Exact penalty functions in proximal bundle methods for constrained convex nondifferentiable minimization

This paper presents new versions of proximal bundle methods for solving convex constrained nondifferentiable minimization problems. The methods employϑ1 orϑ∞ exact penalty functions with new penalty updates that limit unnecessary penalty growth. In contrast to other methods, some of them are insensitive to problem function scaling. Global convergence of the methods is established, as well as finite termination for polyhedral problems. Some encouraging numerical experience is reported. The ideas presented may also be used in variable metric methods for smooth nonlinear programming.

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

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

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

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

[5]  Dimitri P. Bertsekas,et al.  Constrained Optimization and Lagrange Multiplier Methods , 1982 .

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

[7]  C. Lemaréchal.  Nondifferentiable optimization , 1989 .

[8]  B. N. Pshenichnyi,et al.  Numerical Methods in Extremal Problems. , 1978 .

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

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

[11]  O. Mangasarian.  Sufficiency of exact penalty minimization , 1985 .

[12]  Krzysztof C. Kiwiel,et al.  Proximity control in bundle methods for convex nondifferentiable minimization , 1990, Math. Program..

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

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

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

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

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

[18]  D. Mayne,et al.  On the Extension of Constrained Optimization Algorithms from Differentiable to Nondifferentiable Problems , 1983 .

[19]  A. Auslender,et al.  Penalty-proximal methods in convex programming , 1987 .

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