×

Subsmooth semi-infinite and infinite optimization problems. (English) Zbl 1259.90141

The article of Xi Yin Zheng and Kung Fu Ng is a very valuable contribution to the field of nonlinear and continuous optimization with a great promise for generalized frameworks of the usual constrained programs by allowing (i) an infinite number of inequality constraints to be given, or, including the objective function now also, (ii) to be defined over an infinite dimensional space. Those optimization problems are called (i) semi-infinite and (ii) infinite, respectively. They play an increasing role in the representation, treatment and solution of problems from (i) approximation of real-world phenomena, e.g. through Chebychev approximation, in design, manufacturing, robotics, computational biology, etc., and (ii) calculus of variations and data mining, etc., respectively. Both (i) and (ii) are useful in optimal control and, e.g. via the new “infinite kernel learning” in machine learning. On the other hand, the article also contributes to nonsmooth optimization; in fact, it lies in the definition of problems from (i) and (ii), that max- or sup-type functions occur, which are nonsmooth usually. Of course, such nonsmooth functions are still rather well-structured and -behaving ones, compared with even more general nonsmooth functions. In fact, the authors apply the concept of subsmoothness (introduced for sets and function families as well) to rigorously represent and study the problems from (i) and (ii). This study is done in a wide functional-analytic setting and with great rigor. Here, a particular interest of the authors exists in strong isolated and strong unique local minimizers; they use the notion of a sharp minimum in their related investigations.
First, the authors consider subsmoothness for a function family and present formulas of the subdifferential of the pointwise supremum of a such a family. Then, they consider subsmooth infinite and semi-infinite optimization problems. Especially, they are able to state several dual and primal characterizations for a point to be a sharp minimum of a weak sharp minimum for such optimization problems.
The five sections of the article are as follows: 1. Introduction, 2. Preliminaries, 3. Subsmoothness for a function family, 4. Subsmooth infinite optimization problem, and 5. Subsmooth semi-infinite optimization problem.
Further deep results and also methods may be expected in the future, initiated and fostered by this research paper. Such advances could then support and stimulate additional advances in science, especially, in data mining and statistics, in engineering, economics and social-political decision making, in finance and OR, in healthcare and medicine, and, herewith, to improvements in the living conditions of the peoples on earth.

MSC:

90C30 Nonlinear programming
90C34 Semi-infinite programming
49J52 Nonsmooth analysis
65K10 Numerical optimization and variational techniques
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Auslander A., Goberna M., López M.A.: Penalty and smoothning methods for convex semi-infinite programming. Math. Oper. Res. 34, 303–319 (2009) · Zbl 1218.90199 · doi:10.1287/moor.1080.0362
[2] Aussel D., Daniilidis A., Thibault L.: Subsmooth sets: functional characterizations and related concepts. Trans. Am. Math. Soc. 357, 1275–1301 (2005) · Zbl 1094.49016 · doi:10.1090/S0002-9947-04-03718-3
[3] Brosowski B.: Parametric Semi-Infinite Optimization. Verlag Peter Lang, Frankfurt (1982) · Zbl 0502.49001
[4] Burke J.V., Deng S.: Weak sharp minima revisited. Part I: basic theory. Control Cybern. 31, 439–469 (2002) · Zbl 1105.90356
[5] Burke J.V., Ferris M.C.: Weak sharp minima in mathematical programming. SIAM J. Control Optim. 31, 1340–1359 (1993) · Zbl 0791.90040 · doi:10.1137/0331063
[6] Clarke F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983) · Zbl 0582.49001
[7] Clarke F.H., Stern R., Wolenski P.: Proximal smoothness and the lower-C2 property. J. Convex Anal. 2, 117–144 (1995) · Zbl 0881.49008
[8] Clarke F.H., Ledyaev Y., Stern R., Wolenski P.: Nonsmmoth Analysis and Control Theory. Springer, New York (1998) · Zbl 1047.49500
[9] Cromme L.: Strong uniqueness, a far-reaching criterion for the convergence analysis of iterative procedures. Numer. Math. 29, 179–193 (1978) · Zbl 0352.65012 · doi:10.1007/BF01390337
[10] Ferris, M.C.: Weak Sharp Minima and Penalty Fucntions in Mathematical Programming. Ph.D. Thesis, University of Cambridge, Cambridge (1988)
[11] Goberna A., López M.A.: Linear Semi-Infinite Optimization. Wiley, Chichster (1998) · Zbl 0909.90257
[12] Goberna, A., López, M.A. (eds): Semi-Infinite Programming–Recent Advances. Kluwer, Boston (2001) · Zbl 0978.00025
[13] Hantoute A., López M.A., Zalinescu C.: Subdifferential calculus rules in convex analysis: a unifying approach via pointwise supremum functions. SIAM J. Optim. 19, 863–882 (2008) · Zbl 1163.49014 · doi:10.1137/070700413
[14] Henrion R., Outrata J.: Calmness of constraint systems with applications. Math. Program. 104, 437–464 (2005) · Zbl 1093.90058 · doi:10.1007/s10107-005-0623-2
[15] Hettich R., Kortanek K.O.: Semi-infinite programming: theory, methods, and applications. SIAM Rev. 35, 380–429 (1993) · Zbl 0784.90090 · doi:10.1137/1035089
[16] Jittorntrum K., Osborne M.R.: Strong uniqueness and second order convergence in nonlinear discrete approximation. Numer. Math. 34, 439–455 (1980) · Zbl 0486.65008 · doi:10.1007/BF01403680
[17] Jongen H.T., Ruuckmann J.-J., Stein O.: Generalized semi-infinite optimization: a first order optimality condition and examples. Math. Program. 83, 145–158 (1998) · Zbl 0949.90090
[18] Klatte D., Henrion R.: Regularity and stability in nonlinear semi-infinite optimization. Semi Infin Program Nonconvex Optim. Appl. 25, 69–102 (1998) · Zbl 0911.90330
[19] Lewis, A., Pang, J.S.: Error bounds for convex inequality systems. In: Crouzeix, J.-P., Martinez-Legaz, J.-E., Volle, M. (eds.) Generalized Convexity, Generalized Monotonicity: Recent Results, Proceedings of the 5th Symposium on Generalized Convexity, Luminy, 1996, pp. 75-0110. Kluwer, Dordrecht (1997)
[20] López M.A., Vercher E.: Optimality conditions for nondifferentiable convex semi-infinite programming. Math. Program. 27, 307–319 (1983) · Zbl 0527.49029 · doi:10.1007/BF02591906
[21] López M.A., Volle M.: A formula for the set of optimal solutions of a relaxed minimization problem, applications to subdifferential calculus. J. Convx Anal. 17, 1057–1075 (2010) · Zbl 1219.90184
[22] Mordukhovich B.S.: Variational Analysis and Generalized Differentiation, vol. I/II, Basic Theory. Springer, Berlin (2006)
[23] Nurnberger G.: Global unicity in semi-infinite optimization. Numer. Funct. Anal. Optim. 8, 173–191 (1985) · Zbl 0545.41037 · doi:10.1080/01630568508816209
[24] Osborne M.R., Womersley R.S.: Strong uniqueness in sequential linear programming. J. Austral. Math. Soc. Ser. B 31, 379–384 (1990) · Zbl 0709.90091 · doi:10.1017/S0334270000006731
[25] Pang J.S.: Error bounds in mathematical programming. Math. Program. 79, 299–332 (1997) · Zbl 0887.90165
[26] Polak E.: Optimization. Springer, New York (1997)
[27] Poliquin R., Rockafellar R.T.: Prox-regular functions in variational analysis. Trans. Am. Math. Soc. 348(5), 1805–1838 (1996) · Zbl 0861.49015 · doi:10.1090/S0002-9947-96-01544-9
[28] Reemtsen, R., Ruckmann, J.-J. (eds): Semi-Infinite Programming. Kluwer, Boston (1998)
[29] Rockafellar R.T., Wets R.J.-B.: Variational Analysis. Springer, Berlin (1998)
[30] Stein O.: Bi-Level Strategies in Semi-Infinite Programming. Kluwer, Boston (2003) · Zbl 1103.90094
[31] Studniarski M., Ward D.E.: Weak sharp minima: characterizations and sufficient conditions. SIAM J. Control Optim. 38, 219–236 (1999) · Zbl 0946.49011 · doi:10.1137/S0363012996301269
[32] Tapia R.A., Trosset M.W.: An extension of the Karush–Kuhn–Tucker necessity conditions to infinite programming. SIAM Rev. 36, 1–17 (1994) · Zbl 0805.49014 · doi:10.1137/1036001
[33] Vazquez F.G., Ruckmann J.J.: Extensions of the Kuhn–Tucker constraint qualification to generalized semi-infinite programming. SIAM J. Optim. 15, 926–937 (2005) · Zbl 1114.90141 · doi:10.1137/S1052623403431500
[34] Zalinescu, C.: Weak sharp minima, well-behaving functions and global error bounds for convex inequalities in Banach spaces. In: Proceedings of 12th Baikal International Conference on Optimization Methods and Their Applications, Irkutsk, pp. 272–284 (2001)
[35] Zalinescu C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)
[36] Zheng X.Y., Ng K.F.: Metric regularity and constraint qualifications for convex inequalities on Banach spaces. SIAM J. Optim. 14, 757–772 (2003) · Zbl 1079.90103 · doi:10.1137/S1052623403423102
[37] Zheng X.Y., Ng K.F.: Linear regularity for a collection of subsmooth sets in Banach spaces. SIAM J. Optim. 19, 62–76 (2008) · Zbl 1190.90231 · doi:10.1137/060659132
[38] Zheng X.Y., Yang X.Q.: Lagrange multipliers in nonsmooth semi-infinite optimization problems. Math. Oper. Res. 32, 168–181 (2007) · Zbl 1278.90411 · doi:10.1287/moor.1060.0234
[39] Zheng X.Y., Yang X.Q.: Weak sharp minima for semi-infinite optimization problems with applications. SIAM J. Optim. 18, 573–588 (2007) · Zbl 1190.90251 · doi:10.1137/060670213
[40] Zheng X.Y., Yang X.Q.: Global weak sharp minima for convex (semi-)infinite optimization problems. J. Math. Anal. Appl. 248, 1021–1028 (2008) · Zbl 1279.90176 · doi:10.1016/j.jmaa.2008.07.052
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.