Proximity control in bundle methods for convex nondifferentiable minimization

AbstractProximal bundle methods for minimizing a convex functionf generate a sequence {xk} by takingxk+1 to be the minimizer of $$\hat f^k (x) + u^k |x - x^k |^2 /2$$ , where $$\hat f^k $$ is a sufficiently accurate polyhedral approximation tof anduk > 0. The usual choice ofuk = 1 may yield very slow convergence. A technique is given for choosing {uk} adaptively that eliminates sensitivity to objective scaling. Some encouraging numerical experience is reported.

[1]  H. Kunzi,et al.  Lectu re Notes in Economics and Mathematical Systems , 1975 .

[2]  R. Rockafellar.  Monotone Operators and the Proximal Point Algorithm , 1976 .

[3]  C. Lemaréchal.  Nonsmooth Optimization and Descent Methods , 1978 .

[4]  Timothy J. Lowe,et al.  A subgradient algorithm for certain minimax and minisum problems , 1978, Math. Program..

[5]  Klaus Schittkowski,et al.  More test examples for nonlinear programming codes , 1981 .

[6]  Willi Hock,et al.  Lecture Notes in Economics and Mathematical Systems , 1981 .

[7]  D. Hearn,et al.  A Subgradient Algorithm for Certain Minimax and Minisum Problems—The Constrained Case , 1982 .

[8]  R. Mifflin.  A modification and an extension of Lemarechal’s algorithm for nonsmooth minimization , 1982 .

[9]  Krzysztof C. Kiwiel,et al.  An aggregate subgradient method for nonsmooth convex minimization , 1983, Math. Program..

[10]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[11]  Masao Fukushima,et al.  A descent algorithm for nonsmooth convex optimization , 1984, Math. Program..

[12]  K. Kiwiel.  Methods of Descent for Nondifferentiable Optimization , 1985 .

[13]  K. Kiwiel.  An algorithm for linearly constrained convex nondifferentiable minimization problems , 1985 .

[14]  V. F. Dem'yanov,et al.  Nondifferentiable Optimization , 1985 .

[15]  K. Kiwiel.  An Exact Penalty Function Algorithm for Non-smooth Convex Constrained Minimization Problems , 1985 .

[16]  Roy L. Streit,et al.  ALGORITHM 635: an algorithm for the solution of systems of complex linear equations in the L∞ norm with constraints on the unknowns , 1985, TOMS.

[17]  Krzysztof C. Kiwiel.  A method of linearizations for linearly constrained nonconvex nonsmooth minimization , 1986, Math. Program..

[18]  C. Lemaréchal.  Constructing Bundle Methods for Convex Optimization , 1986 .

[19]  R. Streit.  Solution of systems of complex linear equations in the l ∞ 0E norm with constraints on the unknowns , 1986 .

[20]  Andrzej Ruszczynski,et al.  A regularized decomposition method for minimizing a sum of polyhedral functions , 1986, Math. Program..

[21]  K. Kiwiel.  A constraint linearization method for nondifferentiable convex minimization , 1987 .

[22]  A. Auslender.  Numerical methods for nondifferentiable convex optimization , 1987 .

[23]  K. Kiwiel.  A Dual Method for Certain Positive Semidefinite Quadratic Programming Problems , 1989 .