Evaluating an element of the clarke generalized jacobian of a composite piecewise differentiable function

ACM Transactions on Mathematical Software

Volumn 39, Issue 4, 2013, Pages

  Khan, Kamil A ^a   Barton, Paul I ^a  

^a Massachusetts Institute of Technology Cambridge   (United States)

Author keywords

Algorithms; Performance; Theory

Indexed keywords

DIFFERENTIABLE FUNCTIONS; DIRECTIONAL DERIVATIVE; GENERALIZED JACOBIAN; NON-SMOOTH FUNCTIONS; NONSMOOTH OPTIMIZATION; PERFORMANCE; SEMISMOOTH NEWTON METHOD; THEORY;

ALGORITHMS; NEWTON-RAPHSON METHOD;

FUNCTION EVALUATION;

EID: 84880812848     PISSN: 00983500     EISSN: 15577295     Source Type: Journal    
DOI: 10.1145/2491491.2491493     Document Type: Article

References (18)

1. BERTSIMAS, D. AND TSITSIKLIS, J. N. 1997. Introduction to Linear Optimization. Athena Scientific and Dynamic Ideas, Belmont, MA.
2. BOYD, S. AND VANDENBERGHE, L. 2004. Convex Optimization. Cambridge University Press, Cambridge.
3. CLARKE, F. H. 1990. Optimization and Nonsmooth Analysis. SIAM, Philadelphia, PA.
4. DURAN, M. A. AND GROSSMANN, I. E. 1986. Simultaneous optimization and heat integration of chemical processes. AIChE J. 32, 1, 123-138. DOI:http://dx.doi.org/10.1002/aic.690320114.
5. FACCHINEI, F. AND PANG, J.-S. 2003. Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer.
6. GRIEWANK, A. 1994. Automatic directional differentiation of nonsmooth composite functions. In Proceedings of the French-German Conference on Optimization.
7. GRIEWANK, A. AND WALTHER, A. 2008. Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation 2nd Ed. SIAM, Philadelphia, PA.
8. HUANG, Z.-D. AND MA, G.-C. 2010. On the computation of an element of Clarke generalized Jacobian for a vector-valued max function. Nonlinear Anal.-Theor. 72, 2, 998-1009. DOI:http://dx.doi.org/10.1016/j.na.2009.07.040.
9. KHAN, K. A. AND BARTON, P. I. 2012. Evaluating an element of the Clarke generalized Jacobian of a piecewise differentiable function. In Recent Advances in Algorithmic Differentiation. S. Forth, P. Hovland, E. Phipps, J. Utke, and A.Walther, Eds., Springer, 115-125. DOI:http://dx.doi.org/10.1007/978-3-642- 30023-311.
10. KIWIEL, K. C. 1985. Methods of Descent for Nondifferentiable Optimization. Springer.
11. KOJIMA, M. AND SHINDO, S. 1986. Extension of Newton and quasi-Newton methods to systems of PC1 equations. J. Oper. Res. Soc. Jpn 29, 4, 352-375.
12. LEMARÉCHAL, C., STRODIOT, J.-J., AND BIHAIN, A. 1981. On a bundle algorithm for nonsmooth optimization. In Nonlinear Programming 4. O. L. Mangasarian, R. R. Meyer, and S. M. Robinson, Eds., Academic Press, New York, NY.
13. LUKŠAN, L. AND VLČEK, J. 1998. A bundle-Newton method for nonsmooth unconstrained minimization. Math. Program. 83, 373-391. DOI:http://dx.doi.org/10.1007/BF02680566.
14. NESTEROV, Y. 2005. Lexicographic differentiation of nonsmooth functions. Math. Program. B 104, 669-700. DOI:http://dx.doi.org/10.1007/s10107-005-0633-0.
15. QI, L. AND SUN, J. 1993. A nonsmooth version of Newton's method.Math. Program. 58, 353-367. DOI:http://dx.doi.org/10.1007/BF01581275.
16. SCHOLTES, S. 1994. Introduction to piecewise differentiable equations. Habilitation Thesis, Institut für Statistik und Mathematische Wirtschaftstheorie, University of Karlsruhe.
17. WALTER, J., KOCH, M., WINKLER, G., AND BELLOT, D. 2010. BOOST Basic Linear Algebra Library. http://www.boost.org/doc/libs/1480/libs/numeric/ublas/ doc/index.htm.
18. ZIEGLER, G. M. 1998. Lectures on Polytopes. Springer.