# Path following in the exact penalty method of convex programming

## Author

Listed:
• Hua Zhou

()

• Kenneth Lange

## Abstract

Classical penalty methods solve a sequence of unconstrained problems that put greater and greater stress on meeting the constraints. In the limit as the penalty constant tends to $$\infty$$ ∞ , one recovers the constrained solution. In the exact penalty method, squared penalties are replaced by absolute value penalties, and the solution is recovered for a finite value of the penalty constant. In practice, the kinks in the penalty and the unknown magnitude of the penalty constant prevent wide application of the exact penalty method in nonlinear programming. In this article, we examine a strategy of path following consistent with the exact penalty method. Instead of performing optimization at a single penalty constant, we trace the solution as a continuous function of the penalty constant. Thus, path following starts at the unconstrained solution and follows the solution path as the penalty constant increases. In the process, the solution path hits, slides along, and exits from the various constraints. For quadratic programming, the solution path is piecewise linear and takes large jumps from constraint to constraint. For a general convex program, the solution path is piecewise smooth, and path following operates by numerically solving an ordinary differential equation segment by segment. Our diverse applications to (a) projection onto a convex set, (b) nonnegative least squares, (c) quadratically constrained quadratic programming, (d) geometric programming, and (e) semidefinite programming illustrate the mechanics and potential of path following. The final detour to image denoising demonstrates the relevance of path following to regularized estimation in inverse problems. In regularized estimation, one follows the solution path as the penalty constant decreases from a large value. Copyright Springer Science+Business Media New York 2015

## Suggested Citation

• Hua Zhou & Kenneth Lange, 2015. "Path following in the exact penalty method of convex programming," Computational Optimization and Applications, Springer, vol. 61(3), pages 609-634, July.
• Handle: RePEc:spr:coopap:v:61:y:2015:i:3:p:609-634
DOI: 10.1007/s10589-015-9732-x
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File URL: http://hdl.handle.net/10.1007/s10589-015-9732-x

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## References listed on IDEAS

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1. Hua Zhou & Yichao Wu, 2014. "A Generic Path Algorithm for Regularized Statistical Estimation," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(506), pages 686-699, June.
2. Elmor L. Peterson, 1976. "Fenchel's Duality Thereom in Generalized Geometric Programming," Discussion Papers 252, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
3. Elmor L. Peterson, 1976. "Optimality Conditions in Generalized Geometric Programming," Discussion Papers 221, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
4. Willard I. Zangwill, 1967. "Non-Linear Programming Via Penalty Functions," Management Science, INFORMS, vol. 13(5), pages 344-358, January.
5. Hua Zhou & Kenneth L. Lange, 2010. "On the Bumpy Road to the Dominant Mode," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(4), pages 612-631.
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### Keywords

Constrained convex optimization; Exact penalty; Geometric programming; Ordinary differential equation; Quadratically constrained quadratic programming; Regularization; Semidefinite programming; 65K05; 90C25;

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