Abstract
The Frank—Wolfe linearization technique is a popular feasible direction algorithm for the solution of certain linearly constrained nonlinear problems. The popularity of this technique is due in part to its ability to exploit special constraint structures, such as network structures, and in part to the fact that it decomposes nonseparable problems over Cartesian product sets. However, the linearization which induces these advantages is also the source of the main disadvantages of the method: a sublinear rate of convergence and zigzagging behaviour. In order to avoid these disadvantages, a regularization penalty term is added to the objective of the direction generating subproblem. This results in a generic feasible direction method which also includes certain known nonlinear programming methods.
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References
M.S. Bazaraa and C.M. Shetty,Nonlinear programming—Theory and Algorithms (John Wiley & Sons, New York, NY, 1979).
D.P. Bertsekas, “On the Goldstein—Levitin—Polyak gradient projection method,”IEEE Transactions on Automatic Control AC-21 (1976) 174–184.
D.C. Cantor and M. Gerla, “Optimal routing in a packet-switched computer network,”IEEE Transactions on Computers C-23 (1974) 1062–1069.
S.C. Dafermos and F.T. Sparrow, “The traffic assignment problem for a general network,”Journal of RNBS 73B (1969) 91–118.
C.F. Daganzo, “On the traffic assignment problem with flow dependent cost,”Transportation Research 11 (1977) 433–441.
J.P. Dussault, J. Ferland and B. Lemaire, “Convex quadratic programming with one constraint and bounded variables,”Mathematical Programming 36 (1986) 90–104.
M. Frank and P. Wolfe, “An algorithm for quadratic programming,”Naval Research Logistic Quarterly 3 (1956) 95–110.
L. Fratta, M. Gerla and L. Kleinrock, “The flow deviation method. An approach to store-and-forward communication network design,”Networks 3 (1973) 97–133.
D.W. Hearn, “The gap function of a convex program,”Operations Research Letters 1 (1982) 67–71.
B. von Hohenbalken, “A finite algorithm to maximize certain pseudoconcave functions on polytopes,”Mathematical Programming 8 (1975) 189–206.
C.A. Holloway, “An extension of the Frank and Wolfe method of feasible direction,”Mathematical Programming 8 (1974) 14–27.
J.L. Kennington and R.V. Helgason,Algorithms for Network Programming (John Wiley & Sons, New York, NY, 1980).
D. Kinderlehrer and G. Stampacchia,An Introduction to Variational Inequalities (Academic Press, New York, NY, 1980).
T. Larsson and A. Migdalas, “An algorithm for nonlinear programs over Cartesian product sets,”Optimization 21 (1990) 535–542.
T. Larsson, A. Migdalas and M. Patriksson, “The application of partial linearization algorithm to the traffic assignment problem,”Optimization 28 (1993) 47–61.
L.J. LeBlanc, E.K. Morlok and W.P. Pierskalla, “An efficient approach to solving the road network equilibrium traffic assignment problem,”Transportation Research 9 (1975) 308–318.
L.J. LeBlanc, R.V. Helgason and D.E. Boyce, “Improved efficiency of the Frank—Wolfe algorithm for convex network programs,”Transportation Science 19 (1985) 445–462.
E.S. Levitin and B.T. Polyak, “Constrained minimization methods,”USSR Computational Mathematics and Mathematical Physics 6 (1966) 787–823.
B. Martos,Nonlinear Programming: Theory and Methods (North-Holland, Amsterdam, 1975).
J.D. Murchland, “Gleich-gewichtsverteilung des Verkehrs im Strassennetz,”Operations Research Verfahren 8 (1970) 143–183.
S. Nguyen, “An algorithm for traffic assignment problem,”Transportation Science 8 (1974) 203–216.
E.R. Petersen, “A primal-dual traffic assignment algorithm,”Management Science 22 (1975) 87–95.
B.N. Pshenichny and Y.M. Danilin,Numerical Methods in Extremal Problems (Mir Publishers, Moscow, 1978).
R.T. Rockafellar, “Augmented Lagrangians and applications of the proximal point algorithm in convex programming,”Mathematics of Operations Research 1 (1976) 97–116.
P. Wolfe, “Convergence theory in nonlinear programming,” in: J. Abadie, ed.,Integer and Nonlinear Programming (North-Holland, Amsterdam, 1970) pp. 1–36.
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Migdalas, A. A regularization of the Frank—Wolfe method and unification of certain nonlinear programming methods. Mathematical Programming 65, 331–345 (1994). https://doi.org/10.1007/BF01581701
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DOI: https://doi.org/10.1007/BF01581701