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Journal articles on the topic 'Nonsmooth critical point theory'

Author: Grafiati

Published: 4 June 2021

Last updated: 11 March 2023

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1

Degiovanni, Marco. "Nonsmooth critical point theory and applications." Nonlinear Analysis: Theory, Methods & Applications 30, no. 1 (December 1997): 89–99. http://dx.doi.org/10.1016/s0362-546x(97)00259-9.

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2

Liu, Jiaquan, and Yuxia Guo. "Critical point theory for nonsmooth functionals." Nonlinear Analysis: Theory, Methods & Applications 66, no. 12 (June 2007): 2731–41. http://dx.doi.org/10.1016/j.na.2006.04.003.

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3

Degiovanni, Marco, and Marco Marzocchi. "A critical point theory for nonsmooth functional." Annali di Matematica Pura ed Applicata 167, no. 1 (December 1994): 73–100. http://dx.doi.org/10.1007/bf01760329.

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4

Campa, Ines, and Marco Degiovanni. "Subdifferential Calculus and Nonsmooth Critical Point Theory." SIAM Journal on Optimization 10, no. 4 (January 2000): 1020–48. http://dx.doi.org/10.1137/s1052623499353169.

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5

Livrea, Roberto, and Giovanni Molica Bisci. "Some remarks on nonsmooth critical point theory." Journal of Global Optimization 37, no. 2 (August 5, 2006): 245–61. http://dx.doi.org/10.1007/s10898-006-9047-7.

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6

Kourogenis, Nikolaos C., and Nikolaos S. Papageorgiou. "Nonsmooth critical point theory and nonlinear elliptic equations at resonance." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 69, no. 2 (October 2000): 245–71. http://dx.doi.org/10.1017/s1446788700002202.

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AbstractIn this paper we complete two tasks. First we extend the nonsmooth critical point theory of Chang to the case where the energy functional satisfies only the weaker nonsmooth Cerami condition and we also relax the boundary conditions. Then we study semilinear and quasilinear equations (involving the p-Laplacian). Using a variational approach we establish the existence of one and of multiple solutions. In simple existence theorems, we allow the right hand side to be discontinuous. In that case in order to have an existence theory, we pass to a multivalued approximation of the original problem by, roughly speaking, filling in the gaps at the discontinuity points.

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7

Kourogenis, Nikolaos C., and Nikolaos S. Papageorgiou. "Nonsmooth critical point theory and nonlinear elliptic equations at resonance." Kodai Mathematical Journal 23, no. 1 (2000): 108–35. http://dx.doi.org/10.2996/kmj/1138044160.

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8

Corvellec, J. N., V. V. Motreanu, and C. Saccon. "Doubly resonant semilinear elliptic problems via nonsmooth critical point theory." Journal of Differential Equations 248, no. 8 (April 2010): 2064–91. http://dx.doi.org/10.1016/j.jde.2009.11.005.

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9

Bartels, Sven G., Ludwig Kuntz, and Stefan Scholtes. "Continuous selections of linear functions and nonsmooth critical point theory." Nonlinear Analysis: Theory, Methods & Applications 24, no. 3 (February 1995): 385–407. http://dx.doi.org/10.1016/0362-546x(95)91645-6.

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10

Costea, Nicuşor, Mihály Csirik, and Csaba Varga. "Linking-Type Results in Nonsmooth Critical Point Theory and Applications." Set-Valued and Variational Analysis 25, no. 2 (August 18, 2016): 333–56. http://dx.doi.org/10.1007/s11228-016-0383-6.

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11

Yang, Bian-Xia, and Hong-Rui Sun. "Periodic Solutions for Semilinear Fourth-Order Differential Inclusions via Nonsmooth Critical Point Theory." Journal of Function Spaces 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/816490.

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Three periodic solutions with prescribed wavelength for a class of semilinear fourth-order differential inclusions are obtained by using a nonsmooth version critical point theorem. Some results of previous related literature are extended.

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12

Gao, Dongdong, and Jianli Li. "Three Solutions for Fourth-Order Impulsive Differential Inclusions via Nonsmooth Critical Point Theory." Journal of Function Spaces 2018 (September 6, 2018): 1–9. http://dx.doi.org/10.1155/2018/1871453.

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An existence of at least three solutions for a fourth-order impulsive differential inclusion will be obtained by applying a nonsmooth version of a three-critical-point theorem. Our results generalize and improve some known results.

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13

Squassina, Marco. "Weak solutions to general Euler's equations via nonsmooth critical point theory." Annales de la faculté des sciences de Toulouse Mathématiques 9, no. 1 (2000): 113–31. http://dx.doi.org/10.5802/afst.956.

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14

Filippakis, Michael E., and Nikolaos S. Papageorgiou. "Solutions for nonlinear variational inequalities with a nonsmooth potential." Abstract and Applied Analysis 2004, no. 8 (2004): 635–49. http://dx.doi.org/10.1155/s1085337504312017.

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First we examine a resonant variational inequality driven by thep-Laplacian and with a nonsmooth potential. We prove the existence of a nontrivial solution. Then we use this existence theorem to obtain nontrivial positive solutions for a class of resonant elliptic equations involving thep-Laplacian and a nonsmooth potential. Our approach is variational based on the nonsmooth critical point theory for functionals of the formφ=φ1+φ2withφ1locally Lipschitz andφ2proper, convex, lower semicontinuous.

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15

Kourogenis, Nikolaos C., and Nikolaos S. Papageorgiou. "Multiple solutions for nonlinear discontinuous strongly resonant elliptic problems." Abstract and Applied Analysis 5, no. 2 (2000): 119–35. http://dx.doi.org/10.1155/s1085337500000269.

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We consider quasilinear strongly resonant problems with discontinuous right-hand side. To develop an existence theory we pass to a multivalued problem by, roughly speaking, filling in the gaps at the discontinuity points. We prove the existence of at least three nontrivial solutions. Our approach uses the nonsmooth critical point theory for locally Lipschitz functionals due to Chang (1981) and a generalized version of the Ekeland variational principle. At the end of the paper we show that the nonsmooth Palais-Smale (PS)-condition implies the coercivity of the functional, extending this way a well-known result of the “smooth” case.

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16

Ning, Yan, Daowei Lu, and Anmin Mao. "Existence and subharmonicity of solutions for nonsmooth $ p $-Laplacian systems." AIMS Mathematics 6, no. 10 (2021): 10947–63. http://dx.doi.org/10.3934/math.2021636.

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<abstract><p>In this paper we study nonlinear periodic systems driven by the vectorial $ p $-Laplacian with a nonsmooth locally Lipschitz potential function. Using variational methods based on nonsmooth critical point theory, some existence of periodic and subharmonic results are obtained, which improve and extend related works.</p></abstract>

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17

Conti, Monica, and Filippo Gazzola. "Positive entire solutions of quasilinear elliptic problems via nonsmooth critical point theory." Topological Methods in Nonlinear Analysis 8, no. 2 (December 1, 1996): 275. http://dx.doi.org/10.12775/tmna.1996.033.

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18

Kyritsi, Sophia Th, and Nikolaos S. Papageorgiou. "Nonsmooth critical point theory on closed convex sets and nonlinear hemivariational inequalities." Nonlinear Analysis: Theory, Methods & Applications 61, no. 3 (May 2005): 373–403. http://dx.doi.org/10.1016/j.na.2004.12.001.

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19

Iannizzotto, Antonio. "Three Solutions for a Partial Differential Inclusion Via Nonsmooth Critical Point Theory." Set-Valued and Variational Analysis 19, no. 2 (July 28, 2010): 311–27. http://dx.doi.org/10.1007/s11228-010-0145-9.

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20

He, Xiumei. "Multiplicity of Solutions for a Modified Schrödinger-Kirchhoff-Type Equation in RN." Discrete Dynamics in Nature and Society 2015 (2015): 1–9. http://dx.doi.org/10.1155/2015/179540.

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21

Filippakis, Michael E., and Nikolaos S. Papageorgiou. "Existence of Positive Solutions for Nonlinear Noncoercive Hemivariational Inequalities." Canadian Mathematical Bulletin 50, no. 3 (September 1, 2007): 356–64. http://dx.doi.org/10.4153/cmb-2007-034-6.

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AbstractIn this paper we investigate the existence of positive solutions for nonlinear elliptic problems driven by the p-Laplacian with a nonsmooth potential (hemivariational inequality). Under asymptotic conditions that make the Euler functional indefinite and incorporate in our framework the asymptotically linear problems, using a variational approach based on nonsmooth critical point theory, we obtain positive smooth solutions. Our analysis also leads naturally to multiplicity results.

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22

Papalini, Francesca. "Nonlinear Periodic Systems with thep-Laplacian: Existence and Multiplicity Results." Abstract and Applied Analysis 2007 (2007): 1–23. http://dx.doi.org/10.1155/2007/80394.

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We study second-order nonlinear periodic systems driven by the vectorp-Laplacian with a nonsmooth, locally Lipschitz potential function. Under minimal and natural hypotheses on the potential and using variational methods based on the nonsmooth critical point theory, we prove existence theorems and a multiplicity result. We conclude the paper with an existence theorem for the scalar problem, in which the energy functional is indefinite (unbounded from both above and below).

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23

YANG, BIAN-XIA, and HONG-RUI SUN. "EXISTENCE OF SOLUTION FOR A FRACTIONAL DIFFERENTIAL INCLUSION VIA NONSMOOTH CRITICAL POINT THEORY." Korean Journal of Mathematics 23, no. 4 (December 30, 2015): 537–55. http://dx.doi.org/10.11568/kjm.2015.23.4.537.

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24

Aouaoui, Sami. "Multiplicity result for some nonlocal anisotropic equation via nonsmooth critical point theory approach." Applied Mathematics and Computation 218, no. 2 (September 2011): 532–41. http://dx.doi.org/10.1016/j.amc.2011.05.097.

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25

Degla, Guy, Cyrille Dansou, and Fortuné Dohemeto. "On Nonsmooth Global Implicit Function Theorems for Locally Lipschitz Functions from Banach Spaces to Euclidean Spaces." Abstract and Applied Analysis 2022 (July 28, 2022): 1–19. http://dx.doi.org/10.1155/2022/1021461.

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In this paper, we establish a generalization of the Galewski-Rădulescu nonsmooth global implicit function theorem to locally Lipschitz functions defined from infinite dimensional Banach spaces into Euclidean spaces. Moreover, we derive, under suitable conditions, a series of results on the existence, uniqueness, and possible continuity of global implicit functions that parametrize the set of zeros of locally Lipschitz functions. Our methods rely on a nonsmooth critical point theory based on a generalization of the Ekeland variational principle.

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26

Gasiński, Leszek, and Nikolaos S. Papageorgiou. "Nonlinear hemivariational inequalities at resonance." Bulletin of the Australian Mathematical Society 60, no. 3 (December 1999): 353–64. http://dx.doi.org/10.1017/s0004972700036546.

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In this paper we consider nonlinear hemivariational inequalities involving the p-Laplacian at resonance. We prove the existence of a nontrivial solution. Our approach is variational based on the critical point theory for nonsmooth, locally Lipschitz functionals due to Chang.

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27

Zhang, Liang, and Peng Zhang. "Periodic Solutions of Second-Order Differential Inclusions Systems with -Laplacian." Abstract and Applied Analysis 2012 (2012): 1–24. http://dx.doi.org/10.1155/2012/475956.

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The existence of periodic solutions for nonautonomous second-order differential inclusion systems with -Laplacian is considered. We get some existence results of periodic solutions for system, a.e. , , by using nonsmooth critical point theory. Our results generalize and improve some theorems in the literature.

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28

Lei, Chun-Yu, and Jia-Feng Liao. "Positive radial symmetric solutions for a class of elliptic problems with critical exponent and -1 growth." Advances in Nonlinear Analysis 10, no. 1 (January 1, 2021): 1222–34. http://dx.doi.org/10.1515/anona-2020-0174.

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Abstract In this paper, we consider a class of semilinear elliptic equation with critical exponent and -1 growth. By using the critical point theory for nonsmooth functionals, two positive solutions are obtained. Moreover, the symmetry and monotonicity properties of the solutions are proved by the moving plane method. Our results improve the corresponding results in the literature.

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29

Grossi, Massimo, Angela Pistoia, and Juncheng Wei. "Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory." Calculus of Variations and Partial Differential Equations 11, no. 2 (September 2000): 143–75. http://dx.doi.org/10.1007/pl00009907.

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30

Ge, Bin, and Ji-Hong Shen. "Multiple Solutions for a Class of Differential Inclusion System Involving the(p(x),q(x))-Laplacian." Abstract and Applied Analysis 2012 (2012): 1–19. http://dx.doi.org/10.1155/2012/971243.

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We consider a differential inclusion system involving the(p(x),q(x))-Laplacian with Dirichlet boundary condition on a bounded domain and obtain two nontrivial solutions under appropriate hypotheses. Our approach is variational and it is based on the nonsmooth critical point theory for locally Lipschitz functions.

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31

Kourogenis, Nikolaos C., and Nikolaos S. Papageorgiou. "Existence theorems for elliptic hemivariational inequalities involving thep-Laplacian." Abstract and Applied Analysis 7, no. 5 (2002): 259–77. http://dx.doi.org/10.1155/s1085337502000908.

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We study quasilinear hemivariational inequalities involving thep-Laplacian. We prove two existence theorems. In the first, we allow “crossing” of the principal eigenvalue by the generalized potential, while in the second, we incorporate problems at resonance. Our approach is based on the nonsmooth critical point theory for locally Lipschitz energy functionals.

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32

Filippakis, Michael, Leszek Gasiński, and Nikolaos S. Papageorgiou. "Multiplicity Results for Nonlinear Neumann Problems." Canadian Journal of Mathematics 58, no. 1 (February 1, 2006): 64–92. http://dx.doi.org/10.4153/cjm-2006-004-6.

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AbstractIn this paper we study nonlinear elliptic problems of Neumann type driven by the p-Laplacian differential operator. We look for situations guaranteeing the existence of multiple solutions. First we study problems which are strongly resonant at infinity at the first (zero) eigenvalue. We prove five multiplicity results, four for problems with nonsmooth potential and one for problems with a C1-potential. In the last part, for nonsmooth problems in which the potential eventually exhibits a strict super-p-growth under a symmetry condition, we prove the existence of infinitely many pairs of nontrivial solutions. Our approach is variational based on the critical point theory for nonsmooth functionals. Also we present some results concerning the first two elements of the spectrum of the negative p-Laplacian with Neumann boundary condition.

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33

Tian, Yu, and Johnny Henderson. "Three anti-periodic solutions for second-order impulsive differential inclusions via nonsmooth critical point theory." Nonlinear Analysis: Theory, Methods & Applications 75, no. 18 (December 2012): 6496–505. http://dx.doi.org/10.1016/j.na.2012.07.025.

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34

Littig, Samuel, and Friedemann Schuricht. "Perturbation results involving the 1-Laplace operator." Advances in Calculus of Variations 12, no. 3 (July 1, 2019): 277–302. http://dx.doi.org/10.1515/acv-2017-0006.

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AbstractWe consider perturbed eigenvalue problems of the 1-Laplace operator and verify the existence of a sequence of solutions. It is shown that the eigenvalues of the perturbed problem converge to the corresponding eigenvalue of the unperturbed problem as the perturbation becomes small. The results rely on nonsmooth critical point theory based on the weak slope.

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35

D'AVENIA, PIETRO, EUGENIO MONTEFUSCO, and MARCO SQUASSINA. "ON THE LOGARITHMIC SCHRÖDINGER EQUATION." Communications in Contemporary Mathematics 16, no. 02 (April 2014): 1350032. http://dx.doi.org/10.1142/s0219199713500326.

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In the framework of the nonsmooth critical point theory for lower semi-continuous functionals, we propose a direct variational approach to investigate the existence of infinitely many weak solutions for a class of semi-linear elliptic equations with logarithmic nonlinearity arising in physically relevant situations. Furthermore, we prove that there exists a unique positive solution which is radially symmetric and nondegenerate.

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36

Degiovanni, Marco, and Friedemann Schuricht. "Buckling of nonlinearly elastic rods in the presence of obstacles treated by nonsmooth critical point theory." Mathematische Annalen 311, no. 4 (August 1, 1998): 675–728. http://dx.doi.org/10.1007/s002080050206.

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37

FILIPPAKIS, MICHAEL E., and NIKOLAOS S. PAPAGEORGIOU. "MULTIPLE SOLUTIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH A DISCONTINUOUS NONLINEARITY." Analysis and Applications 04, no. 01 (January 2006): 1–18. http://dx.doi.org/10.1142/s021953050600067x.

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We consider a nonlinear elliptic equation driven by the p-Laplacian with a discontinuous nonlinearity. Such problems have a "multivalued" and a "single-valued" interpretation. We are interested in the latter and we prove the existence of at least two distinct solutions, both smooth and one strictly positive. Our approach is variational based on the nonsmooth critical point theory for locally Lipschitz functions, coupled with penalization and truncation techniques.

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38

Dinu, T. L. "ENTIRE SOLUTIONS OF SCHRÖDINGER ELLIPTIC SYSTEMS WITH DISCONTINUOUS NONLINEARITY AND SIGN‐CHANGING POTENTIAL." Mathematical Modelling and Analysis 11, no. 3 (September 30, 2006): 229–42. http://dx.doi.org/10.3846/13926292.2006.9637315.

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We establish the existence of an entire solution for a class of stationary Schrodinger systems with subcritical discontinuous nonlinearities and lower bounded potentials that blow‐up at infinity. The proof is based on the critical point theory in the sense of Clarke and we apply the Mountain Pass Lemma for locally Lipschitz functionals. Our result generalizes in a nonsmooth framework the result of Rabinowitz [16] on the existence of entire solutions of the nonlinear Schrodinger equation.

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39

Dinu, Teodora-Liliana. "Standing wave solutions of Schrödinger systems with discontinuous nonlinearity in anisotropic media." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–13. http://dx.doi.org/10.1155/ijmms/2006/73619.

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We establish the existence of an entire solution for a class of stationary Schrödinger systems with subcritical discontinuous nonlinearities and lower bounded potentials that blow up at infinity. The proof is based on the critical point theory in the sense of Clarke and we apply Chang's version of the mountain pass lemma for locally Lipschitz functionals. Our result generalizes in a nonsmooth framework a result of Rabinowitz (1992) related to entire solutions of the Schrödinger equation.

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40

Dinu, Teodora-Liliana. "Ground State Solutions of Nonlinear Stationary Schrödinger Systems with Discontinuous Nonlinearity and Variable Potential." gmj 13, no. 3 (September 2006): 433–45. http://dx.doi.org/10.1515/gmj.2006.433.

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Abstract We establish the existence of an entire solution for a class of stationary Schrödinger systems with subcritical discontinuous nonlinearities and lower bounded potentials that blow-up at infinity. The proof is based on the critical point theory in the sense of Clarke and we apply the Mountain Pass Lemma for locally Lipschitz functionals. Our result generalizes in a nonsmooth framework a result of Rabinowitz [Z. Angew. Math. Phys. 43: 270–291, 1992] on the existence of entire solutions of the nonlinear Schrödinger equation.

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41

Duan, Lian, Lihong Huang, and Zuowei Cai. "ON EXISTENCE OF THREE SOLUTIONS FOR $p(x)$-KIRCHHOFF TYPE DIFFERENTIAL INCLUSION PROBLEM VIA NONSMOOTH CRITICAL POINT THEORY." Taiwanese Journal of Mathematics 19, no. 2 (March 2015): 397–418. http://dx.doi.org/10.11650/tjm.19.2015.4097.

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42

Peng, Linyan, Hongmin Suo, Deke Wu, Hongxi Feng, and Chunyu Lei. "Multiple positive solutions for a logarithmic Schrödinger–Poisson system with singular nonelinearity." Electronic Journal of Qualitative Theory of Differential Equations, no. 90 (2021): 1–15. http://dx.doi.org/10.14232/ejqtde.2021.1.90.

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In this article, we devote ourselves to investigate the following logarithmic Schrödinger–Poisson systems with singular nonlinearity { − Δ u + ϕ u = | u | p−2 u log ⁡ | u | + λ u γ , i n Ω , − Δ ϕ = u 2 , i n Ω , u = ϕ = 0 , o n ∂ Ω , where Ω is a smooth bounded domain with boundary 0 < γ < 1 , p ∈ ( 4 , 6 ) and λ > 0 is a real parameter. By using the critical point theory for nonsmooth functional and variational method, the existence and multiplicity of positive solutions are established.

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43

Carl, Siegfried, and Dumitru Motreanu. "Sign-Changing Solutions for Nonlinear Elliptic Problems Depending on Parameters." International Journal of Differential Equations 2010 (2010): 1–33. http://dx.doi.org/10.1155/2010/536236.

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The study of multiple solutions for quasilinear elliptic problems under Dirichlet or nonlinear Neumann type boundary conditions has received much attention over the last decades. The main goal of this paper is to present multiple solutions results for elliptic inclusions of Clarke's gradient type under Dirichlet boundary condition involving the -Laplacian which, in general, depend on two parameters. Assuming different structure and smoothness assumptions on the nonlinearities generating the multivalued term, we prove the existence of multiple constant-sign and sign-changing (nodal) solutions for parameters specified in terms of the Fučik spectrum of the -Laplacian. Our approach will be based on truncation techniques and comparison principles (sub-supersolution method) for elliptic inclusions combined with variational and topological arguments for, in general, nonsmooth functionals, such as, critical point theory, Mountain Pass Theorem, Second Deformation Lemma, and the variational characterization of the “beginning”of the Fu\v cik spectrum of the -Laplacian. In particular, the existence of extremal constant-sign solutions and their variational characterization as global (resp., local) minima of the associated energy functional will play a key-role in the proof of sign-changing solutions.

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44

Qing-Mei, Zhou, and Ge Bin. "Three Solutions for Inequalities Dirichlet Problem Driven byp(x)-Laplacian-Like." Abstract and Applied Analysis 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/575328.

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A class of nonlinear elliptic problems driven byp(x)-Laplacian-like with a nonsmooth locally Lipschitz potential was considered. Applying the version of a nonsmooth three-critical-point theorem, existence of three solutions of the problem is proved.

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45

LI, Zhouxin, and Yaotian SHEN. "Nonsmooth critical point theorems and its applications to quasilinear schrödinger equations." Acta Mathematica Scientia 36, no. 1 (January 2016): 73–86. http://dx.doi.org/10.1016/s0252-9602(15)30079-5.

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46

TENG, KAIMIN. "EXISTENCE AND MULTIPLICITY OF NONTRIVIAL SOLUTIONS FOR DEGENERATE HEMIVARIATIONAL INEQUALITIES INVOLVING LERAY–LIONS TYPE OPERATOR WITH CRITICAL GROWTH." Analysis and Applications 11, no. 01 (January 2013): 1350007. http://dx.doi.org/10.1142/s0219530513500073.

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In this paper, we investigate a hemivariational inequality involving Leray–Lions type operator with critical growth. Some existence and multiple results are obtained through using the concentration compactness principle of P. L. Lions and some nonsmooth critical point theorems.

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47

Jongen, H. Th, Jan J. Rückmann, and V. Shikhman. "MPCC: Critical Point Theory." SIAM Journal on Optimization 20, no. 1 (January 2009): 473–84. http://dx.doi.org/10.1137/080733693.

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48

Wu, Qiong, Jin-He Wang, Hong-Wei Zhang, Shuang Wang, and Li-Ping Pang. "Nonsmooth Optimization Method for H∞ Output Feedback Control." Asia-Pacific Journal of Operational Research 36, no. 03 (June 2019): 1950015. http://dx.doi.org/10.1142/s0217595919500155.

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This paper proposes a nonsmooth optimization method for [Formula: see text] output feedback control problem of linear time-invariant(LTI) systems based on bundle technique. We formulate this problem as a nonconvex and nonsmooth semi-infinite constrained optimization problem by quantifying both internal stability of closed-loop system and measurement of system performance, where [Formula: see text] norm of closed-loop transfer function and a stabilization channel is used. Our method uses progress function and bundle technique to solve the resulting problem which has a composite structure. We prove the convergence to a critical point from a feasible initial point and test some benchmarks to demonstrate the effectiveness of this method.

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49

Ferrara, Massimiliano, Giuseppe Caristi, and Amjad Salari. "Existence of Infinitely Many Periodic Solutions for Perturbed Semilinear Fourth-Order Impulsive Differential Inclusions." Abstract and Applied Analysis 2016 (2016): 1–12. http://dx.doi.org/10.1155/2016/5784273.

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This paper discusses the existence of infinitely many periodic solutions for a semilinear fourth-order impulsive differential inclusion with a perturbed nonlinearity and two parameters. The approach is based on a critical point theorem for nonsmooth functionals.

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50

Kazemi, Ehsan, and Liqiang Wang. "Asynchronous Delay-Aware Accelerated Proximal Coordinate Descent for Nonconvex Nonsmooth Problems." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 1528–35. http://dx.doi.org/10.1609/aaai.v33i01.33011528.

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Nonconvex and nonsmooth problems have recently attracted considerable attention in machine learning. However, developing efficient methods for the nonconvex and nonsmooth optimization problems with certain performance guarantee remains a challenge. Proximal coordinate descent (PCD) has been widely used for solving optimization problems, but the knowledge of PCD methods in the nonconvex setting is very limited. On the other hand, the asynchronous proximal coordinate descent (APCD) recently have received much attention in order to solve large-scale problems. However, the accelerated variants of APCD algorithms are rarely studied. In this paper, we extend APCD method to the accelerated algorithm (AAPCD) for nonsmooth and nonconvex problems that satisfies the sufficient descent property, by comparing between the function values at proximal update and a linear extrapolated point using a delay-aware momentum value. To the best of our knowledge, we are the first to provide stochastic and deterministic accelerated extension of APCD algorithms for general nonconvex and nonsmooth problems ensuring that for both bounded delays and unbounded delays every limit point is a critical point. By leveraging Kurdyka-Łojasiewicz property, we will show linear and sublinear convergence rates for the deterministic AAPCD with bounded delays. Numerical results demonstrate the practical efficiency of our algorithm in speed.

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