Abstract
This paper presents a globally convergent multiplier method which utilizes an explicit formula for the multiplier. The algorithm solves finite dimensional optimization problems with equality constraints. A unique feature of the algorithm is that it automatically calculates a value for the penalty coefficient, which, under certain assumptions, leads to global convergence.
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References
K.J. Arrow and R.M. Solow, “Gradient methods for constrained maxima with weakened assumptions”, in: K.J. Arrow, L. Hurwicz and H. Uzawa, eds.,Studies in linear and nonlinear programming (Stanford University Press, Stanford, Calif., 1958) pp. 166–176.
D.P. Bertsekas, “Combined primal-dual and penalty methods for constrained minimization”,SIAM Journal on Control, to appear.
D.P. Bertsekas, “On penalty and multiplier methods for constrained minimization”,SIAM Journal on Control, to appear.
J.D. Buys, “Dual algorithm for constrained optimization problems”, Doctoral Dissertation, University of Leiden (June 1972).
R. Fletcher, “A class of methods for nonlinear programming with termination and convergence properties”, in: J. Abadie, ed.,Integer and nonlinear programming (North-Holland, Amsterdam, 1970) pp. 157–175.
R. Fletcher and S. Lill, “A class of methods for nonlinear programming II: computational experience”, in: J.B. Rosen, O.L. Mangasarian and K. Ritter, eds.,Nonlinear programming (Academic Press, New York, 1971).
R. Fletcher, “A class of methods for nonlinear programming III: rate of convergence”, in: F.A. Lootsma, ed.,Numerical methods for nonlinear optimization (Academic Press, New York, 1972).
P.C. Haarhoff and J.D. Buys, “A new method for the optimization of a nonlinear function subject to nonlinear constraints”,The Computer Journal 13 (1970) 178–184.
M.R. Hestenes, “The Weierstrass E-function in the calculus of variations”,Transactions of the American Mathematical Society 60 (1946) 51–71.
M.R. Hestenes, “Multiplier and gradient methods”,Journal of Optimization Theory and Applications 4 (1969) 303–320.
K. Levenberg, “A method for the solution of certain nonlinear problems in least squares”,Quarterly of Applied Mathematics 2 (1944) 164–168.
D. Marquardt, “An algorithm for least squares estimation of nonlinear parameters”,SIAM Journal on Applied Mathematics 11 (1963) 431–441.
K. Mârtensson, “A new approach to constrained function optimization”,Journal of Optimization Theory and Applications 12 (1973) 531–554.
A. Miele, P.E. Moseley and E.E. Cragg, “Numerical experiments on Hestenes' method of multipliers for mathematical programming problems”, Aero-Astronautics Rept. No. 85, Rice University, Houston, Texas (1971).
A. Miele, P.E. Moseley and E.E. Cragg, “A modification of the method of multipliers for mathematical programming problems”, in: A.V. Balakrishnan, ed.,Techniques of optimization (Academic Press, New York, 1972).
E. Polak, R.W.H. Sargent and D.J. Sebastian, “On the convergence of sequential minimization algorithms”,Journal of Optimization Theory and Applications 14 (1974) 439–442.
M.J.D. Powell, “A method for nonlinear constraints in minimization problems”, in: R. Fletcher, ed.,Optimization (Academic Press, New York, 1969).
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Research sponsored by the Joint Services Electronics Program, Contract F44620-71-C-0087 and the National Science Foundation, Grant GK-37672.
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Mukai, H., Polak, E. A quadratically convergent primal-dual algorithm with global convergence properties for solving optimization problems with equality constraints. Mathematical Programming 9, 336–349 (1975). https://doi.org/10.1007/BF01681354
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DOI: https://doi.org/10.1007/BF01681354