Abstract
Directional derivatives play one of the major roles in optimization. Optimality conditions can be described in terms of these objects. These conditions, however, are not constructive. To overcome this problem, one has to represent the directional derivative in special forms. Two such forms are quasidifferentials and exhausters proposed by V.F. Demyanov. Quasidifferentials were introduced in 1980s. Optimality conditions in terms of these objects were developed by L.N. Polyakova and V.F. Demyanov. It was described how to find directions of steepest descent and ascent when these conditions are not satisfied. This paved a way for constructing new optimization algorithms. Quasidifferentials allow one to treat a wide class of functions. V.F. Demyanov introduced the notion of exhausters in 2000s to expand the class of functions that can be treated. It should be noted that a great contribution to the emergence of this notion was made by B.N. Pshenichny and A.M. Rubinov. In this work it is shown that exhausters not only allow one to treat a wider class of functions than quasidifferentials (since every quasidifferentiable function has exhausters) but is also preferable even for quasidifferentiable functions when solving nonsmooth optimization problems.
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References
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Demyanov, V.F., Polyakova, L.N., Rubinov, A.M.: On one generalization of the concept of subdifferential. In: Abstracts. All-Union Conference on Dynamical Control. Sverdlovsk, pp. 79–84 (1979)
Demyanov, V.F., Rubinov, A.M.: On quasidifferentiable functionals. Sov. Math. Dokl. 21, 13–17 (1980)
Demyanov, V.F., Vasiliev, L.V.: Nondifferentiable Optimization. Springer-Optimization Software, New York (1985)
Demyanov, V.F., Polyakova, L.N.: The minimum conditions of a quasidifferentiable function on a quasidifferentiable set (in Russian). Comput. Math. Math. Phys. 20, 849–856 (1980)
Gorokhovik, V.V.: On the quasidifferentiability of real functions and the conditions of local extrema. Sib. Math. J. 25, 62–70 (1984)
Qi, L.: Quasidifferentials and maximal normal operators. Math. Program. 49, 263–271 (1990)
Pallaschke, D., Urbanski, R.: Reduction of quasidifferentials and minimal representations. Math. Program. 66, 161–180 (1994)
Dempe, S., Pallaschke, D.: Quasidifferentiability of optimal solutions in parametric nonlinear optimization. Optimization 40, 1–24 (1997)
Stavroulakis, G.E.: Quasidifferentiable optimization: calculus of quasidifferentials. In: Pardalos, P.N., Floudas, C.A. (eds.) Encyclopedia of Optimization, pp. 3190–3194. Springer, Berlin (2009)
Basaeva, E.K., Kusraev, A.G., Kutateladze, S.S.: Quaisidifferentials in Kantorovich spaces. J. Optim. Theory Appl. 171, 365–383 (2016)
Pshenichny, B.N.: Convex Analysis and Extremal Problems. Nauka, Moscow (1980). (in Russian)
Demyanov, V.F., Rubinov, A.M.: Constructive Nonsmooth Analysis. Approximation and Optimization, vol. 7. Peter Lang, Frankfurt am Main (1995)
Demyanov, V.F., Rubinov, A.M.: Exhaustive families of approximations revisited. In: From Convexity to Nonconvexity. Nonconvex Optim. Appl., vol. 55, pp. 43–50. Kluwer Academic, Dordrecht (2001)
Demyanov, V.F.: Exhausters af a positively homogeneous function. Optimization 45, 13–29 (1999)
Demyanov, V.F.: Exhausters and convexificators—new tools in nonsmooth analysis. In: Demyanov, V., Rubinov, A. (eds.) Quasidifferentiability and Related Topics, pp. 85–137. Kluwer Academic Publishers, Dordrecht (2000)
Demyanov, V.F.: Proper exhausters and coexhausters in nonsmooth analysis. Optimization 61, 1347–1368 (2012)
Demyanov, V.F., Roshchina, V.A.: Constrained optimality conditions in terms of proper and adjoint exhausters. Appl. Comput. Math. 4, 144–124 (2005)
Demyanov, V.F., Roshchina, V.A.: Optimality conditions in terms of upper and lower exhausters. Optimization 55, 525–540 (2006)
Roshchina, V.A.: Reducing exhausters. J. Optim. Theory Appl. 136, 261–273 (2008)
Abbasov, M.E., Demyanov, V.F.: Extremum conditions for a nonsmooth function in terms of exhausters and coexhausters. Proc. Steklov Inst. Math. 269, 6–15 (2010)
Abbasov, M.E., Demyanov, V.F.: Proper and adjoint exhausters in nonsmooth analysis: optimality conditions. J. Glob. Optim. 56, 569–585 (2013)
Abbasov, M.E.: Generalized exhausters: existence, construction, optimality conditions. J. Glob. Optim. 11, 217–230 (2015)
Ishizuka, Y.: Optimality conditions for quasidifferentiable programs with application to two-level optimization. SIAM J. Control Optim. 26, 1388–1398 (1988)
Glover, B.M., Ishizuka, Y., Jeyakumar, V., Tuan, H.D.: Complete charactirizations of global optimality for problems involving the pointwise minimum of sublinear functions. SIAM J. Optim. 6, 362–372 (1996)
Castellani, M.: A dual representation for proper positively homogeneous functions. J. Glob. Optim. 16, 393–400 (2000)
Murzabekova, G.E.: Implicit function theorem for nonsmooth systems by means of exhausters. J. Comput. Syst. Sci. Int. 48, 574–580 (2009)
Küçük, M., Urbanski, R., Grzybowski, J., et al.: Reduction of weak exhausters and optimality conditions via reduced weak exhausters. J. Optim. Theory Appl. 165, 693–707 (2015)
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The reported study was partially supported by RFBR, Research Project No. 16-31-00056.
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Abbasov, M.E. Comparison Between Quasidifferentials and Exhausters. J Optim Theory Appl 175, 59–75 (2017). https://doi.org/10.1007/s10957-017-1167-3
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DOI: https://doi.org/10.1007/s10957-017-1167-3