Abstract
In this paper, we characterize a class of feasible direction methods in nonlinear programming through the concept of partial linearization of the objective function. Based on a feasible point, the objective is replaced by an arbitrary convex and continuously differentiable function, and the error is taken into account by a first-order approximation of it. A new feasible point is defined through a line search with respect to the original objective, toward the solution of the approximate problem. Global convergence results are given for exact and approximate line searches, and possible interpretations are made. We present some instances of the general algorithm and discuss extensions to nondifferentiable programming.
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References
Bazaraa, M. S., andShetty, C. M.,Nonlinear Programming: Theory and Algorithms, John Wiley and Sons, New York, New York, 1979.
Larsson, T., andMigdalas, A.,An Algorithm for Nonlinear Programs over Cartesian Product Sets, Optimization, Vol. 21, pp. 535–542, 1990.
Larsson, T., andPatriksson, M.,A Class of Gap Functions for Variational Inequalities, Report LiTH-MAT-R-90-18, Department of Mathematics, Linköping Institute of Technology, Linköping, Sweden, 1990. (To appear in Mathematical Programming.)
Martos, B.,Nonlinear Programming Theory and Methods, North-Holland, Amsterdam, Holland, 1975.
Rockafellar, R. T.,The Theory of Subgradients and Its Applications to Problems of Optimization: Convex and Nonconvex Functions, Heldermann-Verlag, Berlin, Germany, 1981.
Armijo, L.,Minimization of Functions Having Lipschitz Continuous First Partial Derivatives, Pacific Journal of Mathematics, Vol. 16, pp. 1–3, 1964.
Pshenichny, B. N., andDanilin, Yu. M.,Numerical Methods in Extremal Problems, Mir, Moscow, Russia, 1978.
Frank, M., andWolfe, P.,An Algorithm for Quadratic Programming, Naval Research Logistics Quarterly, Vol. 3, pp. 95–110, 1956.
Migdalas, A.,A Regularization of the Frank-Wolfe Algorithm, Report LiTH-MAT-R-90-10, Department of Mathematics, Linköping Institute of Technology, Linköping, Sweden, 1990.
Larsson, T., Migdalas, A., andPatriksson, M.,A Partial Linearization Method for the Traffic Assignment Problem, Report LiTH-MAT-R-89-06 Department of Mathematics, Linköping Institute of Technology, Linköping, Sweden, 1989. (To appear in optimization)
Patriksson, M.,A Unified Description of Iterative Algorithms for Traffic Equilibria, European Journal of Operational Research (to appear).
Polyak, B. T.,Introduction to Optimization, Optimization Software, New York, New York, 1987.
Rockafellar, R. T.,Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming, Mathematics of Operations Research, Vol. 1, pp. 97–116, 1976.
Mine, H., andFukushima, M.,A Minimization Method for the Sum of a Convex Function and a Continuously Differentiable Function, Journal of Optimization Theory and Applications, Vol. 33, pp. 9–23, 1981.
Clarke, F. H.,Generalized Gradients and Applications, Transactions of the American Mathematical Society, Vol. 205, pp. 247–262, 1975.
Fukushima, M., andMine, H.,A Generalized Proximal Point Algorithm for Certain Nonconvex Minimization Problems, International Journal of Systems Sciences, Vol. 12, pp. 989–1000, 1981.
Fukushima, M.,A Successive Quadratic Programming Method for a Class of Constrained Nonsmooth Optimization Problems, Mathematical Programming, Vol. 49, pp. 231–251, 1990.
Kiwiel, K. C.,A Method for Minimizing the Sum of a Convex Function and a Continuously Differentiable Function, Journal of Optimization Theory and Applications, Vol. 48, pp. 437–449, 1986.
Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.
Rudin, W.,Principles of Mathematical Analysis, 3rd Edition, McGraw-Hill, Singapore, Republic of Singapore, 1985.
Shor, N. Z.,Minimization Methods for Nondifferentiable Functions, Springer-Verlag, Berlin, Germany, 1985.
Canon, M. D., andCullum, C. D.,A Tight Upper Bound on the Rate of Convergence of the Frank-Wolfe Algorithm, SIAM Journal on Control, Vol. 6, pp. 509–516, 1968.
Florian, M., andNguyen, S.,A Method for Computing Network Equilibrium with Elastic Demands, Transportation Science, Vol. 8, pp. 321–332, 1974.
Holmberg, K., andJörnsten, K. O.,Cross Decomposition Applied to the Stochastic Transportation Problem, European Journal of Operational Research, Vol. 17, pp. 361–368, 1984.
Fukushima, M.,Equivalent Differentiable Optimization Problems and Descent Methods for Asymmetric Variational Inequality Problems, Mathematical Programming, Vol. 53, pp. 99–110, 1992.
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Communicated by D. G. Luenberger
The author wishes to thank Drs. K. Holmberg, T. Larsson, and A. Migdalas for their helpful comments.
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Patriksson, M. Partial linearization methods in nonlinear programming. J Optim Theory Appl 78, 227–246 (1993). https://doi.org/10.1007/BF00939668
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DOI: https://doi.org/10.1007/BF00939668