Abstract
In this paper, an entropy-like proximal method for the minimization of a convex function subject to positivity constraints is extended to an interior algorithm in two directions. First, to general linearly constrained convex minimization problems and second, to variational inequalities on polyhedra. For linear programming, numerical results are presented and quadratic convergence is established.
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References
A. Auslender,Optimisation — Methodes Numériques (Masson, Paris, 1976).
H. Brezis and A. Haraux, “Image d'une somme d'operateurs monotones et applications,”Israel Journal on Mathematics 23 (1976) 165–186.
Y. Censor and S. Zenios, “The proximal minimization algorithm with D-functions,”Journal of Optimization Theory and Applications 73 (1992) 451–464.
G. Chen and M. Teboulle, “Convergence analysis of proximal-like optimization algorithm using Bregman functions,”SIAM Journal on Optimization 3 (1993) 538–543.
R. Cominetti and J. San Martín, “Asymptotic analysis of the exponential penalty trajectory in linear programming,”Mathematical Programming 67 (2) (1994) 169–187.
J. Eckstein, “Nonlinear proximal point algorithms using Bregman functions, with applications to convex programming,”Mathematics of Operations Research 18 (1993) 202–226.
P.P.B. Eggermont, “Multiplicative iterative algorithms for convex programming,”Linear Algebra and its Applications 130 (1990) 25–42.
K.R. Frisch, “The logarithmic potential method of convex programming,” Memorandum, University Institute of Economics, Oslo, Norway (1955).
P.E. Gill, W. Murray, M.A. Saunders, J.A. Tomlin and M.H. Wright, “On projected Newton barrier methods for linear programming and an equivalence to Karmarkar's projective method,”Mathematical Programming 36 (1986) 183–209.
C.C. Gonzaga, “Interior point algorithm for linear programming with inequality constraints,”Mathematical Programming 52 (1991) 209–226.
C.C. Gonzaga, “Path-following methods for linear programming,”SIAM Review 34 (1992) 167–224.
O. Güler, “Existence of interior points and interior paths in nonlinear monotone complementarity problems,”Mathematics of Operations Research 18 (1993) 128–147.
A.J. Hoffman, “On approximate solutions of systems of linear inequalities,”Journal of the National Bureau of Standards 49 (1952) 263–265.
P. Huard, “Resolution of mathematical programming with nonlinear constraints by the method of centers nonlinear programming,” in: J. Abadie, ed.,Nonlinear Programming (North-Holland, Amsterdam, 1967) pp. 207–219.
A.N. Iusem, B.F. Svaiter and M. Teboulle, “Entropy-like proximal methods in convex programming,”Mathematics of Operations Research 19 (1994) 790–814.
N. Iusem and M. Teboulle, “Convergence rate analysis of nonquadratic proximal and augmented Lagrangian methods for convex and linear programming,” Technical Report 92-17, Department of Mathematics and Statistics, University of Maryland, Baltimore, MD (1992).
N. Karmarkar, “A new polynomial time algorithm for linear programming,”Combinatorica 4 (1984) 373–395.
M. Kojima, S. Mizuno and A. Yoshise, “A polynomial time algorithm for linear complementarity problems,”Mathematical Programming 44 (1989) 1–26.
B. Martinet, “Perturbation des methodes d' optimisation, applications,”R.A.I.R.O. Analyse Numérique (1978) 153–171.
N. Megiddo, “Pathways to the optimal set in linear programming,” in: N. Megiddo, ed.,Interior Points and Related Methods (Springer, New York, 1989) pp. 131–158.
G.J.M. Minty, “On the maximal domain of a monotone function,”Michigan Mathematical Journal 3 (1967) 135–137.
Y. Nesterov and A. Nemirovsky, “Path-following polynomial time algorithm for monotone variational inequalities,” Research Report 4224, Central Economic and Mathematical Institute, Mathematical Departement, USSR Academy of Sciences (1991).
J. Renegar, “A polynomial-time algorithm, based on Newton's method, for linear programming,”Mathematical Programming 40 (1988) 59–94.
R.T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,”Transactions of the American Mathematical Society 149 (1970) 75–88.
R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970).
C. Roos, “New trajectory-following polynomial-time algorithm for linear programming problems,”Journal of Optimization Theory and Applications 3 (1989) 433–458.
G.Y. Sonnevend, “An analytic center for polyhedrons and new class of global algorithms for linear (smooth, convex) programming,” in: Lecture Notes in Control and Information Sciences, Vol. 84 (Springer, New York, 1985) pp. 866–876.
M. Teboulle, “Entropic proximal mappings with applications to nonlinear programming,”Mathematics of Operations Research 17 (1992) 670–690.
P.M. Vaidya, “An algorithm for linear programming which requires O((m + n)n 2 + (m + n)1,5 n)L) arithmetic operations,”Mathematical Programming 47 (1990) 175–202.
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Corresponding author. His research has been supported by C.E.E grants: CI1* CT 92-0046.
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Auslender, A., Haddou, M. An interior-proximal method for convex linearly constrained problems and its extension to variational inequalities. Mathematical Programming 71, 77–100 (1995). https://doi.org/10.1007/BF01592246
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DOI: https://doi.org/10.1007/BF01592246