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

A Variant of the Topkis—Veinott Method for Solving Inequality Constrained Optimization Problems 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 ∈ N(z) \ {z } , f 0 (y) ≠ f 0 (z) , where f 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

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

, Volume 41 (3) – Feb 1, 2042
22 pages

/lp/springer_journal/a-variant-of-the-topkis-veinott-method-for-solving-inequality-ElEfoI1ypO
Publisher
Springer-Verlag
Copyright © Inc. by 2000 Springer-Verlag New York
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s002459911015
Publisher site
See Article on Publisher Site

### 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 ∈ N(z) \ {z } , f 0 (y) ≠ f 0 (z) , where f 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.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Feb 1, 2042

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