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On almost smooth functions and piecewise smooth functions. (English) Zbl 1125.26019
The paper contains contributions on the study of the class of piecewise smooth functions (PS) and other classes of semismooth functions, $f:O\to{\Bbb R},\;O\subset {\Bbb R}^n,$ $O$ open. Denote by $X_f$ the set of smooth points of $f$. One of the obtained main results is the following: if $f$ is a PS function, then $X_f$ is not locally connected around a point $x\in O\setminus X_f$. Using this criteria one obtains that a large class of semismooth functions, like the $p$-norms functions, NCP functions, smoothing/penalty and integral functions are not PS functions. In connections with this property the authors introduced the concept of almost smooth functions (AS), namely a function $f$ is AS function if for any $x\in O\setminus X_f$, there is $\varepsilon>0$, such that $B_{\overline{\varepsilon}}(x)\cap X_f$ is connected for any $0<\overline{\varepsilon}<\varepsilon$. In addition there are introduced some variants of AS functions. A discussion, completed by many examples, about the relationships between these notions and the above classes of semismooth functions is made.

26B05Continuity and differentiation questions (several real variables)
26A27Nondifferentiability of functions of one real variable; discontinuous derivatives
26B25Convexity and generalizations (several real variables)
26B35Special properties of functions of several real variables, Hölder conditions, etc.
49J52Nonsmooth analysis (other weak concepts of optimality)
52A41Convex functions and convex programs (convex geometry)
65D15Algorithms for functional approximation
90C25Convex programming
Full Text: DOI
[1] Agrachev, A.; Pallaschke, D.; Scholtes, S.: On Morse theory for piecewise smooth functions. J. dynam. Control systems 3, 449-469 (1997) · Zbl 0951.49024
[2] Andersson, L. E.; Elfving, T.; Iliev, G.; Vlachkova, K.: Interpolation of convex scattered data in R3 based upon an edged convex minimum norm network. J. approx. Theory 80, 299-320 (1995) · Zbl 0822.41017
[3] Auslender, A.: Penalty and barrier methods: a unified framework. SIAM J. Optim. 10, 211-230 (1999) · Zbl 0953.90045
[4] Bertsekas, D. P.: Constrained optimization and Lagrange multiplier methods. (1982) · Zbl 0572.90067
[5] Chen, B.; Harker, P. T.: A non-interior-point continuation method for linear complementarity problems. SIAM J. Matrix anal. Appl. 14, 1168-1190 (1993) · Zbl 0788.65073
[6] Chen, B.; Xiu, N.: A global linear and local quadratic non-interior continuation method for nonlinear complementarity problems based on Chen--mangasarian smoothing function. SIAM J. Optim. 9, 605-623 (1999) · Zbl 1037.90052
[7] Chen, C.; Mangasarian, O. L.: Smoothing methods for convex inequalities and linear complementarity problems. Math. program. 71, 51-69 (1995) · Zbl 0855.90124
[8] Chen, C.; Mangasarian, O. L.: A class of smoothing functions for nonlinear and mixed complementarity problems. Comp. optim. Appl. 5, 97-138 (1996) · Zbl 0859.90112
[9] Chen, X.; Qi, L.; Sun, D.: Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities. Math. comp. 67, 519-540 (1998) · Zbl 0894.90143
[10] Clarke, F. H.: Optimization and nonsmooth analysis. (1983) · Zbl 0582.49001
[11] De Luca, T.; Facchinei, F.; Kanzow, C.: A semismooth equation approach to the solution of nonlinear complementarity problems. Math. program. 75, 407-439 (1996) · Zbl 0874.90185
[12] Dontchev, A. L.; Qi, H. D.; Qi, L.: Convergence of Newton’s method for convex best interpolation. Numer. math. 87, 435-456 (2001) · Zbl 0970.65009
[13] Dontchev, A. L.; Qi, H. D.; Qi, L.: Quadratic convergence of Newton’s method for convex interpolation and smoothing. Constr. approx. 19, 123-143 (2003) · Zbl 1014.65011
[14] Facchinei, F.; Kanzow, C.: A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems. Math. program. 76, 493-512 (1997) · Zbl 0871.90096
[15] Facchinei, F.; Pang, J. -S.: Finite-dimensional variational inequalities and complementarity problems. (2003) · Zbl 1062.90002
[16] Fischer, A.: Solution of monotone complementarity problems with locally Lipschitzian functions. Math. program. 76, 513-532 (1997) · Zbl 0871.90097
[17] Hotta, K.; Yoshise, A.: Global convergence of a class of non-interior-point algorithms using Chen--harker--kanzow functions for nonlinear complementarity problems. Math. program. 86, 105-133 (1999) · Zbl 0978.90095
[18] Irvine, L. D.; Marin, S. P.; Smith, P. W.: Constrained interpolation and smoothing. Constr. approx. 2, 129-151 (1986) · Zbl 0596.41012
[19] Jiang, H.; Qi, L.; Chen, X.; Sun, D.: Semismoothness and superlinear convergence in nonsmooth optimization and nonsmooth equations. Nonlinear optimization and applications (Erice, 1995), 197-212 (1996) · Zbl 0991.90123
[20] Kanzow, C.; Kleinmichel, H.: A new class of semismooth Newton-type methods for nonlinear complementarity problems. Comput. optim. Appl. 11, 227-251 (1998) · Zbl 0913.90250
[21] Klatte, D.; Kummer, B.: Nonsmooth equations in optimization. (2002) · Zbl 1173.49300
[22] Kuntz, L.; Scholtes, S.: Structural analysis of nonsmooth mappings, inverse functions, and metric projections. J. math. Anal. appl. 188, 346-386 (1994) · Zbl 0809.49014
[23] Kuntz, L.; Scholtes, S.: Qualitative aspects of the local approximation of a piecewise differentiable function. Nonlinear anal. 25, 197-215 (1995) · Zbl 0834.90121
[24] Luo, Z. Q.; Tseng, P.: A new class of merit functions for the nonlinear complementarity problem. Complementarity and variational problems: state of the art, 204-225 (1997) · Zbl 0886.90158
[25] Mifflin, R.: Semismoothness and semiconvex functions in constrained optimization. SIAM J. Control optim. 15, 959-972 (1977) · Zbl 0376.90081
[26] Mordukhovich, B. S.: Maximum principle in the problem of time optimal response with nonsmooth constraints. J. appl. Math. mech. 40, 960-969 (1976) · Zbl 0362.49017
[27] Mordukhovich, B. S.: Variational analysis and generalized differentiation, I: Basic theory, II: Applications. (2006)
[28] Pang, J. -S.; Ralph, D.: Piecewise smoothness, local invertibility, and parametric analysis of normal maps. Math. oper. Res. 21, 401-426 (1996) · Zbl 0857.90122
[29] Peng, J. M.; Lin, Z.: A non-interior continuation method for generalized linear complementarity problems. Math. program. 86, 533-563 (1999) · Zbl 0987.90081
[30] Qi, H. D.; Liao, L.: A smoothing Newton method for extended vertical linear complementarity problems. SIAM J. Matrix anal. Appl. 21, 45-66 (1999) · Zbl 1017.90114
[31] Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. oper. Res. 18, 227-244 (1993) · Zbl 0776.65037
[32] Qi, L.: Regular pseudo-smooth NCP and BVIP functions and globally and quadratically convergent generalized Newton methods for complementarity and variational inequality problems. Math. oper. Res. 24, 440-471 (1999) · Zbl 0977.90064
[33] Qi, L.; Jiang, H.: Semismooth karush--Kuhn--Tucker equations and convergence analysis of Newton and quasi-Newton methods for solving these equations. Math. oper. Res. 22, 301-325 (1997) · Zbl 0881.65054
[34] Qi, L.; Shapiro, A.; Ling, C.: Differentiability and semismoothness properties of integral functions and their applications. Math. program. 102, 223-248 (2005) · Zbl 1079.90143
[35] Qi, L.; Sun, D.; Zhou, G.: A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities. Math. program. 87, 1-35 (2000) · Zbl 0989.90124
[36] Qi, L.; Sun, J.: A nonsmooth version of Newton’s method. Math. program. 58, 353-367 (1993) · Zbl 0780.90090
[37] Qi, L.; Yin, H.: A strongly semismooth integral function and its application. Comput. optim. Appl. 25, 223-246 (2003) · Zbl 1065.90076
[38] Qi, L.; Zhou, G.: A smoothing Newton method for minimizing a sum of Euclidean norms. SIAM J. Optim. 11, 389-410 (2000) · Zbl 1010.90078
[39] Ralph, D.; Scholtes, S.: Sensitivity analysis of composite piecewise smooth equations. Math. program. 76, 593-612 (1997) · Zbl 0871.90094
[40] Rockafellar, R. T.: Convex analysis. (1970) · Zbl 0193.18401
[41] Rockafellar, R. T.: A property of piecewise smooth functions. Comput. optim. Appl. 25, 247-250 (2003) · Zbl 1042.90045
[42] Rockafellar, R. T.; Wets, R. J. B.: Variational analysis. (1998) · Zbl 0888.49001
[43] S. Scholtes, Introduction to piecewise differentiable equations, Habilitation Thesis, Preprint No. 54/1994, Institut für Statistik und Mathematische Wirtschaftstheorie, University of Karlsruhe, Karlsruhe, Germany, 1994
[44] S. Smale, Algorithms for solving equations, in: Proceedings of the International Congress of Mathematicians, Berkeley, CA, 1986, pp. 172--195
[45] Tseng, P.: Global behaviour of a class of merit functions for the nonlinear complementarity problem. J. optim. Theory appl. 89, 17-37 (1996) · Zbl 0866.90127
[46] Tseng, P.: Analysis of a non-interior continuation method based on Chen--mangasarian smoothing functions for complementarity problems. Reformulation: nonsmooth, piecewise smooth, semismooth and smoothing methods, 381-404 (1999) · Zbl 0928.65078
[47] Watson, G. A.: On the Gauss--Newton method for l1 orthogonal distance regression. IMA J. Numer. anal. 22, 345-357 (2002) · Zbl 1017.65004
[48] Yamashita, N.; Fukushima, M.: Modified Newton methods for solving semismooth reformulations of monotone complementarity problems. Math. program. 76, 469-491 (1997) · Zbl 0872.90102