:

- OSTI Identifier:
- 22769372

- Resource Type:
- Journal Article

- Journal Name:
- Computational and Applied Mathematics

- Additional Journal Information:
- Journal Volume: 37; Journal Issue: 1; Other Information: Copyright (c) 2018 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0101-8205

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; CONVERGENCE; DESIGN; OPTIMIZATION; PROGRAMMING

### Citation Formats

```
Zhu, Zhichuan, and Yu, Bo.
```*A modified homotopy method for solving the principal-agent bilevel programming problem*. United States: N. p., 2018.
Web. doi:10.1007/S40314-016-0361-5.

```
Zhu, Zhichuan, & Yu, Bo.
```*A modified homotopy method for solving the principal-agent bilevel programming problem*. United States. doi:10.1007/S40314-016-0361-5.

```
Zhu, Zhichuan, and Yu, Bo. Thu .
"A modified homotopy method for solving the principal-agent bilevel programming problem". United States. doi:10.1007/S40314-016-0361-5.
```

```
@article{osti_22769372,
```

title = {A modified homotopy method for solving the principal-agent bilevel programming problem},

author = {Zhu, Zhichuan and Yu, Bo},

abstractNote = {In this paper, the principal-agent bilevel programming problem with integral operator is considered, in which the upper-level object is the agent that maximizes its expected utility with respect to an agreed compensation contract. The constraints are the principal’s participation and the agent’s incentive compatibility. The latter is a lower-level optimization problem with respect to its private action. To solve an equivalent single-level nonconvex programming problem with integral operator, a modified homotopy method for solving the Karush–Kuhn–Tucker system is proposed. This method requires only an interior point and, not necessarily, a feasible initial approximation for the constraint shifting set. Global convergence is proven under some mild conditions. Numerical experiments were performed by our homotopy method as well as by fmincon in Matlab, LOQO and MINOS. The experiments showed that: designing a piecewise linear contract is much better than designing a piecewise constant contract and only needs to solve a much lower-dimensional optimization problem and hence needs much less computation time; the optimal value of the principal-agent model with designing piecewise linear contract tends to a limitation, while the discrete segments gradually increase; and finally, the proposed modified homotopy method is feasible and effective.},

doi = {10.1007/S40314-016-0361-5},

journal = {Computational and Applied Mathematics},

issn = {0101-8205},

number = 1,

volume = 37,

place = {United States},

year = {2018},

month = {3}

}