January  2015, 11(1): 217-230. doi: 10.3934/jimo.2015.11.217

Generalized exhausters: Existence, construction, optimality conditions

1. 

Saint Petersburg State University, Universitetskaya nab., 7-9, St. Petersburg, Russian Federation

Received  May 2013 Revised  March 2014 Published  May 2014

In this work a generalization of the notion of exhauster is considered. Exhausters are new tools in nonsmooth analysis introduced in works of Demyanov V.F., Rubinov A.M., Pshenichny B.N. In essence, exhausters are families of convex compact sets, allowing to represent the increments of a function at a considered point in an $\inf\max$ or $\sup\min$ form, the upper exhausters used for the first representation, and the lower one for the second representation. Using this objects one can get new optimality conditions, find descent and ascent directions and thus construct new optimization algorithms. Rubinov A.M. showed that an arbitrary upper or lower semicontinuous positively homogenous function bounded on the unit ball has an upper or lower exhausters respectively. One of the aims of the work is to obtain the similar result under weaker conditions on the function under study, but for this it is necessary to use generalized exhausters - a family of convex (but not compact!) sets, allowing to represent the increments of the function at a considered point in the form of $\inf\sup$ or $\sup\inf$. The resulting existence theorem is constructive and gives a theoretical possibility of constructing these families. Also in terms of these objects optimality conditions that generalize the conditions obtained by Demyanov V.F., Abbasov M.E. are stated and proved. As an illustration of obtained results, an example of $n$-dimensional function, that has a non-strict minimum at the origin, is demonstrated. A generalized upper and lower exhausters for this function at the origin are constructed, the necessary optimality conditions are obtained and discussed.
Citation: Majid E. Abbasov. Generalized exhausters: Existence, construction, optimality conditions. Journal of Industrial & Management Optimization, 2015, 11 (1) : 217-230. doi: 10.3934/jimo.2015.11.217
References:
[1]

M. E. Abbasov, Extremality conditions in terms of adjoint exhausters,, (In Russian) Vestnik of Saint-Petersburg University; Applied mathematics, 10 (2011), 3.   Google Scholar

[2]

M. E. Abbasov and V. F. Demyanov, Extremum Conditions for a Nonsmooth Function in Terms of Exhausters and Coexhausters,, (In Russian) Trudy Instituta Matematiki i Mekhaniki UrO RAN, (2010).   Google Scholar

[3]

M. E. Abbasov and V. F. Demyanov, Proper and adjoint exhausters in Nonsmooth analysis: Optimality conditions,, Journal of Global Optimization, 56 (2013), 569.  doi: 10.1007/s10898-012-9873-8.  Google Scholar

[4]

M. Castellani, A dual representation for proper positively homogeneous functions,, J. Global Optim., 16 (2000), 393.  doi: 10.1023/A:1008394516838.  Google Scholar

[5]

V. F. Demyanov, Optimality Conditions and Variational Calculus,, (In Russian). Moscow, (2005).   Google Scholar

[6]

V. F. Demyanov, Exhausters and convexificators - new tools in nonsmooth analysis,, Nonconvex Optim. Appl., 43 (2000), 85.  doi: 10.1007/978-1-4757-3137-8_4.  Google Scholar

[7]

V. F. Demyanov, Proper Exhausters and Coexhausters in Nonsmooth Analysis,, Optimization, 61 (2012), 1347.  doi: 10.1080/02331934.2012.700929.  Google Scholar

[8]

V. F. Demyanov, Exhausters of a positively homogeneous function,, Optimization, 45 (1999), 13.  doi: 10.1080/02331939908844424.  Google Scholar

[9]

V. F. Demyanov and V. A. Roschina, Optimality conditions in terms of upper and lower exhausters,, Optimization, 55 (2006), 525.  doi: 10.1080/02331930600815777.  Google Scholar

[10]

V. F. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis,, Approximation & Optimization, (1995).   Google Scholar

[11]

V. F. Demyanov and A. M. Rubinov, Exhaustive families of approximations revisited,From convexity to nonconvexity,, Nonconvex Optim. Appl., 55 (2001), 43.  doi: 10.1007/978-1-4613-0287-2_4.  Google Scholar

[12]

B. N. Pshenichny, Convex Analysis and Extremal Problems (in Russian)., Nauka, (1980).   Google Scholar

[13]

R. T. Rockafellar, Convex Analysis,, Princeton University Press, (1970).   Google Scholar

[14]

V. A. Roshchina, Limited exhausters and optimality conditions,, Control Processes and Stability: Proceedings of the 36-th international conference of students and graduate students, (2005), 521.   Google Scholar

show all references

References:
[1]

M. E. Abbasov, Extremality conditions in terms of adjoint exhausters,, (In Russian) Vestnik of Saint-Petersburg University; Applied mathematics, 10 (2011), 3.   Google Scholar

[2]

M. E. Abbasov and V. F. Demyanov, Extremum Conditions for a Nonsmooth Function in Terms of Exhausters and Coexhausters,, (In Russian) Trudy Instituta Matematiki i Mekhaniki UrO RAN, (2010).   Google Scholar

[3]

M. E. Abbasov and V. F. Demyanov, Proper and adjoint exhausters in Nonsmooth analysis: Optimality conditions,, Journal of Global Optimization, 56 (2013), 569.  doi: 10.1007/s10898-012-9873-8.  Google Scholar

[4]

M. Castellani, A dual representation for proper positively homogeneous functions,, J. Global Optim., 16 (2000), 393.  doi: 10.1023/A:1008394516838.  Google Scholar

[5]

V. F. Demyanov, Optimality Conditions and Variational Calculus,, (In Russian). Moscow, (2005).   Google Scholar

[6]

V. F. Demyanov, Exhausters and convexificators - new tools in nonsmooth analysis,, Nonconvex Optim. Appl., 43 (2000), 85.  doi: 10.1007/978-1-4757-3137-8_4.  Google Scholar

[7]

V. F. Demyanov, Proper Exhausters and Coexhausters in Nonsmooth Analysis,, Optimization, 61 (2012), 1347.  doi: 10.1080/02331934.2012.700929.  Google Scholar

[8]

V. F. Demyanov, Exhausters of a positively homogeneous function,, Optimization, 45 (1999), 13.  doi: 10.1080/02331939908844424.  Google Scholar

[9]

V. F. Demyanov and V. A. Roschina, Optimality conditions in terms of upper and lower exhausters,, Optimization, 55 (2006), 525.  doi: 10.1080/02331930600815777.  Google Scholar

[10]

V. F. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis,, Approximation & Optimization, (1995).   Google Scholar

[11]

V. F. Demyanov and A. M. Rubinov, Exhaustive families of approximations revisited,From convexity to nonconvexity,, Nonconvex Optim. Appl., 55 (2001), 43.  doi: 10.1007/978-1-4613-0287-2_4.  Google Scholar

[12]

B. N. Pshenichny, Convex Analysis and Extremal Problems (in Russian)., Nauka, (1980).   Google Scholar

[13]

R. T. Rockafellar, Convex Analysis,, Princeton University Press, (1970).   Google Scholar

[14]

V. A. Roshchina, Limited exhausters and optimality conditions,, Control Processes and Stability: Proceedings of the 36-th international conference of students and graduate students, (2005), 521.   Google Scholar

[1]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25

[2]

Ethan Akin, Julia Saccamano. Generalized intransitive dice II: Partition constructions. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021005

[3]

Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225

[4]

Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209

[5]

Samir Adly, Oanh Chau, Mohamed Rochdi. Solvability of a class of thermal dynamical contact problems with subdifferential conditions. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 91-104. doi: 10.3934/naco.2012.2.91

[6]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533

[7]

Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 307-320. doi: 10.3934/naco.2020027

[8]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810

[9]

Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973

[10]

Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005

[11]

Graziano Crasta, Philippe G. LeFloch. Existence result for a class of nonconservative and nonstrictly hyperbolic systems. Communications on Pure & Applied Analysis, 2002, 1 (4) : 513-530. doi: 10.3934/cpaa.2002.1.513

[12]

Enkhbat Rentsen, Battur Gompil. Generalized nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022

[13]

Sohana Jahan. Discriminant analysis of regularized multidimensional scaling. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 255-267. doi: 10.3934/naco.2020024

[14]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327

[15]

Mats Gyllenberg, Jifa Jiang, Lei Niu, Ping Yan. On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 615-650. doi: 10.3934/dcds.2018027

[16]

Rabiaa Ouahabi, Nasr-Eddine Hamri. Design of new scheme adaptive generalized hybrid projective synchronization for two different chaotic systems with uncertain parameters. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2361-2370. doi: 10.3934/dcdsb.2020182

[17]

José Raúl Quintero, Juan Carlos Muñoz Grajales. On the existence and computation of periodic travelling waves for a 2D water wave model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 557-578. doi: 10.3934/cpaa.2018030

[18]

Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115

[19]

Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1

[20]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (26)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]