Nonconvex Optimization: Gradient Flows and Deformation

H. Attouch, X. Goudou, and P. Redont, The heavy ball with friction method, I. The continuous dynamical system: global exploration of the local minima of a real-valued function by asymptotic analysis of a dissipative dynamical system. Commun. Contemp. Math. 2 (2000), 1–34.

J. F. Bonnans and A. Shapiro, Perturbation analysis of optimization problems. Springer, New York, 2000.

A. A. Davydov and H. Th. Jongen, Normal forms in one-parametric optimization. Ann. Oper. Res. (to appear).

P. Dupuis and A. Nagurney, Dynamical systems and variational inequalities. Ann. Oper. Res. 44 (1993), 9–42.

I. V. Girsanov, Lectures on mathematical theory of extremum problems. In: Lecture Notes in Econom. and Math. Systems, Vol. 67, Springer, Berlin, 1972.

T. J. Graettinger and B. H. Krogh, The acceleration radius: a global performance measure for robotic manipulators. IEEE J. Robotics and Automation 4 (1988), 60–69.

A. O. Griewank, Generalized descent for global optimization. J. Optim. Theory Appl. 34 (1981), 11–39.

J. Guddat, F. Guerra Vasquez, and H. Th. Jongen, Parametric optimization: singularities, pathfollowing and jumps. Wiley, Chichester, Teubner Stuttgart, 1990.

J. Guddat, H. Th. Jongen, and J.-J. Rückmann, On stability and stationary points in nonlinear optimization. J. Austral. Math. Soc. Ser. B 28 (1986), 36–56.

H. Günzel, H. Th. Jongen, Global optimization: on pathlengths in minmax graphs. J. Global Optim. 17 (2000), 161–165.

R. Hettich and H. Th. Jongen, Semi-infinite programming: conditions of optimality and applications. In: Optim. Techniques, (J. Stoer, ed)., Part 2, Lecture Notes in Control and Inform. Sci., Vol. 7, Springer, Berlin, 1978, 1–11.

R. Hettich and K. O. Kortanek, Semi-infinite programming: theory, methods, and applications. SIAM Rev. 35 (1993), 380–429.

R. Hettich and G. Still, Second order optimality conditions for generalized semi-infinite programming problems. Optimization 34 (1995), 195–211.

R. Hettich and P. Zencke, Numerische Methoden der Approximation und semi-infiniten Optimierung. Teubner Studienbücher, Stuttgart, 1982.

F. John, Extremum problems with inequalities as subsidiary conditions. In: Studies and Essays, R. Courant Anniversary Volume, Interscience, New York (1948), 187–204.

H. Th. Jongen, P. Jonker, and F. Twilt, Nonlinear optimization in ℝn, I: Morse theory, Chebyshev approximation. Peter Lang Verlag, Frankfurt/Main, 1983.

_____, Nonlinear optimization in ℝn, II: Transversality, flows, parametric aspects. Peter Lang Verlag, Frankfurt/Main, 1986.

_____, One-parameter families of optimization problems: equality constraints. J. Optim. Theory Appl. 48 (1986), 141–161.

_____, Critical sets in parametric optimization. Math. Program. 34 (1986), 333–353.

_____, Nonlinear optimization in finite dimensions. Kluwer, Dordrecht, 2000.

H. Th. Jongen and J.-J. Rückmann, On stability and deformation in semi-infinite optimization. In: Semi-Infinite Programming (R. Reemtsen and J.-J. Rückmann, eds.), Kluwer Academic Publishers (1998), 29–67.

H. Th. Jongen, J.-J. Rückmann, and O. Stein, Disjunctive optimization: critical point theory. J. Optim. Theory Appl. 93 (1997), 321–336.

_____, Generalized semi-infinite optimization: a first order optimality condition and examples. Math. Program. 83 (1998), 145–158.

H. Th. Jongen and A. Ruiz Jhones, Nonlinear optimization: on the minmax digraph and global smoothing. In: Calculus of Variations and Differential Equations (A. Ioffe, S. Reich, I. Shafrir, eds.), Chapman & Hall / CRC Res. Notes Math. 410 (1999), 119–135.

H. Th. Jongen and O. Stein, On generic one-parametric semi-infinite optimization. SIAM J. Optim. 7 (1997), 1103–1137.

A. Kaplan and R. Tichatschke, On a class of terminal variational problems. In: J. Guddat, H. Th. Jongen, Parametric optimization and related topics IV. (F. Nožička, G. Still, F. Twilt, eds.), Peter Lang, Frankfurt/Main (1997), 185–199.

P. J. M. Laarhoven and E. H. L. Aarts, Simulated annealing: theory and applications. D. Reidel Publishing, 1987.

V. H. Nguyen and J. J. Strodiot, Computing a global optimal solution to a design centering problem. Math. Program. 53 (1992), 111–123.

E. Polak, An implementable algorithm for the optimal design centering, tolerancing and tuning problem. J. Optim. Theory Appl. 37 (1982), 45–67.

Semi-infinite programming. (R. Reemtsen, J.-J. Rückmann, eds.), Kluwer Academic Publishers, Boston, 1998.

J.-J. Rückmann and A. Shapiro, First-order optimality conditions in generalized semi-infinite programming. J. Optim. Theory Appl. 101 (1999), 677–691.

J.-J. Rückmann and O. Stein, On linear and linearized generalized semi-infinite optimization problems. Ann. Oper. Res. (to appear).

J. Schropp, A dynamical systems approach to constrained minimization. Numer. Funct. Anal. Optim. 21 (2000), 537–551.

S. Smale, On gradient dynamical systems. Ann. Math., II. Ser. 74 (1961), 199–206.

_____, Trap-doors in the solution set of semi-infinite optimization problems. In: Recent Advances in Optimization. (P. Gritzmann, R. Horst, E. Sachs, and R. Tichatschke, eds.), Springer, Berlin (1997), 348–355.

_____, On parametric semi-infinite optimization. Shaker, Aachen, 1997.

_____, On level sets of marginal functions. Optimization 48 (2000), 43–67.

_____, First order optimality conditions for degenerate index sets in generalized semi-infinite programming. Math. Oper. Res. (to appear).

_____, The feasible set in generalized semi-infinite programming. In: Approximation, Optimization and Mathematical Economics, (M. Lassonde, ed)., Springer (2001), 313–331.

O. Stein and G. Still, On optimality conditions for generalized semi-infinite programming problems. J. Optim. Theory Appl. 104 (2000), 443–458.

K. Tanabe, A geometric method in nonlinear programming. J. Optim. Theory Appl. 30 (1980), 181–210.

G.-W. Weber, Generalized semi-infinite optimization and related topics. Habilitation Thesis, Darmstadt University of Technology, 1999.

W. Wetterling, Definitheitsbedingungen für relative Extrema bei Optimierungs-und Approximationsaufgaben. Numer. Math. 15 (1970), 122–136.