# A Variant of the Topkis-Veinott Method for Solving Inequality Constrained Optimization Problems

## Abstract

In this paper we give a variant of the Topkis-Veinott method for solving inequality constrained optimization problems. This method uses a linearly constrained positive semidefinite quadratic problem to generate a feasible descent direction at each iteration. Under mild assumptions, the algorithm is shown to be globally convergent in the sense that every accumulation point of the sequence generated by the algorithm is a Fritz-John point of the problem. We introduce a Fritz-John (FJ) function, an FJ1 strong second-order sufficiency condition (FJ1-SSOSC), and an FJ2 strong second-order sufficiency condition (FJ2-SSOSC), and then show, without any constraint qualification (CQ), that (i) if an FJ point z satisfies the FJ1-SSOSC, then there exists a neighborhood N(z) of z such that, for any FJ point y element of N(z) {l_brace}z {r_brace} , f{sub 0}(y) {ne} f{sub 0}(z) , where f{sub 0} is the objective function of the problem; (ii) if an FJ point z satisfies the FJ2-SSOSC, then z is a strict local minimum of the problem. The result (i) implies that the entire iteration point sequence generated by the method converges to an FJ point. We also show that if the parameters are chosen large enough, a unit step length can be acceptedmore »

- Authors:

- Robert R. McCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL 60208 (United States)
- School of Mathematics, University of New South Wales, Sydney, New South Wales 2052 (Australia)

- Publication Date:

- OSTI Identifier:
- 21067527

- Resource Type:
- Journal Article

- Journal Name:
- Applied Mathematics and Optimization

- Additional Journal Information:
- Journal Volume: 41; Journal Issue: 3; Other Information: DOI: 10.1007/s002459911015; Copyright (c) 2000 Springer-Verlag New York Inc.; Article Copyright (c) Inc. 2000 Springer-Verlag New York; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0095-4616

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGORITHMS; FUNCTIONS; MATHEMATICAL SOLUTIONS; OPTIMIZATION

### Citation Formats

```
Birge, J. R., Qi, L., and Wei, Z.
```*A Variant of the Topkis-Veinott Method for Solving Inequality Constrained Optimization Problems*. United States: N. p., 2000.
Web. doi:10.1007/S002459911015.

```
Birge, J. R., Qi, L., & Wei, Z.
```*A Variant of the Topkis-Veinott Method for Solving Inequality Constrained Optimization Problems*. United States. doi:10.1007/S002459911015.

```
Birge, J. R., Qi, L., and Wei, Z. Mon .
"A Variant of the Topkis-Veinott Method for Solving Inequality Constrained Optimization Problems". United States. doi:10.1007/S002459911015.
```

```
@article{osti_21067527,
```

title = {A Variant of the Topkis-Veinott Method for Solving Inequality Constrained Optimization Problems},

author = {Birge, J. R. and Qi, L. and Wei, Z.},

abstractNote = {In this paper we give a variant of the Topkis-Veinott method for solving inequality constrained optimization problems. This method uses a linearly constrained positive semidefinite quadratic problem to generate a feasible descent direction at each iteration. Under mild assumptions, the algorithm is shown to be globally convergent in the sense that every accumulation point of the sequence generated by the algorithm is a Fritz-John point of the problem. We introduce a Fritz-John (FJ) function, an FJ1 strong second-order sufficiency condition (FJ1-SSOSC), and an FJ2 strong second-order sufficiency condition (FJ2-SSOSC), and then show, without any constraint qualification (CQ), that (i) if an FJ point z satisfies the FJ1-SSOSC, then there exists a neighborhood N(z) of z such that, for any FJ point y element of N(z) {l_brace}z {r_brace} , f{sub 0}(y) {ne} f{sub 0}(z) , where f{sub 0} is the objective function of the problem; (ii) if an FJ point z satisfies the FJ2-SSOSC, then z is a strict local minimum of the problem. The result (i) implies that the entire iteration point sequence generated by the method converges to an FJ point. We also show that if the parameters are chosen large enough, a unit step length can be accepted by the proposed algorithm.},

doi = {10.1007/S002459911015},

journal = {Applied Mathematics and Optimization},

issn = {0095-4616},

number = 3,

volume = 41,

place = {United States},

year = {2000},

month = {5}

}