Generalized Derivatives for Solutions of Parametric Ordinary Differential Equations with Non-differentiable Right-Hand Sides

Journal of Optimization Theory and Applications

Volumn 163, Issue 2, 2014, Pages 355-386

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

^a Massachusetts Institute of Technology Cambridge   (United States)

Author keywords

Generalized Jacobians; Nonsmooth analysis; Ordinary differential equations; Sensitivity analysis

Indexed keywords

EMBEDDED SYSTEMS; NUMERICAL METHODS; ORDINARY DIFFERENTIAL EQUATIONS;

DIFFERENTIAL EQUATION SYSTEMS; EQUATIONS SOLVING; GENERALIZED DERIVATIVES; GENERALIZED JACOBIAN; LIPSCHITZ CONTINUOUS; LOCALLY LIPSCHITZ; NON-DIFFERENTIABLE; NON-SMOOTH ANALYSIS; OPTIMISATIONS; POST OPTIMALITY ANALYSIS;

SENSITIVITY ANALYSIS;

EID: 84918832561     PISSN: 00223239     EISSN: 15732878     Source Type: Journal    
DOI: 10.1007/s10957-014-0539-1     Document Type: Article

References (25)

1. Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM, Philadelphia (1990)
2. Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)
3. Kojima, M., Shindo, S.: Extension of Newton and quasi-Newton methods to systems of (Formula presented.) equations. J. Oper. Res. Soc. Jpn 29, 352–375 (1986)
4. Kiwiel, K.C.: Methods of Descent for Nondifferentiable Optimization. Lecture Notes in Mathematics. Springer, Berlin (1985)
5. Lemaréchal, C., Strodiot, J.J., Bihain, A.: On a bundle algorithm for nonsmooth optimization. In: Mangasarian, O.L., Meyer, R.R., Robinson, S.M. (eds.) Nonlinear Programming 4. Academic Press, New York (1981)
6. Luksǎn, L., Vlček, J.: A bundle-Newton method for nonsmooth unconstrained minimization. Math. Program. 83, 373–391 (1998)
7. Khan, K.A., Barton, P.I.: Evaluating an element of the Clarke generalized Jacobian of a composite piecewise differentiable function. ACM Trans. Math. Softw. 39, 23–28 (2013)
8. Griewank, A.: On stable piecewise linearization and generalized algorithmic differentiation. Optim. Method Softw. 28, 1139–1178 (2013)
9. Hartman, P.: Ordinary Differential Equations, 2nd edn. SIAM, Philadelphia (2002)
10. Galán, S., Feehery, W.F., Barton, P.I.: Parametric sensitivity functions for hybrid discrete/continuous systems. Appl. Numer. Math. 31, 17–47 (1999)
11. Scott, J.K., Barton, P.I.: Improved relaxations for the parametric solutions of ODEs using differential inequalities. J. Glob. Optim. 57, 143–176 (2013)
12. Yunt, M., Khan, K.A., Barton, P.I.: Parametric sensitivity analysis of dynamic systems using nonsmooth analysis: I. Theory (submitted) (2013)
13. Pang, J.S., Stewart, D.E.: Solution dependence on initial conditions in differential variational inequalities. Math. Program. B 116, 429–460 (2009)
14. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)
15. Yunt, M.: Nonsmooth dynamic optimization of systems with varying structure. Ph.D. Thesis, Massachusetts Institute of Technology (2011)
16. Sweetser, T.: A minimal set-valued strong derivative for vector-valued Lipschitz functions. J. Optim. Theory Appl. 23, 549–562 (1977)
17. Xu, H.: Set-valued approximations and Newton’s methods. Math. Program. 84, 401–420 (1999)
18. Hiriart-Urruty, J.B.: Characterizations of the plenary hull of the generalized Jacobian matrix. Math. Program. Stud. 17, 1–12 (1982)
19. Nesterov, Y.: Lexicographic differentiation of nonsmooth functions. Math. Program. B 104, 669–700 (2005)
20. Scholtes, S.: Introduction to Piecewise Differentiable Equations (SpringerBriefs in Optimization). Springer, New York (2012)
21. Khan, K.A., Barton, P.I.: On a vector forward AD mode for generalized Jacobian element evaluation (in preparation) (2013)
22. Imbert, C.: Support functions of the Clarke generalized Jacobian and of its plenary hull. Nonlinear Anal. Theor. 49, 1111–1125 (2002)
23. Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Kluwer, Dordrecht (1988)
24. Willett, D., Wong, J.S.W.: On the discrete analogues of some generalizations of Gronwall’s inequality. Monatsh. Math. 69, 362–367 (1965)
25. Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw Hill, New York (1955)