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Volumn 35, Issue 2, 2010, Pages 438-457

Proximal alternating minimization and projection methods for nonconvex problems: An approach based on the kurdyka-lojasiewicz inequality

Author keywords

Alternating minimization algorithms; Alternating projections algorithms; Convergence rate; Finite; Kurdyka Lojasiewicz inequality; Nonconvex optimization; O minimal structures; Proximal algorithms; Tame optimization

Indexed keywords

ALTERNATING MINIMIZATION ALGORITHMS; ALTERNATING PROJECTIONS; ALTERNATING PROJECTIONS ALGORITHMS; CONVERGENCE RATES; NONCONVEX OPTIMIZATION; O-MINIMAL STRUCTURE; O-MINIMAL STRUCTURES; PROXIMAL ALGORITHM;

EID: 77953092582     PISSN: 0364765X     EISSN: 15265471     Source Type: Journal    
DOI: 10.1287/moor.1100.0449     Document Type: Article
Times cited : (1125)

References (49)
  • 1
    • 33646735617 scopus 로고    scopus 로고
    • Convergence of the iterates of descent methods for analytic cost functions
    • Absil, P.-A., R. Mahony, B. Andrews. 2005. Convergence of the iterates of descent methods for analytic cost functions. SIAM J. Optim. 16(2) 31-547.
    • (2005) SIAM J. Optim. , vol.16 , Issue.2 , pp. 31-547
    • Absil, P.-A.1    Mahony, R.2    Andrews, B.3
  • 3
    • 45749101721 scopus 로고    scopus 로고
    • On the convergence of the proximal algorithm for nonsmooth functions involving analytic features
    • Attouch, H., J. Bolte. 2009. On the convergence of the proximal algorithm for nonsmooth functions involving analytic features. Math. Programming Ser. B 116(1-2) 5-16.
    • (2009) Math. Programming Ser. B , vol.116 , Issue.1 , pp. 5-16
    • Attouch, H.1    Bolte, J.2
  • 4
    • 33747712643 scopus 로고    scopus 로고
    • Inertia and reactivity in decision making as cognitive variational inequalities
    • Attouch, H., A. Soubeyran. 2006. Inertia and reactivity in decision making as cognitive variational inequalities. J. Convex Anal. 13(2) 207-224.
    • (2006) J. Convex Anal. , vol.13 , Issue.2 , pp. 207-224
    • Attouch, H.1    Soubeyran, A.2
  • 5
    • 48549105307 scopus 로고    scopus 로고
    • A new class of alternating proximal minimization algorithms with costs to move
    • Attouch, H., P. Redont, A. Soubeyran. 2007. A new class of alternating proximal minimization algorithms with costs to move. SIAM J. Optim. 18(3) 1061-1081.
    • (2007) SIAM J. Optim. , vol.18 , Issue.3 , pp. 1061-1081
    • Attouch, H.1    Redont, P.2    Soubeyran, A.3
  • 6
    • 48549093619 scopus 로고    scopus 로고
    • Alternating proximal algorithms for weakly coupled convex minimization problems. Applications to dynamical games and PDE's
    • Attouch, H., J. Bolte, P. Redont, A. Soubeyran. 2008. Alternating proximal algorithms for weakly coupled convex minimization problems. Applications to dynamical games and PDE's. J. Convex Anal. 15(3) 485-506.
    • (2008) J. Convex Anal. , vol.15 , Issue.3 , pp. 485-506
    • Attouch, H.1    Bolte, J.2    Redont, P.3    Soubeyran, A.4
  • 9
    • 0030246542 scopus 로고    scopus 로고
    • On projection algorithms for solving convex feasibility problems
    • Bauschke, H. H., J. M. Borwein. 1996. On projection algorithms for solving convex feasibility problems. SIAM Rev. 38(3) 367-426.
    • (1996) SIAM Rev. , vol.38 , Issue.3 , pp. 367-426
    • Bauschke, H.H.1    Borwein, J.M.2
  • 10
    • 41849132222 scopus 로고    scopus 로고
    • Joint minimization with alternating Bregman proximity operators
    • Bauschke, H. H., P. L. Combettes, D. Noll. 2006. Joint minimization with alternating Bregman proximity operators. Pacific J. Optim. 2(3) 401-424.
    • (2006) Pacific J. Optim. , vol.2 , Issue.3 , pp. 401-424
    • Bauschke, H.H.1    Combettes, P.L.2    Noll, D.3
  • 14
    • 36249023228 scopus 로고    scopus 로고
    • The Lojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems
    • Bolte, J., A. Daniilidis, A. Lewis. 2006. The Lojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems. SIAM J. Optim. 17(4) 1205-1223.
    • (2006) SIAM J. Optim. , vol.17 , Issue.4 , pp. 1205-1223
    • Bolte, J.1    Daniilidis, A.2    Lewis, A.3
  • 15
    • 33646703294 scopus 로고    scopus 로고
    • A nonsmooth Morse-Sard theorem for subanalytic functions
    • Bolte, J., A. Daniilidis, A. Lewis. 2006. A nonsmooth Morse-Sard theorem for subanalytic functions. J. Math. Anal. Appl. 321(2) 729-740.
    • (2006) J. Math. Anal. Appl. , vol.321 , Issue.2 , pp. 729-740
    • Bolte, J.1    Daniilidis, A.2    Lewis, A.3
  • 16
    • 35348955638 scopus 로고    scopus 로고
    • Clarke subgradients of stratifiable functions
    • Bolte, J., A. Daniilidis, A. Lewis, M. Shiota. 2007. Clarke subgradients of stratifiable functions. SIAM J. Optim. 18(2) 556-572.
    • (2007) SIAM J. Optim. , vol.18 , Issue.2 , pp. 556-572
    • Bolte, J.1    Daniilidis, A.2    Lewis, A.3    Shiota, M.4
  • 17
    • 77951625247 scopus 로고    scopus 로고
    • Characterizations of Lojasiewicz inequalities and applications: Subgradient flows, talweg, convexity
    • Bolte, J., A. Daniilidis, O. Ley, L. Mazet. 2010. Characterizations of Lojasiewicz inequalities and applications: Subgradient flows, talweg, convexity. Trans. Amer. Math. Soc. 362 3319-3363.
    • (2010) Trans. Amer. Math. Soc. , Issue.362 , pp. 3319-3363
    • Bolte, J.1    Daniilidis, A.2    Ley, O.3    Mazet., L.4
  • 18
    • 59749104367 scopus 로고    scopus 로고
    • From sparse solutions of systems of equations to sparse modeling of signals and images
    • Bruckstein, A., D. Donoho, M. Elad. 2009. From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Rev. 51(1) 34-81.
    • (2009) SIAM Rev. , vol.51 , Issue.1 , pp. 34-81
    • Bruckstein, A.1    Donoho, D.2    Elad, M.3
  • 19
    • 0035565698 scopus 로고    scopus 로고
    • Proximity function minimization using multiple Bregman projections, with applications to split feasibility and Kullback-Leibler distance minimization
    • Byrne, C., Y. Censor. 2001. Proximity function minimization using multiple Bregman projections, with applications to split feasibility and Kullback-Leibler distance minimization. Ann. Oper. Res. 105(1-4) 77-98.
    • (2001) Ann. Oper. Res. , vol.105 , Issue.1 , pp. 77-98
    • Byrne, C.1    Censor, Y.2
  • 21
    • 0027579732 scopus 로고
    • Signal recovery by best feasible approximation
    • Combettes, P. L. 1993. Signal recovery by best feasible approximation. IEEE Trans. Image Processing 2(2) 269-271.
    • (1993) IEEE Trans. Image Processing , vol.2 , Issue.2 , pp. 269-271
    • Combettes, P.L.1
  • 22
    • 16244375249 scopus 로고    scopus 로고
    • Proximal methods for cohypomonotone operators
    • Combettes, P. L., T. Pennanen. 2004. Proximal methods for cohypomonotone operators. SIAM J. Control Optim. 43(2) 731-742.
    • (2004) SIAM J. Control Optim. , vol.43 , Issue.2 , pp. 731-742
    • Combettes, P.L.1    Pennanen, T.2
  • 23
    • 30844438177 scopus 로고    scopus 로고
    • Signal recovery by proximal forward-backward splitting
    • Combettes, P. L., V. Wajs. 2005. Signal recovery by proximal forward-backward splitting. Multiscale Model. Simulation 4(4) 1168-1200.
    • (2005) Multiscale Model. Simulation , vol.4 , Issue.4 , pp. 1168-1200
    • Combettes, P.L.1    Wajs, V.2
  • 24
    • 0001560954 scopus 로고
    • Information geometry and alternating minimization procedures
    • Csiszár, I., G. Tusnády. 1984. Information geometry and alternating minimization procedures. Statist. Decisions (Suppl. 1) 205-237.
    • (1984) Statist. Decisions (Suppl. 1) , pp. 205-237
    • Csiszár, I.1    Tusnády, G.2
  • 27
    • 0032216898 scopus 로고    scopus 로고
    • The geometry of algorithms with orthogonality constraints
    • Edelman, A., A. Arias, S. T. Smith. 1999. The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20(2) 303-353.
    • (1999) SIAM J. Matrix Anal. Appl. , vol.20 , Issue.2 , pp. 303-353
    • Edelman, A.1    Arias, A.2    Smith, S.T.3
  • 28
    • 45749096678 scopus 로고    scopus 로고
    • Computing proximal points of nonconvex functions
    • Hare, W, C. Sagastizábal. 2009. Computing proximal points of nonconvex functions. Math. Program. Ser. B 116(1-2) 221-258.
    • (2009) Math. Program. Ser. B , vol.116 , Issue.1 , pp. 221-258
    • Hare, W.1    Sagastizábal, C.2
  • 29
    • 0030555447 scopus 로고    scopus 로고
    • Complements of subanalytic sets and existential formulas for analytic functions
    • Gabrielov, A. 1996. Complements of subanalytic sets and existential formulas for analytic functions. Inventiones Math. 125(1) 1-12.
    • (1996) Inventiones Math. , vol.125 , Issue.1 , pp. 1-12
    • Gabrielov, A.1
  • 30
    • 33847264561 scopus 로고    scopus 로고
    • Efficient rank reduction of correlation matrices
    • Grubisić, I., R. Pietersz. 2007. Efficient rank reduction of correlation matrices. Linear Algebra Its Appl. 422(2-3) 629-653.
    • (2007) Linear Algebra Its Appl. , vol.422 , Issue.2 , pp. 629-653
    • Grubisić, I.1    Pietersz, R.2
  • 31
    • 0000841855 scopus 로고    scopus 로고
    • Metric regularity and subdifferential calculus (Russian)
    • Ioffe, A. D. 2000. Metric regularity and subdifferential calculus. (Russian) Uspekhi Mat. Nauk 55(3) (333) 103-162.
    • (2000) Uspekhi Mat. Nauk , vol.55 , Issue.3 , pp. 103-162
    • Ioffe, A.D.1
  • 32
    • 0034361329 scopus 로고    scopus 로고
    • Surveys
    • Translation in 2000. Russian Math. Surveys 55 501-558.
    • (2000) Russian Math , vol.55 , pp. 501-558
  • 33
    • 0242679720 scopus 로고    scopus 로고
    • Inexact variants of the proximal point algorithm without monotonicity
    • Iusem, A. N., T. Pennanen, B. F. Svaiter. 2003. Inexact variants of the proximal point algorithm without monotonicity. SIAM J. Optim. 13(4) 1080-1097.
    • (2003) SIAM J. Optim. , vol.13 , Issue.4 , pp. 1080-1097
    • Iusem, A.N.1    Pennanen, T.2    Svaiter, B.F.3
  • 34
    • 0032385816 scopus 로고    scopus 로고
    • On gradients of functions definable in o-minimal structures
    • Kurdyka, K. 1998. On gradients of functions definable in o-minimal structures. Ann. Inst. Fourier 48(3) 769-783.
    • (1998) Ann. Inst. Fourier , vol.48 , Issue.3 , pp. 769-783
    • Kurdyka, K.1
  • 35
    • 48549094732 scopus 로고    scopus 로고
    • Alternating projection on manifolds
    • Lewis, A. S., J. Malick. 2008. Alternating projection on manifolds. Math. Oper. Res. 33(1) 216-234.
    • (2008) Math. Oper. Res. , vol.33 , Issue.1 , pp. 216-234
    • Lewis, A.S.1    Malick, J.2
  • 36
    • 70349387487 scopus 로고    scopus 로고
    • Local linear convergence for alternating and averaged nonconvex projections
    • Lewis, A. S., D. R. Luke, J. Malick. 2009. Local linear convergence for alternating and averaged nonconvex projections. Foundations Computational Math. 9(4) 485-513.
    • (2009) Foundations Computational Math. , vol.9 , Issue.4 , pp. 485-513
    • Lewis, A.S.1    Luke, D.R.2    Malick, J.3
  • 37
    • 0003121787 scopus 로고
    • On the Schwarz alternating method. III. A variant for nonoverlapping subdomains
    • T. F. Chan, R. Glowinski, J. Périaux, O. Widlund, eds, SIAM, Philadelphia
    • Lions, P. L. 1990. On the Schwarz alternating method. III. A variant for nonoverlapping subdomains. T. F. Chan, R. Glowinski, J. Périaux, O. Widlund, eds. Third Internat. Sympos. Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, 202-231.
    • (1990) Third Internat. Sympos. Domain Decomposition Methods for Partial Differential Equations , pp. 202-231
    • Lions, P.L.1
  • 38
    • 0003366108 scopus 로고
    • Une propriété topologique des sous-ensembles analytiques réels.
    • Éditions du centre National de la Recherche Scientifique, Paris
    • Lojasiewicz, S. 1963. Une propriété topologique des sous-ensembles analytiques réels. Les Équations aux Dérivées Partielles. Éditions du centre National de la Recherche Scientifique, Paris, 87-89.
    • (1963) Les Équations Aux Dérivées Partielles , pp. 87-89
    • Lojasiewicz, S.1
  • 39
    • 0001228050 scopus 로고
    • Sur la géométrie semi- et sous-analytique
    • Lojasiewicz, S. 1993. Sur la géométrie semi- et sous-analytique. Ann. Inst. Fourier 43 1575-1595.
    • (1993) Ann. Inst. Fourier , vol.43 , pp. 1575-1595
    • Lojasiewicz, S.1
  • 40
    • 50049097331 scopus 로고    scopus 로고
    • Finding best approximation pairs relative to a convex and prox-regular set in a Hilbert space
    • Luke, D. R. Finding best approximation pairs relative to a convex and prox-regular set in a Hilbert space. SIAM J. Optim. 19(2) 714-739.
    • SIAM J. Optim. , vol.19 , Issue.2 , pp. 714-739
    • Luke, D.R.1
  • 41
    • 34249303170 scopus 로고    scopus 로고
    • Variational analysis and generalized differentiation. I. basic theory
    • Springer-Verlag, Berlin
    • Mordukhovich, B. 2006. Variational analysis and generalized differentiation. I. basic theory. Grundlehren der Mathematischen Wissenschaften, Vol.330. Springer-Verlag, Berlin.
    • (2006) Grundlehren der Mathematischen Wissenschaften , vol.330
    • Mordukhovich, B.1
  • 42
    • 0003696537 scopus 로고    scopus 로고
    • Introductory lectures on convex optimization. A basic course
    • Kluwer Academic Publishers, Boston
    • Nesterov, Yu. 2004. Introductory lectures on convex optimization. A basic course. Applied Optimization, Vol.87. Kluwer Academic Publishers, Boston.
    • (2004) Applied Optimization , vol.87
    • Yu, N.1
  • 44
    • 0002613659 scopus 로고    scopus 로고
    • Geometry of subanalytic and semialgebraic sets
    • Birkhäuser, Boston
    • Shiota, M. 1997. Geometry of subanalytic and semialgebraic sets. Progress in Mathematics, Vol.150. Birkhäuser, Boston.
    • (1997) Progress in Mathematics , vol.150
    • Shiota, M.1
  • 45
    • 0035533631 scopus 로고    scopus 로고
    • Convergence of a block coordinate descent method for nondifferentiable minimization
    • Tseng, P. 2001. Convergence of a block coordinate descent method for nondifferentiable minimization. J. Optim. Theory Appl. 109(3) 475-494.
    • (2001) J. Optim. Theory Appl. , vol.109 , Issue.3 , pp. 475-494
    • Tseng, P.1
  • 46
    • 0003286956 scopus 로고    scopus 로고
    • Tame topology and o-minimal structures
    • Lecture Note Series. Cambridge University Press, Cambridge, UK
    • van den Dries, L. 1998. Tame topology and o-minimal structures. London Mathematical Society, Vol.248, Lecture Note Series. Cambridge University Press, Cambridge, UK.
    • (1998) London Mathematical Society , vol.248
    • Van Den Dries, L.1
  • 47
    • 0042224996 scopus 로고
    • Functional operators
    • Princeton University Press, Princeton, NJ
    • von Neumann, J. 1950. Functional operators. Annals ofMathematics Studies, Vol.22. Princeton University Press, Princeton, NJ.
    • (1950) Annals OfMathematics Studies , vol.22
    • Von Neumann, J.1
  • 49
    • 0030503308 scopus 로고    scopus 로고
    • Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function
    • Wilkie, A. J. 1996. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Amer. Math. Soc. 9(4) 1051-1094.
    • (1996) J. Amer. Math. Soc. , vol.9 , Issue.4 , pp. 1051-1094
    • Wilkie, A.J.1


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