Abstract
For many kinds of optimization problems, convexity properties are very important, and when they are present in a thorough form they lead to an interesting kind of duality. This duality is sometimes useful in methods of computation, but it also has theoretical applications, such as in the analysis of economic models where dual variables can be interpreted as prices. The study of duality, even though it may pertain to a special subclass of problems often aids in the general development of a subject by suggesting alternative ways of looking at things.
In the classical calculus of variations, convexity and duality first enter the picture in the correspondence between Lagrangian and Hamiltonian functions and in the way this is connected with necessary conditions and the existence of solutions. Expressed in terms of the Hamiltonian, the optimality conditions for an arc x pair it with an "adjoint" arc p. The pairing carries over to problems of optimal control via the maximum principle. Duality theory in this context aims at uncovering and analyzing cases where p happens to solve a dual problem for which x is in turn the adjoint arc. But although this is the principal motivation, a number of side issues have to be explored along the way, and these suggest new approaches even to problems where duality is not at stake.
Research sponsored by the Air Force Office of Scientific Research, Air Force Systems Command, United States Air Force, under AF-AFUSR grant number 77-0546 at the University of Washington, Seattle.
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References
Viorel Barbu, "Convex control problems of Bolza in Hilbert spaces", SIAM J. Control 13 (1975), 754–771.
Viorel Barbu, "On the control problem of Bolza in Hilbert spaces", SIAM J. Control 13 (1975), 1062–1076.
V. Barbu, "Convex control problems for linear differential systems of retarded type", Ricerche Mat. 26 (1977), 3–26.
Viorel Barbu, "Constrained control problems with convex cost in Hilbert space", J. Math. Anal. Appl. 56 (1976), 502–528.
V. Barbu, "On convex control problems on infinite intervals", submitted.
Jean-Michel Bismut, "Conjugate convex functions in optimal stochastic control", J. Math. Anal. Appl. 44 (1973), 384–404.
Charles Castaing, "Sur les équations différentielles multivoques", C.R. Acad. Sci. Paris Sér. A-B 263 (1966), A63–A66.
Lamberto Cesari, "Existence theorems for weak and usual optimal solutions in Lagrange problems with unilateral constraints. I", Trans. Amer. Math. Soc. 124 (1966), 369–412.
Frank H. Clarke, "Admissible relaxation in variational and control problems", J. Math. Anal. Appl. 51 (1975), 557–576.
Frank H. Clarke, "Generalized gradients and applications", Trans. Amer. Math. Soc. 205 (1975), 247–262.
Frank H. Clarke, "The Euler-Lagrange differential inclusion", J. Differential Equations 19 (1975), 80–90.
Frank H. Clarke, "La condition hamiltonienne d'optimalité", C.R. Acad. Sci. Paris Sér. A-B 280 (1975), A1205–A1207.
Frank H. Clarke, "The generalized problem of Bolza", SIAM J. Control Optimization 14 (1976), 682–699.
Frank H. Clarke, "Necessary conditions for a general control problem", Calculus of Variations and Control Theory (Symposium, University Wisconsin, Maidson, Wisconsin, 1975, 257–278. Academic Press, New York, San Francisco, London, 1976).
Frank H. Clarke, "The maximum principle under minimal hypothesis", SIAM J. Control Optimization 14 (1976), 1078–1091.
F.H. Clarke, "Generalized gradients of Lipschitz functionals", submitted.
Ivar Ekeland and Roger Temam, Convex Analysis and Variational Problems (Studies in Mathematics and its Applications, 1. North-Holland, Amsterdam, Oxford, New York, 1976).
W. Fenchel, "On conjugate convex functions", Canad. J. Math. 1 (1949), 73–77.
Czeslaw Olech, "Existence theorems for optimal problems with vector-valued cost function", Trans. Amer. Math. Soc. 136 (1969), 159–180.
R. Tyrrell Rockafellar, Convex Analysis (Princeton Mathematical Series, 28. Princeton University Press, Princeton, New Jersey, 1970).
R.T. Rockafellar, "Conjugate convex functions in optimal control and the calculus of variations", J. Math. Anal. Appl. 32 (1970), 174–222.
R. Tyrrell Rockafellar, "Generalized Hamiltonian equations for convex problems of Lagrange", Pacific J. Math. 33 (1970), 411–427.
R.T. Rockafellar, "Existence and duality theorems for convex problems of Bolza", Trans. Amer. Math. Soc. 159 (1971), 1–40.
R. Tyrrell Rockafellar, "State constraints in convex control problems of Bolza", SIAM J. Control 10 (1972), 691–715.
R.T. Rockafellar, "Saddle points of Hamiltonian systems in convex problems of Lagrange", J. Optimization Theory Appl. 12 (1973), 367–390.
R. Tyrrell Rockafellar, Conjugate Duality and Optimization (Conference Board of the Mathematical Sciences, Regional Conference Series in Applied Math., 16. Society for Industrial and Applied Mathematics, Philadelphia, 1974).
R. Tyrrell Rockafellar, "Existence theorems for general control problems of Bolza and Lagrange", Advances in Math. 15 (1975), 312–333.
R. Tyrrell Rockafellar, "Semigroups of convex bifunctions generated by Lagrange problems in the calculus of variations", Math. Scand. 36 (1975), 137–158.
R. Tyrrell Rockafellar, "Integral functionals, normal integrands and measurable selections", Nonlinear Operators and the Calculus of Variations (Lecture Notes in Mathematics, 543, 157–207. Springer-Verlag, Berlin, Heidelberg, New York. 1976)
R. Tyrrell Rockafellar, "Dual problems of Lagrange for arcs of bounded variation", Calculus of Variations and Control Theory (Symposium, University Wisconsin, Madison, 1975, 155–192. Academic Press, New York, San Francisco, London, 1976).
R. Tyrrell Rockafellar, "Saddle points of Hamiltonian systems in convex Lagrange problems having a nonzero discount rate. Hamiltonian Dynamics in economics", J. Econom. Theory 12 (1976), 71–113.
D.H. Wagner, "Survey of measurable selection theorems", SIAM J. Control Optimization 15 (1977), 859–903.
J. Warga, "Relaxed variational problems", J. Math. Anal. Appl. 4 (1962), 111–128.
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Rockafellar, R.T. (1978). Duality in optimal control. In: Coppel, W.A. (eds) Mathematical Control Theory. Lecture Notes in Mathematics, vol 680. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0065318
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DOI: https://doi.org/10.1007/BFb0065318
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