The directed and Rubinov subdifferentials of quasidifferentiable functions. I: Definition and examples. (English) Zbl 1236.49031

Summary: We extend the definition of the directed subdifferential, originally introduced in R. Baier, E. Farkhi [The directed subdifferential of DC functions, Leizarowitz, Arie (ed.) et al., Nonlinear analysis and optimization II. Optimization. A conference in celebration of Alex Ioffe’s 70th and Simeon Reich’s 60th birthdays, June 18–24, 2008, Haifa, Israel. Providence, RI: American Mathematical Society (AMS); Ramat-Gan: Bar-Ilan University (ISBN 978-0-8218-4835-7/pbk). Contemporary Mathematics 514; Israel Mathematical Conference Proceedings, 27-43 (2010; Zbl 1222.49020)], for Differences of Convex (DC) functions to the wider class of quasidifferentiable functions. Such generalization efficiently captures differential properties of a wide class of functions including amenable and lower/upper-\(C^k\) functions. While preserving the most important properties of the quasidifferential, such as exact calculus rules, the directed subdifferential lacks the major drawbacks of quasidifferential: non-uniqueness and ”inflation in size” of the two convex sets representing the quasidifferential after applying calculus rules. The Rubinov subdifferential is defined as the visualization of the directed subdifferential.


49J52 Nonsmooth analysis
26B25 Convexity of real functions of several variables, generalizations
90C26 Nonconvex programming, global optimization


Zbl 1222.49020
Full Text: DOI


[1] Baier, R.; Farkhi, E., The directed subdifferential of DC functions, (), 27-43 · Zbl 1222.49020
[2] Baier, R.; Farkhi, E., Differences of convex compact sets in the space of directed sets, part I: the space of directed sets, Set-valued anal., 9, 3, 217-245, (2001) · Zbl 1097.49507
[3] Baier, R.; Farkhi, E., Differences of convex compact sets in the space of directed sets, part II: visualization of directed sets, Set-valued anal., 9, 3, 247-272, (2001) · Zbl 1097.49508
[4] Rådström, H., An embedding theorem for spaces of convex sets, Proc. amer. math. soc., 3, 165-169, (1952) · Zbl 0046.33304
[5] Demyanov, V.F.; Rubinov, A.M., (), Russian original, Foundations of nonsmooth analysis, and quasidifferential calculus, published in Nauka, Moscow, 1990 · Zbl 0728.49001
[6] Pallaschke, D.; Urbański, R., ()
[7] Baier, R.; Farkhi, E.; Roshchina, V., On computing the mordukhovich subdifferential using directed sets in two dimensions, (), 59-93 · Zbl 1216.49014
[8] Baier, R.; Farkhi, E.; Roshchina, V., The directed and rubinov subdifferentials of quasidifferentiable functions. part II: calculus, Nonlinear anal., 75, 3, 1058-1073, (2012) · Zbl 1242.49033
[9] Mordukhovich, B.S., ()
[10] Rockafellar, R.T., Favorable classes of Lipschitz-continuous functions in subgradient optimization, (), 125-143
[11] Rockafellar, R.T.; Wets, R.J.-B., ()
[12] Hadwiger, H., Minkowskische addition und subtraktion beliebiger punktmengen und die theoreme von erhard Schmidt, Math. Z., 53, 3, 210-218, (1950) · Zbl 0040.38301
[13] Pontryagin, L.S., Linear differential games. ii, Sov. math., dokl., 8, 4, 910-912, (1967) · Zbl 0157.16401
[14] Rubinov, A.M.; Akhundov, I.S., Difference of compact sets in the sense of demyanov and its application to non-smooth analysis, Optimization, 23, 3, 179-188, (1992) · Zbl 0816.52002
[15] Diamond, P.; Kloeden, P.; Rubinov, A.; Vladimirov, A., Comparative properties of three metrics in the space of compact convex sets, Set-valued anal., 5, 3, 267-289, (1997) · Zbl 0895.90151
[16] Pallaschke, D.; Urbański, R., Minimal pairs of compact convex sets, with application to quasidifferential calculus, (), 173-213 · Zbl 0997.49014
[17] Hiriart-Urruty, J.-B.; Lemaréchal, C., ()
[18] A. Shapiro, On functions representable as a difference of two convex functions in inequality constrained optimization. Research report University of South Africa, 1983.
[19] Xianfu Wang, Pathological Lipschitz Functions in \(\mathbb{R}^N\), Ph.D. Thesis, Simon Fraser University, Department of Mathematics & Statistics, Master of Science thesis, Burnaby, BC, Canada, June 1975. http://ir.lib.sfu.ca/bitstream/1892/8022/1/b17501684.pdf.
[20] Hewitt, E.; Stromberg, K., (), Third printing, first published in 1965
[21] E. Nurminski, Subtraction of convex sets and its application in \(\varepsilon\)-subdifferential calculus, in: IIASA Working Paper WP-82-083, Internat. Inst. Appl. Systems Anal., IIASA, Laxenburg, September 1982. 32 pp.
[22] Hiriart-Urruty, J.-B., Generalized differentiability, duality and optimization for problems dealing with differences of convex functions, (), 37-70 · Zbl 0591.90073
[23] Horst, R.; Thoai, N.V., DC programming: overview, J.optim. theory appl., 103, 1, 1-43, (1999) · Zbl 1073.90537
[24] Horst, R.; Pardalos, P.M.; Thoai, N.V., ()
[25] Kaucher, E., Interval analysis in the extended interval space \(\mathbf{I} \mathbf{R}\), (), 33-49
[26] Markov, S., On directed interval arithmetic and its applications, J. ucs, 1, 7, 514-526, (1995), (electronic) · Zbl 0960.65561
[27] R. Baier, M. Dellnitz, M. Hessel-von Molo, I.G. Kevrekidis, S. Sertl, The computation of invariant sets via Newton’s method, May 2010, 21 pages (submitted). · Zbl 1306.65207
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